PR contents

(last sync)

feat(set_theory/cardinal/finite): prove lemmas to handle part_enat.card (#19198)

Prove some lemmas that allow to handle part_enat.card of finite cardinals, analogous to those about nat.card

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

Diff

@@ -1215,6 +1215,10 @@ by rw [to_nat_apply_of_lt_aleph_0 h, ← classical.some_spec (lt_aleph_0.1 h)]
 lemma cast_to_nat_of_aleph_0_le {c : cardinal} (h : ℵ₀ ≤ c) : ↑c.to_nat = (0 : cardinal) :=
 by rw [to_nat_apply_of_aleph_0_le h, nat.cast_zero]
 
+lemma to_nat_eq_iff_eq_of_lt_aleph_0 {c d : cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
+  c.to_nat = d.to_nat ↔ c = d :=
+by rw [←nat_cast_inj, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
+
 lemma to_nat_le_iff_le_of_lt_aleph_0 {c d : cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
   c.to_nat ≤ d.to_nat ↔ c ≤ d :=
 by rw [←nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
@@ -1357,10 +1361,84 @@ to_part_enat_apply_of_aleph_0_le (infinite_iff.1 h)
 @[simp] theorem aleph_0_to_part_enat : to_part_enat ℵ₀ = ⊤ :=
 to_part_enat_apply_of_aleph_0_le le_rfl
 
+lemma to_part_enat_eq_top_iff_le_aleph_0 {c : cardinal} :
+  to_part_enat c = ⊤ ↔ aleph_0 ≤ c :=
+begin
+  cases lt_or_ge c aleph_0 with hc hc,
+  simp only [to_part_enat_apply_of_lt_aleph_0 hc, part_enat.coe_ne_top, false_iff, not_le, hc],
+  simp only [to_part_enat_apply_of_aleph_0_le hc, eq_self_iff_true, true_iff],
+  exact hc,
+end
+
+lemma to_part_enat_le_iff_le_of_le_aleph_0 {c c' : cardinal} (h : c ≤ aleph_0) :
+  to_part_enat c ≤ to_part_enat c' ↔ c ≤ c' :=
+begin
+  cases lt_or_ge c aleph_0 with hc hc,
+  rw to_part_enat_apply_of_lt_aleph_0 hc,
+  cases lt_or_ge c' aleph_0 with hc' hc',
+  { rw to_part_enat_apply_of_lt_aleph_0 hc',
+    rw part_enat.coe_le_coe,
+    exact to_nat_le_iff_le_of_lt_aleph_0 hc hc', },
+  { simp only [to_part_enat_apply_of_aleph_0_le hc',
+    le_top, true_iff],
+    exact le_trans h hc', },
+  { rw to_part_enat_apply_of_aleph_0_le hc,
+    simp only [top_le_iff, to_part_enat_eq_top_iff_le_aleph_0,
+    le_antisymm h hc], },
+end
+
+lemma to_part_enat_le_iff_le_of_lt_aleph_0 {c c' : cardinal} (hc' : c' < aleph_0) :
+  to_part_enat c ≤ to_part_enat c' ↔ c ≤ c' :=
+begin
+  cases lt_or_ge c aleph_0 with hc hc,
+  { rw to_part_enat_apply_of_lt_aleph_0 hc,
+    rw to_part_enat_apply_of_lt_aleph_0 hc',
+    rw part_enat.coe_le_coe,
+    exact to_nat_le_iff_le_of_lt_aleph_0 hc hc', },
+  { rw to_part_enat_apply_of_aleph_0_le hc,
+    simp only [top_le_iff, to_part_enat_eq_top_iff_le_aleph_0],
+    rw [← not_iff_not, not_le, not_le],
+    simp only [hc', lt_of_lt_of_le hc' hc], },
+end
+
+lemma to_part_enat_eq_iff_eq_of_le_aleph_0 {c c' : cardinal}
+  (hc : c ≤ aleph_0) (hc' : c' ≤ aleph_0) :
+  to_part_enat c = to_part_enat c' ↔ c = c' := by
+rw [le_antisymm_iff, le_antisymm_iff,
+  to_part_enat_le_iff_le_of_le_aleph_0 hc, to_part_enat_le_iff_le_of_le_aleph_0 hc']
+
+lemma to_part_enat_mono {c c' : cardinal} (h : c ≤ c') :
+  to_part_enat c ≤ to_part_enat c' :=
+begin
+  cases lt_or_ge c aleph_0 with hc hc,
+  rw to_part_enat_apply_of_lt_aleph_0 hc,
+  cases lt_or_ge c' aleph_0 with hc' hc',
+  rw to_part_enat_apply_of_lt_aleph_0 hc',
+  simp only [part_enat.coe_le_coe],
+  exact to_nat_le_of_le_of_lt_aleph_0 hc' h,
+  rw to_part_enat_apply_of_aleph_0_le hc',
+  exact le_top,
+  rw [to_part_enat_apply_of_aleph_0_le hc,
+  to_part_enat_apply_of_aleph_0_le (le_trans hc h)],
+end
+
 lemma to_part_enat_surjective : surjective to_part_enat :=
 λ x, part_enat.cases_on x ⟨ℵ₀, to_part_enat_apply_of_aleph_0_le le_rfl⟩ $
   λ n, ⟨n, to_part_enat_cast n⟩
 
+lemma to_part_enat_lift (c : cardinal.{v}) : (lift.{u v} c).to_part_enat = c.to_part_enat :=
+begin
+  cases lt_or_ge c ℵ₀ with hc hc,
+  { rw [to_part_enat_apply_of_lt_aleph_0 hc, cardinal.to_part_enat_apply_of_lt_aleph_0 _],
+    simp only [to_nat_lift],
+    rw [← lift_aleph_0, lift_lt], exact hc },
+  { rw [to_part_enat_apply_of_aleph_0_le hc, cardinal.to_part_enat_apply_of_aleph_0_le _],
+  rw [← lift_aleph_0, lift_le], exact hc }
+end
+
+lemma to_part_enat_congr {β : Type v} (e : α ≃ β) : (#α).to_part_enat = (#β).to_part_enat :=
+by rw [←to_part_enat_lift, lift_mk_eq.mpr ⟨e⟩, to_part_enat_lift]
+
 lemma mk_to_part_enat_eq_coe_card [fintype α] : (#α).to_part_enat = fintype.card α :=
 by simp
 
@@ -1369,7 +1447,8 @@ lemma mk_int : #ℤ = ℵ₀ := mk_denumerable ℤ
 lemma mk_pnat : #ℕ+ = ℵ₀ := mk_denumerable ℕ+
 
 /-- **König's theorem** -/
-theorem sum_lt_prod {ι} (f g : ι → cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
+theorem sum_lt_prod {ι} (f g : ι → cardinal) (H : ∀ i, f i < g i) :
+sum f < prod g :=
 lt_of_not_ge $ λ ⟨F⟩, begin
   haveI : inhabited (Π (i : ι), (g i).out),
   { refine ⟨λ i, classical.choice $ mk_ne_zero_iff.1 _⟩,

chore(set_theory/cardinal/basic): reinstate partial_order (#18781)

This change from #18714 was causing computability issues in Lean 4.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

Diff

@@ -191,13 +191,16 @@ instance : has_le cardinal.{u} :=
 ⟨λ q₁ q₂, quotient.lift_on₂ q₁ q₂ (λ α β, nonempty $ α ↪ β) $
   λ α β γ δ ⟨e₁⟩ ⟨e₂⟩, propext ⟨λ ⟨e⟩, ⟨e.congr e₁ e₂⟩, λ ⟨e⟩, ⟨e.congr e₁.symm e₂.symm⟩⟩⟩
 
-instance : linear_order cardinal.{u} :=
+instance : partial_order cardinal.{u} :=
 { le           := (≤),
   le_refl      := by rintros ⟨α⟩; exact ⟨embedding.refl _⟩,
   le_trans     := by rintros ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩; exact ⟨e₁.trans e₂⟩,
-  le_antisymm  := by { rintros ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩, exact quotient.sound (e₁.antisymm e₂) },
-  le_total     := by { rintros ⟨α⟩ ⟨β⟩, apply embedding.total },
-  decidable_le := classical.dec_rel _ }
+  le_antisymm  := by { rintros ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩, exact quotient.sound (e₁.antisymm e₂) }, }
+
+instance : linear_order cardinal.{u} :=
+{ le_total     := by { rintros ⟨α⟩ ⟨β⟩, apply embedding.total },
+  decidable_le := classical.dec_rel _,
+  ..cardinal.partial_order.{u}, }
 
 theorem le_def (α β : Type u) : #α ≤ #β ↔ nonempty (α ↪ β) :=
 iff.rfl
@@ -475,18 +478,26 @@ instance : canonically_ordered_comm_semiring cardinal.{u} :=
   le_self_add := λ a b, (add_zero a).ge.trans $ add_le_add_left (cardinal.zero_le _) _,
   eq_zero_or_eq_zero_of_mul_eq_zero := λ a b, induction_on₂ a b $ λ α β,
     by simpa only [mul_def, mk_eq_zero_iff, is_empty_prod] using id,
-  ..cardinal.comm_semiring, ..cardinal.linear_order }
+  ..cardinal.comm_semiring, ..cardinal.partial_order }
 
 instance : canonically_linear_ordered_add_monoid cardinal.{u} :=
 { ..cardinal.canonically_ordered_comm_semiring,
   ..cardinal.linear_order }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : canonically_ordered_add_monoid cardinal.{u} :=
+{ ..cardinal.canonically_ordered_comm_semiring }
+
 instance : linear_ordered_comm_monoid_with_zero cardinal.{u} :=
 { mul_le_mul_left := @mul_le_mul_left' _ _ _ _,
   zero_le_one := zero_le _,
   ..cardinal.comm_semiring,
   ..cardinal.linear_order }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : comm_monoid_with_zero cardinal.{u} :=
+{ ..cardinal.canonically_ordered_comm_semiring }
+
 lemma zero_power_le (c : cardinal.{u}) : (0 : cardinal.{u}) ^ c ≤ 1 :=
 by { by_cases h : c = 0, rw [h, power_zero], rw [zero_power h], apply zero_le }

e05ead79

chore(set_theory/cardinal/basic): less awkward placement of theorems (#18771)

Some of the new results on limit cardinals were awkwardly breaking up blocks of code with related theorems, probably due to some botched merge. We reorder them to avoid this.

Ported along other changes to this file in: https://github.com/leanprover-community/mathlib4/pull/3343

Diff

@@ -561,21 +561,11 @@ instance : succ_order cardinal :=
 succ_order.of_succ_le_iff (λ c, Inf {c' | c < c'})
   (λ a b, ⟨lt_of_lt_of_le $ Inf_mem $ exists_gt a, cInf_le'⟩)
 
-/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
-  cardinal by this definition, but `0` isn't.
-
-  Use `is_succ_limit` if you want to include the `c = 0` case. -/
-def is_limit (c : cardinal) : Prop := c ≠ 0 ∧ is_succ_limit c
-
-protected theorem is_limit.ne_zero {c} (h : is_limit c) : c ≠ 0 := h.1
-
-protected theorem is_limit.is_succ_limit {c} (h : is_limit c) : is_succ_limit c := h.2
-
-theorem is_limit.succ_lt {x c} (h : is_limit c) : x < c → succ x < c := h.is_succ_limit.succ_lt
+theorem succ_def (c : cardinal) : succ c = Inf {c' | c < c'} := rfl
 
-theorem is_succ_limit_zero : is_succ_limit (0 : cardinal) := is_succ_limit_bot
+lemma succ_pos : ∀ c : cardinal, 0 < succ c := bot_lt_succ
 
-theorem succ_def (c : cardinal) : succ c = Inf {c' | c < c'} := rfl
+lemma succ_ne_zero (c : cardinal) : succ c ≠ 0 := (succ_pos _).ne'
 
 theorem add_one_le_succ (c : cardinal.{u}) : c + 1 ≤ succ c :=
 begin
@@ -589,9 +579,19 @@ begin
           ... ≤ #β          : (f.option_elim b hb).cardinal_le
 end
 
-lemma succ_pos : ∀ c : cardinal, 0 < succ c := bot_lt_succ
+/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
+  cardinal by this definition, but `0` isn't.
 
-lemma succ_ne_zero (c : cardinal) : succ c ≠ 0 := (succ_pos _).ne'
+  Use `is_succ_limit` if you want to include the `c = 0` case. -/
+def is_limit (c : cardinal) : Prop := c ≠ 0 ∧ is_succ_limit c
+
+protected theorem is_limit.ne_zero {c} (h : is_limit c) : c ≠ 0 := h.1
+
+protected theorem is_limit.is_succ_limit {c} (h : is_limit c) : is_succ_limit c := h.2
+
+theorem is_limit.succ_lt {x c} (h : is_limit c) : x < c → succ x < c := h.is_succ_limit.succ_lt
+
+theorem is_succ_limit_zero : is_succ_limit (0 : cardinal) := is_succ_limit_bot
 
 /-- The indexed sum of cardinals is the cardinality of the
   indexed disjoint union, i.e. sigma type. -/
@@ -1015,6 +1015,24 @@ theorem aleph_0_le {c : cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
   exact (nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n+1)))
 end⟩
 
+theorem is_succ_limit_aleph_0 : is_succ_limit ℵ₀ :=
+is_succ_limit_of_succ_lt $ λ a ha, begin
+  rcases lt_aleph_0.1 ha with ⟨n, rfl⟩,
+  rw ←nat_succ,
+  apply nat_lt_aleph_0
+end
+
+theorem is_limit_aleph_0 : is_limit ℵ₀ := ⟨aleph_0_ne_zero, is_succ_limit_aleph_0⟩
+
+theorem is_limit.aleph_0_le {c : cardinal} (h : is_limit c) : ℵ₀ ≤ c :=
+begin
+  by_contra' h',
+  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩,
+  { exact h.ne_zero.irrefl },
+  { rw nat_succ at h,
+    exact not_is_succ_limit_succ _ h.is_succ_limit }
+end
+
 @[simp] lemma range_nat_cast : range (coe : ℕ → cardinal) = Iio ℵ₀ :=
 ext $ λ x, by simp only [mem_Iio, mem_range, eq_comm, lt_aleph_0]
 
@@ -1035,24 +1053,6 @@ lt_aleph_0_iff_finite.trans finite_coe_iff
 
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 
-theorem is_succ_limit_aleph_0 : is_succ_limit ℵ₀ :=
-is_succ_limit_of_succ_lt $ λ a ha, begin
-  rcases lt_aleph_0.1 ha with ⟨n, rfl⟩,
-  rw ←nat_succ,
-  apply nat_lt_aleph_0
-end
-
-theorem is_limit_aleph_0 : is_limit ℵ₀ := ⟨aleph_0_ne_zero, is_succ_limit_aleph_0⟩
-
-theorem is_limit.aleph_0_le {c : cardinal} (h : is_limit c) : ℵ₀ ≤ c :=
-begin
-  by_contra' h',
-  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩,
-  { exact h.ne_zero.irrefl },
-  { rw nat_succ at h,
-    exact not_is_succ_limit_succ _ h.is_succ_limit }
-end
-
 @[simp] theorem lt_aleph_0_iff_subtype_finite {p : α → Prop} :
   #{x // p x} < ℵ₀ ↔ {x | p x}.finite :=
 lt_aleph_0_iff_set_finite

chore(set_theory/cardinal/basic): missing lemmas about </≤ and lift c/↑n (#18752)

Diff

@@ -911,6 +911,20 @@ lift_injective.eq_iff' (lift_nat_cast n)
   (n : cardinal) = lift.{v} a ↔ (n : cardinal) = a :=
 by rw [←lift_nat_cast.{v} n, lift_inj]
 
+@[simp] lemma lift_le_nat_iff {a : cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n :=
+by simp only [←lift_nat_cast, lift_le]
+
+@[simp] lemma nat_le_lift_iff {n : ℕ} {a : cardinal.{u}} :
+  (n : cardinal) ≤ lift.{v} a ↔ (n : cardinal) ≤ a :=
+by simp only [←lift_nat_cast, lift_le]
+
+@[simp] lemma lift_lt_nat_iff {a : cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n :=
+by simp only [←lift_nat_cast, lift_lt]
+
+@[simp] lemma nat_lt_lift_iff {n : ℕ} {a : cardinal.{u}} :
+  (n : cardinal) < lift.{v} a ↔ (n : cardinal) < a :=
+by simp only [←lift_nat_cast, lift_lt]
+
 theorem lift_mk_fin (n : ℕ) : lift (#(fin n)) = n := by simp
 
 lemma mk_coe_finset {α : Type u} {s : finset α} : #s = ↑(finset.card s) := by simp

e08a42b2

chore(set_theory/cardinal/basic): missing lemmas about lift c < ℵ₀ and ℵ₀ < lift c (#18746)

We already had the le versions lift_le_aleph0 and aleph0_le_lift, this adds the lt ones lift_lt_aleph0 and aleph0_lt_lift.

This turns out to be useful for proving some results about finrank, as well as golfing some existing proofs.

Since they're trivial, this adds the same lemmas about continuum too.

Diff

@@ -891,6 +891,12 @@ by rw [←lift_aleph_0, lift_le]
 @[simp] theorem lift_le_aleph_0 {c : cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ₀ :=
 by rw [←lift_aleph_0, lift_le]
 
+@[simp] theorem aleph_0_lt_lift {c : cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c :=
+by rw [←lift_aleph_0, lift_lt]
+
+@[simp] theorem lift_lt_aleph_0 {c : cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :=
+by rw [←lift_aleph_0, lift_lt]
+
 /-! ### Properties about the cast from `ℕ` -/
 
 @[simp] theorem mk_fin (n : ℕ) : #(fin n) = n := by simp
@@ -1245,9 +1251,9 @@ begin
   apply nat_cast_injective,
   cases lt_or_ge c ℵ₀ with hc hc,
   { rw [cast_to_nat_of_lt_aleph_0, ←lift_nat_cast, cast_to_nat_of_lt_aleph_0 hc],
-    rwa [←lift_aleph_0, lift_lt] },
+    rwa [lift_lt_aleph_0] },
   { rw [cast_to_nat_of_aleph_0_le, ←lift_nat_cast, cast_to_nat_of_aleph_0_le hc, lift_zero],
-    rwa [←lift_aleph_0, lift_le] },
+    rwa [aleph_0_le_lift] },
 end
 
 lemma to_nat_congr {β : Type v} (e : α ≃ β) : (#α).to_nat = (#β).to_nat :=
@@ -1285,8 +1291,8 @@ map_prod to_nat_hom _ _
   (ha : a < ℵ₀) (hb : b < ℵ₀) : ((lift.{v u} a) + (lift.{u v} b)).to_nat = a.to_nat + b.to_nat :=
 begin
   apply cardinal.nat_cast_injective,
-  replace ha : (lift.{v u} a) < ℵ₀ := by { rw ←lift_aleph_0, exact lift_lt.2 ha },
-  replace hb : (lift.{u v} b) < ℵ₀ := by { rw ←lift_aleph_0, exact lift_lt.2 hb },
+  replace ha : (lift.{v u} a) < ℵ₀ := by rwa lift_lt_aleph_0,
+  replace hb : (lift.{u v} b) < ℵ₀ := by rwa lift_lt_aleph_0,
   rw [nat.cast_add, ←to_nat_lift.{v u} a, ←to_nat_lift.{u v} b, cast_to_nat_of_lt_aleph_0 ha,
     cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0 (add_lt_aleph_0 ha hb)]
 end

7c2ce0c2

refactor(set_theory/cardinal/basic): redefine cardinal.is_limit in terms of is_succ_limit (#18523)

We redefine cardinal.is_limit x as x ≠ 0 ∧ is_succ_limit x. This will allow us to make use of the extensive is_succ_limit API in the future.

Diff

@@ -9,7 +9,7 @@ import data.nat.part_enat
 import data.set.countable
 import logic.small.basic
 import order.conditionally_complete_lattice.basic
-import order.succ_pred.basic
+import order.succ_pred.limit
 import set_theory.cardinal.schroeder_bernstein
 import tactic.positivity
 
@@ -30,6 +30,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 * Multiplication `c₁ * c₂` is defined by `cardinal.mul_def : #α * #β = #(α × β)`.
 * The order `c₁ ≤ c₂` is defined by `cardinal.le_def α β : #α ≤ #β ↔ nonempty (α ↪ β)`.
 * Exponentiation `c₁ ^ c₂` is defined by `cardinal.power_def α β : #α ^ #β = #(β → α)`.
+* `cardinal.is_limit c` means that `c` is a (weak) limit cardinal: `c ≠ 0 ∧ ∀ x < c, succ x < c`.
 * `cardinal.aleph_0` or `ℵ₀` is the cardinality of `ℕ`. This definition is universe polymorphic:
   `cardinal.aleph_0.{u} : cardinal.{u}` (contrast with `ℕ : Type`, which lives in a specific
   universe). In some cases the universe level has to be given explicitly.
@@ -560,6 +561,20 @@ instance : succ_order cardinal :=
 succ_order.of_succ_le_iff (λ c, Inf {c' | c < c'})
   (λ a b, ⟨lt_of_lt_of_le $ Inf_mem $ exists_gt a, cInf_le'⟩)
 
+/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
+  cardinal by this definition, but `0` isn't.
+
+  Use `is_succ_limit` if you want to include the `c = 0` case. -/
+def is_limit (c : cardinal) : Prop := c ≠ 0 ∧ is_succ_limit c
+
+protected theorem is_limit.ne_zero {c} (h : is_limit c) : c ≠ 0 := h.1
+
+protected theorem is_limit.is_succ_limit {c} (h : is_limit c) : is_succ_limit c := h.2
+
+theorem is_limit.succ_lt {x c} (h : is_limit c) : x < c → succ x < c := h.is_succ_limit.succ_lt
+
+theorem is_succ_limit_zero : is_succ_limit (0 : cardinal) := is_succ_limit_bot
+
 theorem succ_def (c : cardinal) : succ c = Inf {c' | c < c'} := rfl
 
 theorem add_one_le_succ (c : cardinal.{u}) : c + 1 ≤ succ c :=
@@ -1000,6 +1015,24 @@ lt_aleph_0_iff_finite.trans finite_coe_iff
 
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 
+theorem is_succ_limit_aleph_0 : is_succ_limit ℵ₀ :=
+is_succ_limit_of_succ_lt $ λ a ha, begin
+  rcases lt_aleph_0.1 ha with ⟨n, rfl⟩,
+  rw ←nat_succ,
+  apply nat_lt_aleph_0
+end
+
+theorem is_limit_aleph_0 : is_limit ℵ₀ := ⟨aleph_0_ne_zero, is_succ_limit_aleph_0⟩
+
+theorem is_limit.aleph_0_le {c : cardinal} (h : is_limit c) : ℵ₀ ≤ c :=
+begin
+  by_contra' h',
+  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩,
+  { exact h.ne_zero.irrefl },
+  { rw nat_succ at h,
+    exact not_is_succ_limit_succ _ h.is_succ_limit }
+end
+
 @[simp] theorem lt_aleph_0_iff_subtype_finite {p : α → Prop} :
   #{x // p x} < ℵ₀ ↔ {x | p x}.finite :=
 lt_aleph_0_iff_set_finite

ea050b44

feat(set_theory/cardinal/basic): clean up instances (#18714)

We add a missing linear_ordered_comm_monoid_with_zero instance and reorder others.

Diff

@@ -190,11 +190,13 @@ instance : has_le cardinal.{u} :=
 ⟨λ q₁ q₂, quotient.lift_on₂ q₁ q₂ (λ α β, nonempty $ α ↪ β) $
   λ α β γ δ ⟨e₁⟩ ⟨e₂⟩, propext ⟨λ ⟨e⟩, ⟨e.congr e₁ e₂⟩, λ ⟨e⟩, ⟨e.congr e₁.symm e₂.symm⟩⟩⟩
 
-instance : partial_order cardinal.{u} :=
-{ le          := (≤),
-  le_refl     := by rintros ⟨α⟩; exact ⟨embedding.refl _⟩,
-  le_trans    := by rintros ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩; exact ⟨e₁.trans e₂⟩,
-  le_antisymm := by { rintros ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩, exact quotient.sound (e₁.antisymm e₂) } }
+instance : linear_order cardinal.{u} :=
+{ le           := (≤),
+  le_refl      := by rintros ⟨α⟩; exact ⟨embedding.refl _⟩,
+  le_trans     := by rintros ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩; exact ⟨e₁.trans e₂⟩,
+  le_antisymm  := by { rintros ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩, exact quotient.sound (e₁.antisymm e₂) },
+  le_total     := by { rintros ⟨α⟩ ⟨β⟩, apply embedding.total },
+  decidable_le := classical.dec_rel _ }
 
 theorem le_def (α β : Type u) : #α ≤ #β ↔ nonempty (α ↪ β) :=
 iff.rfl
@@ -472,7 +474,17 @@ instance : canonically_ordered_comm_semiring cardinal.{u} :=
   le_self_add := λ a b, (add_zero a).ge.trans $ add_le_add_left (cardinal.zero_le _) _,
   eq_zero_or_eq_zero_of_mul_eq_zero := λ a b, induction_on₂ a b $ λ α β,
     by simpa only [mul_def, mk_eq_zero_iff, is_empty_prod] using id,
-  ..cardinal.comm_semiring, ..cardinal.partial_order }
+  ..cardinal.comm_semiring, ..cardinal.linear_order }
+
+instance : canonically_linear_ordered_add_monoid cardinal.{u} :=
+{ ..cardinal.canonically_ordered_comm_semiring,
+  ..cardinal.linear_order }
+
+instance : linear_ordered_comm_monoid_with_zero cardinal.{u} :=
+{ mul_le_mul_left := @mul_le_mul_left' _ _ _ _,
+  zero_le_one := zero_le _,
+  ..cardinal.comm_semiring,
+  ..cardinal.linear_order }
 
 lemma zero_power_le (c : cardinal.{u}) : (0 : cardinal.{u}) ^ c ≤ 1 :=
 by { by_cases h : c = 0, rw [h, power_zero], rw [zero_power h], apply zero_le }
@@ -500,13 +512,7 @@ begin
 end
 
 instance : no_max_order cardinal.{u} :=
-{ exists_gt := λ a, ⟨_, cantor a⟩, ..cardinal.partial_order }
-
-instance : canonically_linear_ordered_add_monoid cardinal.{u} :=
-{ le_total     := by { rintros ⟨α⟩ ⟨β⟩, apply embedding.total },
-  decidable_le := classical.dec_rel _,
-  ..(infer_instance : canonically_ordered_add_monoid cardinal),
-  ..cardinal.partial_order }
+{ exists_gt := λ a, ⟨_, cantor a⟩ }
 
 -- short-circuit type class inference
 instance : distrib_lattice cardinal.{u} := by apply_instance
@@ -541,7 +547,6 @@ protected theorem lt_wf : @well_founded cardinal.{u} (<) :=
 end⟩
 
 instance : has_well_founded cardinal.{u} := ⟨(<), cardinal.lt_wf⟩
-instance : well_founded_lt cardinal.{u} := ⟨cardinal.lt_wf⟩
 instance wo : @is_well_order cardinal.{u} (<) := { }
 
 instance : conditionally_complete_linear_order_bot cardinal :=

(first ported)

Changes in mathlib3port

mathlib3

mathlib3port

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -5,9 +5,9 @@ Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 -/
 import Data.Fintype.BigOperators
 import Data.Finsupp.Defs
-import Data.Nat.PartEnat
+import Data.Nat.PartENat
 import Data.Set.Countable
-import Logic.Small.Basic
+import Logic.Small.Defs
 import Order.ConditionallyCompleteLattice.Basic
 import Order.SuccPred.Limit
 import SetTheory.Cardinal.SchroederBernstein
@@ -1591,7 +1591,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ (#α) :
 #print Cardinal.natCast_pow /-
 @[simp, norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(pow m n) : Cardinal) = (m^n) := by
-  induction n <;> simp [pow_succ', power_add, *]
+  induction n <;> simp [pow_succ, power_add, *]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 -/
 
@@ -1854,7 +1854,7 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   simp only [Nat.succ_ne_zero, false_or_iff]
   induction' n with n ih
   · simp
-  rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self_iff]
+  rw [succ_nsmul', add_lt_aleph_0_iff, ih, and_self_iff]
 #align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff
 -/

bump to nightly-2024-03-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

22f01ce4

Diff

@@ -543,7 +543,7 @@ theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   by
   refine' Fintype.induction_empty_option _ _ _ α
   · intro α β h e hα; letI := Fintype.ofEquiv β e.symm
-    rwa [mk_congr e, Fintype.card_congr e] at hα 
+    rwa [mk_congr e, Fintype.card_congr e] at hα
   · rfl
   · intro α h hα; simp [hα]; rfl
 #align cardinal.mk_fintype Cardinal.mk_fintype
@@ -956,7 +956,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
   cases' le_of_lt hlt with f
   have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
-  simp only [surjective, Classical.not_forall] at this 
+  simp only [surjective, Classical.not_forall] at this
   rcases this with ⟨b, hb⟩
   calc
     (#γ) + 1 = (#Option γ) := mk_option.symm
@@ -1067,8 +1067,7 @@ theorem lift_sum {ι : Type u} (f : ι → Cardinal.{v}) :
 #print Cardinal.sum_le_sum /-
 theorem sum_le_sum {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : sum f ≤ sum g :=
   ⟨(Embedding.refl _).sigma_map fun i =>
-      Classical.choice <| by
-        have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this ⟩
+      Classical.choice <| by have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this⟩
 #align cardinal.sum_le_sum Cardinal.sum_le_sum
 -/
 
@@ -1242,7 +1241,7 @@ theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) =
 #print Cardinal.prod_le_prod /-
 theorem prod_le_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : prod f ≤ prod g :=
   ⟨Embedding.piCongrRight fun i =>
-      Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this ⟩
+      Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this⟩
 #align cardinal.prod_le_prod Cardinal.prod_le_prod
 -/
 
@@ -1343,7 +1342,7 @@ theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
   le_antisymm
     (le_of_not_gt fun h => by
       rcases lt_lift_iff.1 h with ⟨b, e, h⟩
-      rw [lt_succ_iff, ← lift_le, e] at h 
+      rw [lt_succ_iff, ← lift_le, e] at h
       exact h.not_lt (lt_succ _))
     (succ_le_of_lt <| lift_lt.2 <| lt_succ a)
 #align cardinal.lift_succ Cardinal.lift_succ
@@ -1378,8 +1377,8 @@ theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSu
   apply ((le_csSup_iff' (bdd_above_image _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
   · by_contra h
     obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
-    simp_rw [lift_le] at h hc 
-    rw [csSup_le_iff' hs] at h 
+    simp_rw [lift_le] at h hc
+    rw [csSup_le_iff' hs] at h
     exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
   · rintro i ⟨j, hj, rfl⟩
     exact lift_le.2 (le_csSup hs hj)
@@ -1642,7 +1641,7 @@ theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n :=
   by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
-  rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn 
+  rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn
   cases' hn with f
   refine' (H <| finset.univ.map f).not_lt _
   rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin]
@@ -1653,7 +1652,7 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
 #print Cardinal.cantor' /-
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
-  rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb 
+  rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
 -/
@@ -1734,7 +1733,7 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
   by_contra! h'
   rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
   · exact h.ne_zero.irrefl
-  · rw [nat_succ] at h 
+  · rw [nat_succ] at h
     exact not_is_succ_limit_succ _ h.is_succ_limit
 #align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
 -/
@@ -1879,7 +1878,7 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
   refine' ⟨fun h => _, _⟩
   · by_cases ha : a = 0; · exact Or.inl ha
     right; by_cases hb : b = 0; · exact Or.inl hb
-    right; rw [← Ne, ← one_le_iff_ne_zero] at ha hb ; constructor
+    right; rw [← Ne, ← one_le_iff_ne_zero] at ha hb; constructor
     · rw [← mul_one a]
       refine' (mul_le_mul' le_rfl hb).trans_lt h
     · rw [← one_mul b]
@@ -1894,7 +1893,7 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
 theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) :=
   by
   let h := (@mul_lt_aleph0_iff a b).Not
-  rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h 
+  rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h
 #align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff
 -/
 
@@ -2513,7 +2512,7 @@ theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅
   by
   constructor
   · intro h
-    rw [mk_eq_zero_iff] at h 
+    rw [mk_eq_zero_iff] at h
     exact eq_empty_iff_forall_not_mem.2 fun x hx => h.elim' ⟨x, hx⟩
   · rintro rfl; exact mk_emptyc _
 #align cardinal.mk_emptyc_iff Cardinal.mk_emptyCollection_iff
@@ -2802,7 +2801,7 @@ theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
     {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({x ∈ s | f x ∈ t} : Set α)) :=
   by
-  rw [image_eq_range] at h ; convert mk_preimage_of_subset_range_lift _ _ h using 1
+  rw [image_eq_range] at h; convert mk_preimage_of_subset_range_lift _ _ h using 1
   rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image_lift Cardinal.mk_subset_ge_of_subset_image_lift
 -/
@@ -2811,7 +2810,7 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
     (#t) ≤ (#({x ∈ s | f x ∈ t} : Set α)) :=
   by
-  rw [image_eq_range] at h ; convert mk_preimage_of_subset_range _ _ h using 1
+  rw [image_eq_range] at h; convert mk_preimage_of_subset_range _ _ h using 1
   rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image Cardinal.mk_subset_ge_of_subset_image
 -/
@@ -2854,7 +2853,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
   by
   rw [mk_eq_two_iff]; constructor
   · rintro ⟨a, b, hne, h⟩
-    simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h 
+    simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
     rcases h x with (rfl | rfl)
     exacts [⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
   · rintro ⟨y, hne, hy⟩

bump to nightly-2024-02-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

42bef5c2

Diff

@@ -2070,33 +2070,31 @@ theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
 -/
 
-#print Cardinal.toNat_le_of_le_of_lt_aleph0 /-
-theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
-    c.toNat ≤ d.toNat :=
+#print Cardinal.toNat_le_toNat /-
+theorem toNat_le_toNat {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) : c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
-#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
+#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_toNat
 -/
 
-#print Cardinal.toNat_lt_of_lt_of_lt_aleph0 /-
-theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
-    c.toNat < d.toNat :=
+#print Cardinal.toNat_lt_toNat /-
+theorem toNat_lt_toNat {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) : c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
-#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
+#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_toNat
 -/
 
-#print Cardinal.toNat_cast /-
+#print Cardinal.toNat_natCast /-
 @[simp]
-theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
+theorem toNat_natCast (n : ℕ) : Cardinal.toNat n = n :=
   by
   rw [to_nat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), ← nat_cast_inj]
   exact (Classical.choose_spec (lt_aleph_0.1 (nat_lt_aleph_0 n))).symm
-#align cardinal.to_nat_cast Cardinal.toNat_cast
+#align cardinal.to_nat_cast Cardinal.toNat_natCast
 -/
 
 #print Cardinal.toNat_rightInverse /-
 /-- `to_nat` has a right-inverse: coercion. -/
 theorem toNat_rightInverse : Function.RightInverse (coe : ℕ → Cardinal) toNat :=
-  toNat_cast
+  toNat_natCast
 #align cardinal.to_nat_right_inverse Cardinal.toNat_rightInverse
 -/
 
@@ -2150,7 +2148,7 @@ theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c
     (cast_toNat_of_lt_aleph0
             (lt_of_not_ge (hn ∘ h.symm.trans ∘ toNat_apply_of_aleph0_le))).symm.trans
       (congr_arg coe h),
-    fun h => (congr_arg toNat h).trans (toNat_cast n)⟩
+    fun h => (congr_arg toNat h).trans (toNat_natCast n)⟩
 #align cardinal.to_nat_eq_iff Cardinal.toNat_eq_iff
 -/
 
@@ -2205,27 +2203,29 @@ theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
 #align cardinal.to_nat_mul Cardinal.toNat_mul
 -/
 
-#print Cardinal.toNatHom /-
+/- warning: cardinal.to_nat_hom clashes with cardinal.to_nat -> Cardinal.toNat
+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_hom Cardinal.toNatₓ'. -/
+#print Cardinal.toNat /-
 /-- `cardinal.to_nat` as a `monoid_with_zero_hom`. -/
 @[simps]
-def toNatHom : Cardinal →*₀ ℕ where
+def toNat : Cardinal →*₀ ℕ where
   toFun := toNat
   map_zero' := zero_toNat
   map_one' := one_toNat
   map_mul' := toNat_mul
-#align cardinal.to_nat_hom Cardinal.toNatHom
+#align cardinal.to_nat_hom Cardinal.toNat
 -/
 
 #print Cardinal.toNat_finset_prod /-
 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
-  map_prod toNatHom _ _
+  map_prod toNat _ _
 #align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prod
 -/
 
-#print Cardinal.toNat_add_of_lt_aleph0 /-
+#print Cardinal.toNat_lift_add_lift /-
 @[simp]
-theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
+theorem toNat_lift_add_lift {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
     (lift.{v, u} a + lift.{u, v} b).toNat = a.toNat + b.toNat :=
   by
   apply Cardinal.natCast_injective
@@ -2233,7 +2233,7 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
   replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph_0]
   rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_to_nat_of_lt_aleph_0 ha,
     cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0 (add_lt_aleph_0 ha hb)]
-#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0
+#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_lift_add_lift
 -/
 
 #print Cardinal.toPartENat /-
@@ -2271,11 +2271,11 @@ theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPa
 #align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_le
 -/
 
-#print Cardinal.toPartENat_cast /-
+#print Cardinal.toPartENat_natCast /-
 @[simp]
-theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
+theorem toPartENat_natCast (n : ℕ) : Cardinal.toPartENat n = n := by
   rw [to_part_enat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), to_nat_cast]
-#align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
+#align cardinal.to_part_enat_cast Cardinal.toPartENat_natCast
 -/
 
 #print Cardinal.mk_toPartENat_of_infinite /-
@@ -2292,8 +2292,6 @@ theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
 #align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
 -/
 
-/- warning: cardinal.to_part_enat_eq_top_iff_le_aleph_0 clashes with to_part_enat_eq_top_iff_le_aleph_0 -> Cardinal.toPartENat_eq_top_iff_le_aleph0
-Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0ₓ'. -/
 #print Cardinal.toPartENat_eq_top_iff_le_aleph0 /-
 theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} : toPartENat c = ⊤ ↔ aleph0 ≤ c :=
   by
@@ -2317,7 +2315,7 @@ theorem toPartENat_le_iff_le_of_le_aleph0 {c c' : Cardinal} (h : c ≤ aleph0) :
   · simp only [to_part_enat_apply_of_aleph_0_le hc', le_top, true_iff_iff]
     exact le_trans h hc'
   · rw [to_part_enat_apply_of_aleph_0_le hc]
-    simp only [top_le_iff, Cardinal.toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]
+    simp only [top_le_iff, Cardinal.toPartENat_eq_top, le_antisymm h hc]
 #align cardinal.to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_le_of_le_aleph0
 
 theorem toPartENat_le_iff_le_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < aleph0) :
@@ -2329,7 +2327,7 @@ theorem toPartENat_le_iff_le_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < aleph0)
     rw [PartENat.coe_le_coe]
     exact to_nat_le_iff_le_of_lt_aleph_0 hc hc'
   · rw [to_part_enat_apply_of_aleph_0_le hc]
-    simp only [top_le_iff, Cardinal.toPartENat_eq_top_iff_le_aleph0]
+    simp only [top_le_iff, Cardinal.toPartENat_eq_top]
     rw [← not_iff_not, not_le, not_le]
     simp only [hc', lt_of_lt_of_le hc' hc]
 #align cardinal.to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_le_of_lt_aleph0
@@ -2357,7 +2355,7 @@ theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') : toPartENat c ≤ toPa
 
 #print Cardinal.toPartENat_surjective /-
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
-  PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
+  PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_natCast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 -/

bump to nightly-2023-12-06-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

d09917f6

Diff

@@ -1731,7 +1731,7 @@ theorem isLimit_aleph0 : IsLimit ℵ₀ :=
 #print Cardinal.IsLimit.aleph0_le /-
 theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
   by
-  by_contra' h'
+  by_contra! h'
   rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
   · exact h.ne_zero.irrefl
   · rw [nat_succ] at h

bump to nightly-2023-12-02-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e637da5d

Diff

@@ -574,7 +574,7 @@ instance : Pow Cardinal.{u} Cardinal.{u} :=
 
 local infixr:0 "^" => @Pow.pow Cardinal Cardinal Cardinal.hasPow
 
-local infixr:80 " ^ℕ " => @Pow.pow Cardinal ℕ Monoid.Pow
+local infixr:80 " ^ℕ " => @Pow.pow Cardinal ℕ Monoid.toNatPow
 
 #print Cardinal.power_def /-
 theorem power_def (α β) : ((#α)^#β) = (#β → α) :=

bump to nightly-2023-11-05-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

acc3325f

Diff

@@ -2860,7 +2860,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
     rcases h x with (rfl | rfl)
     exacts [⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
   · rintro ⟨y, hne, hy⟩
-    exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
+    exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => Classical.or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'
 -/

bump to nightly-2023-10-21-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

a2dc86f8

Diff

@@ -956,7 +956,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
   cases' le_of_lt hlt with f
   have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
-  simp only [surjective, not_forall] at this 
+  simp only [surjective, Classical.not_forall] at this 
   rcases this with ⟨b, hb⟩
   calc
     (#γ) + 1 = (#Option γ) := mk_option.symm

bump to nightly-2023-10-07-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

d191b687

Diff

@@ -806,11 +806,11 @@ instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
     eq_zero_or_eq_zero_of_mul_eq_zero := fun a b =>
       inductionOn₂ a b fun α β => by simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }
 
-instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
+instance : CanonicallyLinearOrderedAddCommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
 
 -- Computable instance to prevent a non-computable one being found via the one above
-instance : CanonicallyOrderedAddMonoid Cardinal.{u} :=
+instance : CanonicallyOrderedAddCommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
 
 instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,15 +3,15 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 -/
-import Mathbin.Data.Fintype.BigOperators
-import Mathbin.Data.Finsupp.Defs
-import Mathbin.Data.Nat.PartEnat
-import Mathbin.Data.Set.Countable
-import Mathbin.Logic.Small.Basic
-import Mathbin.Order.ConditionallyCompleteLattice.Basic
-import Mathbin.Order.SuccPred.Limit
-import Mathbin.SetTheory.Cardinal.SchroederBernstein
-import Mathbin.Tactic.Positivity
+import Data.Fintype.BigOperators
+import Data.Finsupp.Defs
+import Data.Nat.PartEnat
+import Data.Set.Countable
+import Logic.Small.Basic
+import Order.ConditionallyCompleteLattice.Basic
+import Order.SuccPred.Limit
+import SetTheory.Cardinal.SchroederBernstein
+import Tactic.Positivity
 
 #align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"

bump to nightly-2023-08-23-23

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

f612b9e0

Diff

@@ -186,7 +186,7 @@ theorem mk_congr (e : α ≃ β) : (#α) = (#β) :=
 #align cardinal.mk_congr Cardinal.mk_congr
 -/
 
-alias mk_congr ← _root_.equiv.cardinal_eq
+alias _root_.equiv.cardinal_eq := mk_congr
 #align equiv.cardinal_eq Equiv.cardinal_eq
 
 #print Cardinal.map /-
@@ -501,7 +501,7 @@ theorem mk_le_one_iff_set_subsingleton {s : Set α} : (#s) ≤ 1 ↔ s.Subsingle
 #align cardinal.mk_le_one_iff_set_subsingleton Cardinal.mk_le_one_iff_set_subsingleton
 -/
 
-alias mk_le_one_iff_set_subsingleton ↔ _ _root_.set.subsingleton.cardinal_mk_le_one
+alias ⟨_, _root_.set.subsingleton.cardinal_mk_le_one⟩ := mk_le_one_iff_set_subsingleton
 #align set.subsingleton.cardinal_mk_le_one Set.Subsingleton.cardinal_mk_le_one
 
 instance : Add Cardinal.{u} :=
@@ -1777,7 +1777,7 @@ theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
 -/
 
-alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
+alias ⟨_, _root_.set.finite.lt_aleph_0⟩ := lt_aleph_0_iff_set_finite
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
 #print Cardinal.lt_aleph0_iff_subtype_finite /-
@@ -1807,7 +1807,7 @@ theorem le_aleph0_iff_set_countable {s : Set α} : (#s) ≤ ℵ₀ ↔ s.Countab
 #align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable
 -/
 
-alias le_aleph_0_iff_set_countable ↔ _ _root_.set.countable.le_aleph_0
+alias ⟨_, _root_.set.countable.le_aleph_0⟩ := le_aleph_0_iff_set_countable
 #align set.countable.le_aleph_0 Set.Countable.le_aleph0
 
 #print Cardinal.le_aleph0_iff_subtype_countable /-

bump to nightly-2023-08-17-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

60285dcc

Diff

@@ -630,7 +630,7 @@ instance : CommSemiring Cardinal.{u} where
   one_mul a := inductionOn a fun α => mk_congr <| Equiv.punitProd α
   mul_one a := inductionOn a fun α => mk_congr <| Equiv.prodPUnit α
   mul_assoc a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodAssoc α β γ
-  mul_comm := mul_comm'
+  mul_comm := hMul_comm'
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
   npow n c := c^n

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-
-! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 3ff3f2d6a3118b8711063de7111a0d77a53219a8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Fintype.BigOperators
 import Mathbin.Data.Finsupp.Defs
@@ -18,6 +13,8 @@ import Mathbin.Order.SuccPred.Limit
 import Mathbin.SetTheory.Cardinal.SchroederBernstein
 import Mathbin.Tactic.Positivity
 
+#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
+
 /-!
 # Cardinal Numbers

bump to nightly-2023-06-27-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

b5e0c440

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 9bb28972724354ac0574e2b318be896ec252025f
+! leanprover-community/mathlib commit 3ff3f2d6a3118b8711063de7111a0d77a53219a8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -2052,6 +2052,13 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat =
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 -/
 
+#print Cardinal.toNat_eq_iff_eq_of_lt_aleph0 /-
+theorem toNat_eq_iff_eq_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
+    c.toNat = d.toNat ↔ c = d := by
+  rw [← nat_cast_inj, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
+#align cardinal.to_nat_eq_iff_eq_of_lt_aleph_0 Cardinal.toNat_eq_iff_eq_of_lt_aleph0
+-/
+
 #print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
@@ -2288,12 +2295,93 @@ theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
 #align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
 -/
 
+/- warning: cardinal.to_part_enat_eq_top_iff_le_aleph_0 clashes with to_part_enat_eq_top_iff_le_aleph_0 -> Cardinal.toPartENat_eq_top_iff_le_aleph0
+Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0ₓ'. -/
+#print Cardinal.toPartENat_eq_top_iff_le_aleph0 /-
+theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} : toPartENat c = ⊤ ↔ aleph0 ≤ c :=
+  by
+  cases' lt_or_ge c aleph_0 with hc hc
+  simp only [to_part_enat_apply_of_lt_aleph_0 hc, PartENat.natCast_ne_top, false_iff_iff, not_le,
+    hc]
+  simp only [to_part_enat_apply_of_aleph_0_le hc, eq_self_iff_true, true_iff_iff]
+  exact hc
+#align cardinal.to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0
+-/
+
+theorem toPartENat_le_iff_le_of_le_aleph0 {c c' : Cardinal} (h : c ≤ aleph0) :
+    toPartENat c ≤ toPartENat c' ↔ c ≤ c' :=
+  by
+  cases' lt_or_ge c aleph_0 with hc hc
+  rw [to_part_enat_apply_of_lt_aleph_0 hc]
+  cases' lt_or_ge c' aleph_0 with hc' hc'
+  · rw [to_part_enat_apply_of_lt_aleph_0 hc']
+    rw [PartENat.coe_le_coe]
+    exact to_nat_le_iff_le_of_lt_aleph_0 hc hc'
+  · simp only [to_part_enat_apply_of_aleph_0_le hc', le_top, true_iff_iff]
+    exact le_trans h hc'
+  · rw [to_part_enat_apply_of_aleph_0_le hc]
+    simp only [top_le_iff, Cardinal.toPartENat_eq_top_iff_le_aleph0, le_antisymm h hc]
+#align cardinal.to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_le_of_le_aleph0
+
+theorem toPartENat_le_iff_le_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < aleph0) :
+    toPartENat c ≤ toPartENat c' ↔ c ≤ c' :=
+  by
+  cases' lt_or_ge c aleph_0 with hc hc
+  · rw [to_part_enat_apply_of_lt_aleph_0 hc]
+    rw [to_part_enat_apply_of_lt_aleph_0 hc']
+    rw [PartENat.coe_le_coe]
+    exact to_nat_le_iff_le_of_lt_aleph_0 hc hc'
+  · rw [to_part_enat_apply_of_aleph_0_le hc]
+    simp only [top_le_iff, Cardinal.toPartENat_eq_top_iff_le_aleph0]
+    rw [← not_iff_not, not_le, not_le]
+    simp only [hc', lt_of_lt_of_le hc' hc]
+#align cardinal.to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_le_of_lt_aleph0
+
+theorem toPartENat_eq_iff_eq_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ aleph0) (hc' : c' ≤ aleph0) :
+    toPartENat c = toPartENat c' ↔ c = c' := by
+  rw [le_antisymm_iff, le_antisymm_iff, Cardinal.toPartENat_le_iff_of_le_aleph0 hc,
+    Cardinal.toPartENat_le_iff_of_le_aleph0 hc']
+#align cardinal.to_part_enat_eq_iff_eq_of_le_aleph_0 Cardinal.toPartENat_eq_iff_eq_of_le_aleph0
+
+#print Cardinal.toPartENat_mono /-
+theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') : toPartENat c ≤ toPartENat c' :=
+  by
+  cases' lt_or_ge c aleph_0 with hc hc
+  rw [to_part_enat_apply_of_lt_aleph_0 hc]
+  cases' lt_or_ge c' aleph_0 with hc' hc'
+  rw [to_part_enat_apply_of_lt_aleph_0 hc']
+  simp only [PartENat.coe_le_coe]
+  exact to_nat_le_of_le_of_lt_aleph_0 hc' h
+  rw [to_part_enat_apply_of_aleph_0_le hc']
+  exact le_top
+  rw [to_part_enat_apply_of_aleph_0_le hc, to_part_enat_apply_of_aleph_0_le (le_trans hc h)]
+#align cardinal.to_part_enat_mono Cardinal.toPartENat_mono
+-/
+
 #print Cardinal.toPartENat_surjective /-
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 -/
 
+#print Cardinal.toPartENat_lift /-
+theorem toPartENat_lift (c : Cardinal.{v}) : (lift.{u, v} c).toPartENat = c.toPartENat :=
+  by
+  cases' lt_or_ge c ℵ₀ with hc hc
+  · rw [to_part_enat_apply_of_lt_aleph_0 hc, Cardinal.toPartENat_apply_of_lt_aleph0 _]
+    simp only [to_nat_lift]
+    rw [← lift_aleph_0, lift_lt]; exact hc
+  · rw [to_part_enat_apply_of_aleph_0_le hc, Cardinal.toPartENat_apply_of_aleph0_le _]
+    rw [← lift_aleph_0, lift_le]; exact hc
+#align cardinal.to_part_enat_lift Cardinal.toPartENat_lift
+-/
+
+#print Cardinal.toPartENat_congr /-
+theorem toPartENat_congr {β : Type v} (e : α ≃ β) : (#α).toPartENat = (#β).toPartENat := by
+  rw [← to_part_enat_lift, lift_mk_eq.mpr ⟨e⟩, to_part_enat_lift]
+#align cardinal.to_part_enat_congr Cardinal.toPartENat_congr
+-/
+
 #print Cardinal.mk_toPartENat_eq_coe_card /-
 theorem mk_toPartENat_eq_coe_card [Fintype α] : (#α).toPartENat = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -125,7 +125,6 @@ def mk : Type u → Cardinal :=
 #align cardinal.mk Cardinal.mk
 -/
 
--- mathport name: cardinal.mk
 scoped prefix:0 "#" => Cardinal.mk
 
 #print Cardinal.canLiftCardinalType /-
@@ -141,17 +140,21 @@ theorem inductionOn {p : Cardinal → Prop} (c : Cardinal) (h : ∀ α, p (#α))
 #align cardinal.induction_on Cardinal.inductionOn
 -/
 
+#print Cardinal.inductionOn₂ /-
 @[elab_as_elim]
 theorem inductionOn₂ {p : Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
     (h : ∀ α β, p (#α) (#β)) : p c₁ c₂ :=
   Quotient.induction_on₂ c₁ c₂ h
 #align cardinal.induction_on₂ Cardinal.inductionOn₂
+-/
 
+#print Cardinal.inductionOn₃ /-
 @[elab_as_elim]
 theorem inductionOn₃ {p : Cardinal → Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
     (c₃ : Cardinal) (h : ∀ α β γ, p (#α) (#β) (#γ)) : p c₁ c₂ c₃ :=
   Quotient.induction_on₃ c₁ c₂ c₃ h
 #align cardinal.induction_on₃ Cardinal.inductionOn₃
+-/
 
 #print Cardinal.eq /-
 protected theorem eq : (#α) = (#β) ↔ Nonempty (α ≃ β) :=
@@ -416,12 +419,16 @@ theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
 #align cardinal.lift_lt Cardinal.lift_lt
 -/
 
+#print Cardinal.lift_strictMono /-
 theorem lift_strictMono : StrictMono lift := fun a b => lift_lt.2
 #align cardinal.lift_strict_mono Cardinal.lift_strictMono
+-/
 
+#print Cardinal.lift_monotone /-
 theorem lift_monotone : Monotone lift :=
   lift_strictMono.Monotone
 #align cardinal.lift_monotone Cardinal.lift_monotone
+-/
 
 instance : Zero Cardinal.{u} :=
   ⟨#PEmpty⟩
@@ -435,10 +442,12 @@ theorem mk_eq_zero (α : Type u) [IsEmpty α] : (#α) = 0 :=
 #align cardinal.mk_eq_zero Cardinal.mk_eq_zero
 -/
 
+#print Cardinal.lift_zero /-
 @[simp]
 theorem lift_zero : lift 0 = 0 :=
   mk_congr (Equiv.equivPEmpty _)
 #align cardinal.lift_zero Cardinal.lift_zero
+-/
 
 #print Cardinal.lift_eq_zero /-
 @[simp]
@@ -501,28 +510,37 @@ alias mk_le_one_iff_set_subsingleton ↔ _ _root_.set.subsingleton.cardinal_mk_l
 instance : Add Cardinal.{u} :=
   ⟨map₂ Sum fun α β γ δ => Equiv.sumCongr⟩
 
+#print Cardinal.add_def /-
 theorem add_def (α β : Type u) : (#α) + (#β) = (#Sum α β) :=
   rfl
 #align cardinal.add_def Cardinal.add_def
+-/
 
 instance : NatCast Cardinal.{u} :=
   ⟨Nat.unaryCast⟩
 
+#print Cardinal.mk_sum /-
 @[simp]
 theorem mk_sum (α : Type u) (β : Type v) : (#Sum α β) = lift.{v, u} (#α) + lift.{u, v} (#β) :=
   mk_congr (Equiv.ulift.symm.sumCongr Equiv.ulift.symm)
 #align cardinal.mk_sum Cardinal.mk_sum
+-/
 
+#print Cardinal.mk_option /-
 @[simp]
 theorem mk_option {α : Type u} : (#Option α) = (#α) + 1 :=
   (Equiv.optionEquivSumPUnit α).cardinal_eq
 #align cardinal.mk_option Cardinal.mk_option
+-/
 
+#print Cardinal.mk_psum /-
 @[simp]
 theorem mk_psum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
   (mk_congr (Equiv.psumEquivSum α β)).trans (mk_sum α β)
 #align cardinal.mk_psum Cardinal.mk_psum
+-/
 
+#print Cardinal.mk_fintype /-
 @[simp]
 theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   by
@@ -532,18 +550,23 @@ theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   · rfl
   · intro α h hα; simp [hα]; rfl
 #align cardinal.mk_fintype Cardinal.mk_fintype
+-/
 
 instance : Mul Cardinal.{u} :=
   ⟨map₂ Prod fun α β γ δ => Equiv.prodCongr⟩
 
+#print Cardinal.mul_def /-
 theorem mul_def (α β : Type u) : (#α) * (#β) = (#α × β) :=
   rfl
 #align cardinal.mul_def Cardinal.mul_def
+-/
 
+#print Cardinal.mk_prod /-
 @[simp]
 theorem mk_prod (α : Type u) (β : Type v) : (#α × β) = lift.{v, u} (#α) * lift.{u, v} (#β) :=
   mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)
 #align cardinal.mk_prod Cardinal.mk_prod
+-/
 
 private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
   inductionOn₂ a b fun α β => mk_congr <| Equiv.prodComm α β
@@ -552,10 +575,8 @@ private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
 instance : Pow Cardinal.{u} Cardinal.{u} :=
   ⟨map₂ (fun α β => β → α) fun α β γ δ e₁ e₂ => e₂.arrowCongr e₁⟩
 
--- mathport name: cardinal.pow
 local infixr:0 "^" => @Pow.pow Cardinal Cardinal Cardinal.hasPow
 
--- mathport name: cardinal.pow.nat
 local infixr:80 " ^ℕ " => @Pow.pow Cardinal ℕ Monoid.Pow
 
 #print Cardinal.power_def /-
@@ -592,9 +613,11 @@ theorem power_one {a : Cardinal} : (a^1) = a :=
 #align cardinal.power_one Cardinal.power_one
 -/
 
+#print Cardinal.power_add /-
 theorem power_add {a b c : Cardinal} : (a^b + c) = (a^b) * (a^c) :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumArrowEquivProdArrow β γ α
 #align cardinal.power_add Cardinal.power_add
+-/
 
 instance : CommSemiring Cardinal.{u} where
   zero := 0
@@ -617,13 +640,17 @@ instance : CommSemiring Cardinal.{u} where
   npow_zero := @power_zero
   npow_succ n c := show (c^n + 1) = c * (c^n) by rw [power_add, power_one, mul_comm']
 
+#print Cardinal.power_bit0 /-
 theorem power_bit0 (a b : Cardinal) : (a^bit0 b) = (a^b) * (a^b) :=
   power_add
 #align cardinal.power_bit0 Cardinal.power_bit0
+-/
 
+#print Cardinal.power_bit1 /-
 theorem power_bit1 (a b : Cardinal) : (a^bit1 b) = (a^b) * (a^b) * a := by
   rw [bit1, ← power_bit0, power_add, power_one]
 #align cardinal.power_bit1 Cardinal.power_bit1
+-/
 
 #print Cardinal.one_power /-
 @[simp]
@@ -632,13 +659,17 @@ theorem one_power {a : Cardinal} : (1^a) = 1 :=
 #align cardinal.one_power Cardinal.one_power
 -/
 
+#print Cardinal.mk_bool /-
 @[simp]
 theorem mk_bool : (#Bool) = 2 := by simp
 #align cardinal.mk_bool Cardinal.mk_bool
+-/
 
+#print Cardinal.mk_Prop /-
 @[simp]
 theorem mk_Prop : (#Prop) = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
+-/
 
 #print Cardinal.zero_power /-
 @[simp]
@@ -659,60 +690,84 @@ theorem power_ne_zero {a : Cardinal} (b) : a ≠ 0 → (a^b) ≠ 0 :=
 #align cardinal.power_ne_zero Cardinal.power_ne_zero
 -/
 
+#print Cardinal.mul_power /-
 theorem mul_power {a b c : Cardinal} : (a * b^c) = (a^c) * (b^c) :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.arrowProdEquivProdArrow α β γ
 #align cardinal.mul_power Cardinal.mul_power
+-/
 
+#print Cardinal.power_mul /-
 theorem power_mul {a b c : Cardinal} : (a^b * c) = ((a^b)^c) := by rw [mul_comm b c];
   exact induction_on₃ a b c fun α β γ => mk_congr <| Equiv.curry γ β α
 #align cardinal.power_mul Cardinal.power_mul
+-/
 
+#print Cardinal.pow_cast_right /-
 @[simp]
 theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : (a^(↑n : Cardinal.{u})) = a ^ℕ n :=
   rfl
 #align cardinal.pow_cast_right Cardinal.pow_cast_right
+-/
 
+#print Cardinal.lift_one /-
 @[simp]
 theorem lift_one : lift 1 = 1 :=
   mk_congr <| Equiv.ulift.trans Equiv.punitEquivPUnit
 #align cardinal.lift_one Cardinal.lift_one
+-/
 
+#print Cardinal.lift_add /-
 @[simp]
 theorem lift_add (a b) : lift (a + b) = lift a + lift b :=
   inductionOn₂ a b fun α β =>
     mk_congr <| Equiv.ulift.trans (Equiv.sumCongr Equiv.ulift Equiv.ulift).symm
 #align cardinal.lift_add Cardinal.lift_add
+-/
 
+#print Cardinal.lift_mul /-
 @[simp]
 theorem lift_mul (a b) : lift (a * b) = lift a * lift b :=
   inductionOn₂ a b fun α β =>
     mk_congr <| Equiv.ulift.trans (Equiv.prodCongr Equiv.ulift Equiv.ulift).symm
 #align cardinal.lift_mul Cardinal.lift_mul
+-/
 
+#print Cardinal.lift_bit0 /-
 @[simp]
 theorem lift_bit0 (a : Cardinal) : lift (bit0 a) = bit0 (lift a) :=
   lift_add a a
 #align cardinal.lift_bit0 Cardinal.lift_bit0
+-/
 
+#print Cardinal.lift_bit1 /-
 @[simp]
 theorem lift_bit1 (a : Cardinal) : lift (bit1 a) = bit1 (lift a) := by simp [bit1]
 #align cardinal.lift_bit1 Cardinal.lift_bit1
+-/
 
+#print Cardinal.lift_two /-
 theorem lift_two : lift.{u, v} 2 = 2 := by simp
 #align cardinal.lift_two Cardinal.lift_two
+-/
 
+#print Cardinal.mk_set /-
 @[simp]
 theorem mk_set {α : Type u} : (#Set α) = (2^#α) := by simp [Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
+-/
 
+#print Cardinal.mk_powerset /-
 /-- A variant of `cardinal.mk_set` expressed in terms of a `set` instead of a `Type`. -/
 @[simp]
 theorem mk_powerset {α : Type u} (s : Set α) : (#↥(𝒫 s)) = (2^#↥s) :=
   (mk_congr (Equiv.Set.powerset s)).trans mk_set
 #align cardinal.mk_powerset Cardinal.mk_powerset
+-/
 
+#print Cardinal.lift_two_power /-
 theorem lift_two_power (a) : lift (2^a) = (2^lift a) := by simp
 #align cardinal.lift_two_power Cardinal.lift_two_power
+-/
 
 section OrderProperties
 
@@ -726,13 +781,17 @@ protected theorem zero_le : ∀ a : Cardinal, 0 ≤ a := by rintro ⟨α⟩ <;>
 private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d → a + c ≤ b + d := by
   rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.sum_map e₂⟩
 
+#print Cardinal.add_covariantClass /-
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
 #align cardinal.add_covariant_class Cardinal.add_covariantClass
+-/
 
+#print Cardinal.add_swap_covariantClass /-
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
 #align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClass
+-/
 
 instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
   { Cardinal.commSemiring,
@@ -791,6 +850,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 #align cardinal.self_le_power Cardinal.self_le_power
 -/
 
+#print Cardinal.cantor /-
 /-- **Cantor's theorem** -/
 theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
   by
@@ -800,6 +860,7 @@ theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
   rintro ⟨⟨f, hf⟩⟩
   exact cantor_injective f hf
 #align cardinal.cantor Cardinal.cantor
+-/
 
 instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 
@@ -812,12 +873,14 @@ theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
 -/
 
+#print Cardinal.power_le_max_power_one /-
 theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 :=
   by
   by_cases ha : a = 0
   simp [ha, zero_power_le]
   exact (power_le_power_left ha h).trans (le_max_left _ _)
 #align cardinal.power_le_max_power_one Cardinal.power_le_max_power_one
+-/
 
 #print Cardinal.power_le_power_right /-
 theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
@@ -859,28 +922,37 @@ instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
 
+#print Cardinal.sInf_empty /-
 @[simp]
 theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg not_nonempty_empty
 #align cardinal.Inf_empty Cardinal.sInf_empty
+-/
 
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => sInf {c' | c < c'}) fun a b =>
     ⟨lt_of_lt_of_le <| csInf_mem <| exists_gt a, csInf_le'⟩
 
+#print Cardinal.succ_def /-
 theorem succ_def (c : Cardinal) : succ c = sInf {c' | c < c'} :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
+-/
 
+#print Cardinal.succ_pos /-
 theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
   bot_lt_succ
 #align cardinal.succ_pos Cardinal.succ_pos
+-/
 
+#print Cardinal.succ_ne_zero /-
 theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
   (succ_pos _).ne'
 #align cardinal.succ_ne_zero Cardinal.succ_ne_zero
+-/
 
+#print Cardinal.add_one_le_succ /-
 theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   by
   refine' (le_csInf_iff'' (exists_gt c)).2 fun b hlt => _
@@ -893,6 +965,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
     (#γ) + 1 = (#Option γ) := mk_option.symm
     _ ≤ (#β) := (f.option_elim b hb).cardinal_le
 #align cardinal.add_one_le_succ Cardinal.add_one_le_succ
+-/
 
 #print Cardinal.IsLimit /-
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
@@ -916,9 +989,11 @@ protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
 #align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
 -/
 
+#print Cardinal.IsLimit.succ_lt /-
 theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
   h.IsSuccLimit.succ_lt
 #align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
+-/
 
 #print Cardinal.isSuccLimit_zero /-
 theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
@@ -934,16 +1009,21 @@ def sum {ι} (f : ι → Cardinal) : Cardinal :=
 #align cardinal.sum Cardinal.sum
 -/
 
+#print Cardinal.le_sum /-
 theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
   rw [← Quotient.out_eq (f i)] <;>
     exact ⟨⟨fun a => ⟨i, a⟩, fun a b h => eq_of_hEq <| by injection h⟩⟩
 #align cardinal.le_sum Cardinal.le_sum
+-/
 
+#print Cardinal.mk_sigma /-
 @[simp]
 theorem mk_sigma {ι} (f : ι → Type _) : (#Σ i, f i) = sum fun i => #f i :=
   mk_congr <| Equiv.sigmaCongrRight fun i => outMkEquiv.symm
 #align cardinal.mk_sigma Cardinal.mk_sigma
+-/
 
+#print Cardinal.sum_const /-
 @[simp]
 theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
     (sum fun i : ι => a) = lift.{v} (#ι) * lift.{u} a :=
@@ -953,21 +1033,28 @@ theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
         (Σ i : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
         _ ≃ ULift ι × ULift α := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)
 #align cardinal.sum_const Cardinal.sum_const
+-/
 
+#print Cardinal.sum_const' /-
 theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = (#ι) * a := by simp
 #align cardinal.sum_const' Cardinal.sum_const'
+-/
 
+#print Cardinal.sum_add_distrib /-
 @[simp]
 theorem sum_add_distrib {ι} (f g : ι → Cardinal) : sum (f + g) = sum f + sum g := by
   simpa only [mk_sigma, mk_sum, mk_out, lift_id] using
     mk_congr (Equiv.sigmaSumDistrib (Quotient.out ∘ f) (Quotient.out ∘ g))
 #align cardinal.sum_add_distrib Cardinal.sum_add_distrib
+-/
 
+#print Cardinal.sum_add_distrib' /-
 @[simp]
 theorem sum_add_distrib' {ι} (f g : ι → Cardinal) :
     (Cardinal.sum fun i => f i + g i) = sum f + sum g :=
   sum_add_distrib f g
 #align cardinal.sum_add_distrib' Cardinal.sum_add_distrib'
+-/
 
 #print Cardinal.lift_sum /-
 @[simp]
@@ -980,18 +1067,23 @@ theorem lift_sum {ι : Type u} (f : ι → Cardinal.{v}) :
 #align cardinal.lift_sum Cardinal.lift_sum
 -/
 
+#print Cardinal.sum_le_sum /-
 theorem sum_le_sum {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : sum f ≤ sum g :=
   ⟨(Embedding.refl _).sigma_map fun i =>
       Classical.choice <| by
         have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this ⟩
 #align cardinal.sum_le_sum Cardinal.sum_le_sum
+-/
 
+#print Cardinal.mk_le_mk_mul_of_mk_preimage_le /-
 theorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : α → β) (hf : ∀ b : β, (#f ⁻¹' {b}) ≤ c) :
     (#α) ≤ (#β) * c := by
   simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using
     sum_le_sum _ _ hf
 #align cardinal.mk_le_mk_mul_of_mk_preimage_le Cardinal.mk_le_mk_mul_of_mk_preimage_le
+-/
 
+#print Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le /-
 theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v} {c : Cardinal}
     (f : α → β) (hf : ∀ b : β, lift.{v} (#f ⁻¹' {b}) ≤ c) : lift.{v} (#α) ≤ lift.{u} (#β) * c :=
   (mk_le_mk_mul_of_mk_preimage_le fun x : ULift.{v} α => ULift.up.{u} (f x.1)) <|
@@ -1002,6 +1094,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
                 Equiv.ulift.symm)).trans_le
         (hf b)
 #align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
+-/
 
 #print Cardinal.bddAbove_range /-
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
@@ -1058,32 +1151,42 @@ theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbo
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
 -/
 
+#print Cardinal.iSup_le_sum /-
 theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
   ciSup_le' <| le_sum _
 #align cardinal.supr_le_sum Cardinal.iSup_le_sum
+-/
 
+#print Cardinal.sum_le_iSup_lift /-
 theorem sum_le_iSup_lift {ι : Type u} (f : ι → Cardinal.{max u v}) : sum f ≤ (#ι).lift * iSup f :=
   by
   rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
   exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)
 #align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift
+-/
 
+#print Cardinal.sum_le_iSup /-
 theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f := by
   rw [← lift_id (#ι)]; exact sum_le_supr_lift f
 #align cardinal.sum_le_supr Cardinal.sum_le_iSup
+-/
 
+#print Cardinal.sum_nat_eq_add_sum_succ /-
 theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
     Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) :=
   by
   refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _
   simp only [mk_sum, mk_out, lift_id, mk_sigma]
 #align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ
+-/
 
+#print Cardinal.iSup_of_empty /-
 /-- A variant of `csupr_of_empty` but with `0` on the RHS for convenience -/
 @[simp]
 protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
   ciSup_of_empty f
 #align cardinal.supr_of_empty Cardinal.iSup_of_empty
+-/
 
 #print Cardinal.lift_mk_shrink /-
 @[simp]
@@ -1139,18 +1242,24 @@ theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) =
 #align cardinal.prod_const' Cardinal.prod_const'
 -/
 
+#print Cardinal.prod_le_prod /-
 theorem prod_le_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : prod f ≤ prod g :=
   ⟨Embedding.piCongrRight fun i =>
       Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this ⟩
 #align cardinal.prod_le_prod Cardinal.prod_le_prod
+-/
 
+#print Cardinal.prod_eq_zero /-
 @[simp]
 theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 := by
   lift f to ι → Type u using fun _ => trivial; simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi]
 #align cardinal.prod_eq_zero Cardinal.prod_eq_zero
+-/
 
+#print Cardinal.prod_ne_zero /-
 theorem prod_ne_zero {ι} (f : ι → Cardinal) : prod f ≠ 0 ↔ ∀ i, f i ≠ 0 := by simp [prod_eq_zero]
 #align cardinal.prod_ne_zero Cardinal.prod_ne_zero
+-/
 
 #print Cardinal.lift_prod /-
 @[simp]
@@ -1181,6 +1290,7 @@ theorem prod_eq_of_fintype {α : Type u} [Fintype α] (f : α → Cardinal.{v})
 #align cardinal.prod_eq_of_fintype Cardinal.prod_eq_of_fintype
 -/
 
+#print Cardinal.lift_sInf /-
 @[simp]
 theorem lift_sInf (s : Set Cardinal) : lift (sInf s) = sInf (lift '' s) :=
   by
@@ -1188,11 +1298,14 @@ theorem lift_sInf (s : Set Cardinal) : lift (sInf s) = sInf (lift '' s) :=
   · simp
   · exact lift_monotone.map_Inf hs
 #align cardinal.lift_Inf Cardinal.lift_sInf
+-/
 
+#print Cardinal.lift_iInf /-
 @[simp]
 theorem lift_iInf {ι} (f : ι → Cardinal) : lift (iInf f) = ⨅ i, lift (f i) := by unfold iInf;
   convert lift_Inf (range f); rw [range_comp]
 #align cardinal.lift_infi Cardinal.lift_iInf
+-/
 
 #print Cardinal.lift_down /-
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} : b ≤ lift a → ∃ a', lift a' = b :=
@@ -1227,6 +1340,7 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
 -/
 
+#print Cardinal.lift_succ /-
 @[simp]
 theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
   le_antisymm
@@ -1236,6 +1350,7 @@ theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
       exact h.not_lt (lt_succ _))
     (succ_le_of_lt <| lift_lt.2 <| lt_succ a)
 #align cardinal.lift_succ Cardinal.lift_succ
+-/
 
 #print Cardinal.lift_umax_eq /-
 @[simp]
@@ -1245,16 +1360,21 @@ theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
 #align cardinal.lift_umax_eq Cardinal.lift_umax_eq
 -/
 
+#print Cardinal.lift_min /-
 @[simp]
 theorem lift_min {a b : Cardinal} : lift (min a b) = min (lift a) (lift b) :=
   lift_monotone.map_min
 #align cardinal.lift_min Cardinal.lift_min
+-/
 
+#print Cardinal.lift_max /-
 @[simp]
 theorem lift_max {a b : Cardinal} : lift (max a b) = max (lift a) (lift b) :=
   lift_monotone.map_max
 #align cardinal.lift_max Cardinal.lift_max
+-/
 
+#print Cardinal.lift_sSup /-
 /-- The lift of a supremum is the supremum of the lifts. -/
 theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSup (lift.{u} '' s) :=
   by
@@ -1267,26 +1387,34 @@ theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSu
   · rintro i ⟨j, hj, rfl⟩
     exact lift_le.2 (le_csSup hs hj)
 #align cardinal.lift_Sup Cardinal.lift_sSup
+-/
 
+#print Cardinal.lift_iSup /-
 /-- The lift of a supremum is the supremum of the lifts. -/
 theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
     lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by rw [iSup, iSup, lift_Sup hf, ← range_comp]
 #align cardinal.lift_supr Cardinal.lift_iSup
+-/
 
+#print Cardinal.lift_iSup_le /-
 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
 theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
     (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by rw [lift_supr hf]; exact ciSup_le' w
 #align cardinal.lift_supr_le Cardinal.lift_iSup_le
+-/
 
+#print Cardinal.lift_iSup_le_iff /-
 @[simp]
 theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
     {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by rw [lift_supr hf];
   exact ciSup_le_iff' (bdd_above_range_comp hf _)
 #align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff
+-/
 
 universe v' w'
 
+#print Cardinal.lift_iSup_le_lift_iSup /-
 /-- To prove an inequality between the lifts to a common universe of two different supremums,
 it suffices to show that the lift of each cardinal from the smaller supremum
 if bounded by the lift of some cardinal from the larger supremum.
@@ -1298,7 +1426,9 @@ theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardina
   rw [lift_supr hf, lift_supr hf']
   exact ciSup_mono' (bdd_above_range_comp hf' _) fun i => ⟨_, h i⟩
 #align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup
+-/
 
+#print Cardinal.lift_iSup_le_lift_iSup' /-
 /-- A variant of `lift_supr_le_lift_supr` with universes specialized via `w = v` and `w' = v'`.
 This is sometimes necessary to avoid universe unification issues. -/
 theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
@@ -1306,6 +1436,7 @@ theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardin
     (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') :=
   lift_iSup_le_lift_iSup hf hf' h
 #align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'
+-/
 
 #print Cardinal.aleph0 /-
 /-- `ℵ₀` is the smallest infinite cardinal. -/
@@ -1314,7 +1445,6 @@ def aleph0 : Cardinal.{u} :=
 #align cardinal.aleph_0 Cardinal.aleph0
 -/
 
--- mathport name: cardinal.aleph_0
 scoped notation "ℵ₀" => Cardinal.aleph0
 
 #print Cardinal.mk_nat /-
@@ -1373,52 +1503,74 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
 /-! ### Properties about the cast from `ℕ` -/
 
 
+#print Cardinal.mk_fin /-
 @[simp]
 theorem mk_fin (n : ℕ) : (#Fin n) = n := by simp
 #align cardinal.mk_fin Cardinal.mk_fin
+-/
 
+#print Cardinal.lift_natCast /-
 @[simp]
 theorem lift_natCast (n : ℕ) : lift.{u} (n : Cardinal.{v}) = n := by induction n <;> simp [*]
 #align cardinal.lift_nat_cast Cardinal.lift_natCast
+-/
 
+#print Cardinal.lift_eq_nat_iff /-
 @[simp]
 theorem lift_eq_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a = n ↔ a = n :=
   lift_injective.eq_iff' (lift_natCast n)
 #align cardinal.lift_eq_nat_iff Cardinal.lift_eq_nat_iff
+-/
 
+#print Cardinal.nat_eq_lift_iff /-
 @[simp]
 theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a := by rw [← lift_natCast.{v} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
+-/
 
+#print Cardinal.lift_le_nat_iff /-
 @[simp]
 theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
   simp only [← lift_nat_cast, lift_le]
 #align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
+-/
 
+#print Cardinal.nat_le_lift_iff /-
 @[simp]
 theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) ≤ lift.{v} a ↔ (n : Cardinal) ≤ a := by simp only [← lift_nat_cast, lift_le]
 #align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
+-/
 
+#print Cardinal.lift_lt_nat_iff /-
 @[simp]
 theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
   simp only [← lift_nat_cast, lift_lt]
 #align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
+-/
 
+#print Cardinal.nat_lt_lift_iff /-
 @[simp]
 theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) < lift.{v} a ↔ (n : Cardinal) < a := by simp only [← lift_nat_cast, lift_lt]
 #align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
+-/
 
+#print Cardinal.lift_mk_fin /-
 theorem lift_mk_fin (n : ℕ) : lift (#Fin n) = n := by simp
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
+-/
 
+#print Cardinal.mk_coe_finset /-
 theorem mk_coe_finset {α : Type u} {s : Finset α} : (#s) = ↑(Finset.card s) := by simp
 #align cardinal.mk_coe_finset Cardinal.mk_coe_finset
+-/
 
+#print Cardinal.mk_finset_of_fintype /-
 theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
+-/
 
 #print Cardinal.mk_finsupp_lift_of_fintype /-
 @[simp]
@@ -1434,45 +1586,62 @@ theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
 #align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype
 -/
 
+#print Cardinal.card_le_of_finset /-
 theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ (#α) :=
   @mk_coe_finset _ s ▸ mk_set_le _
 #align cardinal.card_le_of_finset Cardinal.card_le_of_finset
+-/
 
+#print Cardinal.natCast_pow /-
 @[simp, norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(pow m n) : Cardinal) = (m^n) := by
   induction n <;> simp [pow_succ', power_add, *]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
+-/
 
+#print Cardinal.natCast_le /-
 @[simp, norm_cast]
 theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
   rw [← lift_mk_fin, ← lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le,
     Fintype.card_fin, Fintype.card_fin]
 #align cardinal.nat_cast_le Cardinal.natCast_le
+-/
 
+#print Cardinal.natCast_lt /-
 @[simp, norm_cast]
 theorem natCast_lt {m n : ℕ} : (m : Cardinal) < n ↔ m < n := by simp [lt_iff_le_not_le, ← not_le]
 #align cardinal.nat_cast_lt Cardinal.natCast_lt
+-/
 
 instance : CharZero Cardinal :=
   ⟨StrictMono.injective fun m n => natCast_lt.2⟩
 
+#print Cardinal.natCast_inj /-
 theorem natCast_inj {m n : ℕ} : (m : Cardinal) = n ↔ m = n :=
   Nat.cast_inj
 #align cardinal.nat_cast_inj Cardinal.natCast_inj
+-/
 
+#print Cardinal.natCast_injective /-
 theorem natCast_injective : Injective (coe : ℕ → Cardinal) :=
   Nat.cast_injective
 #align cardinal.nat_cast_injective Cardinal.natCast_injective
+-/
 
+#print Cardinal.nat_succ /-
 @[simp, norm_cast]
 theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ n :=
   (add_one_le_succ _).antisymm (succ_le_of_lt <| natCast_lt.2 <| Nat.lt_succ_self _)
 #align cardinal.nat_succ Cardinal.nat_succ
+-/
 
+#print Cardinal.succ_zero /-
 @[simp]
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
+-/
 
+#print Cardinal.card_le_of /-
 theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n :=
   by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
@@ -1482,6 +1651,7 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin]
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
+-/
 
 #print Cardinal.cantor' /-
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
@@ -1501,12 +1671,14 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le
 #align cardinal.one_le_iff_ne_zero Cardinal.one_le_iff_ne_zero
 -/
 
+#print Cardinal.nat_lt_aleph0 /-
 theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
   succ_le_iff.1
     (by
       rw [← nat_succ, ← lift_mk_fin, aleph_0, lift_mk_le.{0, 0, u}]
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
+-/
 
 #print Cardinal.one_lt_aleph0 /-
 @[simp]
@@ -1520,6 +1692,7 @@ theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
 #align cardinal.one_le_aleph_0 Cardinal.one_le_aleph0
 -/
 
+#print Cardinal.lt_aleph0 /-
 theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n :=
   ⟨fun h => by
     rcases lt_lift_iff.1 h with ⟨c, rfl, h'⟩
@@ -1531,13 +1704,16 @@ theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n :=
     haveI := infinite.to_subtype h'
     exact ⟨Infinite.natEmbedding S⟩, fun ⟨n, e⟩ => e.symm ▸ nat_lt_aleph0 _⟩
 #align cardinal.lt_aleph_0 Cardinal.lt_aleph0
+-/
 
+#print Cardinal.aleph0_le /-
 theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
   ⟨fun h n => (nat_lt_aleph0 _).le.trans h, fun h =>
     le_of_not_lt fun hn => by
       rcases lt_aleph_0.1 hn with ⟨n, rfl⟩
       exact (Nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
+-/
 
 #print Cardinal.isSuccLimit_aleph0 /-
 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
@@ -1566,14 +1742,18 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
 #align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
 -/
 
+#print Cardinal.range_natCast /-
 @[simp]
 theorem range_natCast : range (coe : ℕ → Cardinal) = Iio ℵ₀ :=
   ext fun x => by simp only [mem_Iio, mem_range, eq_comm, lt_aleph_0]
 #align cardinal.range_nat_cast Cardinal.range_natCast
+-/
 
+#print Cardinal.mk_eq_nat_iff /-
 theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ Fin n) := by
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
+-/
 
 #print Cardinal.lt_aleph0_iff_finite /-
 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
@@ -1641,25 +1821,33 @@ theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
 #align cardinal.le_aleph_0_iff_subtype_countable Cardinal.le_aleph0_iff_subtype_countable
 -/
 
+#print Cardinal.canLiftCardinalNat /-
 instance canLiftCardinalNat : CanLift Cardinal ℕ coe fun x => x < ℵ₀ :=
   ⟨fun x hx =>
     let ⟨n, hn⟩ := lt_aleph0.mp hx
     ⟨n, hn.symm⟩⟩
 #align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNat
+-/
 
+#print Cardinal.add_lt_aleph0 /-
 theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a + b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← Nat.cast_add] <;> apply nat_lt_aleph_0
 #align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0
+-/
 
+#print Cardinal.add_lt_aleph0_iff /-
 theorem add_lt_aleph0_iff {a b : Cardinal} : a + b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ :=
   ⟨fun h => ⟨(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h⟩, fun ⟨h1, h2⟩ =>
     add_lt_aleph0 h1 h2⟩
 #align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iff
+-/
 
+#print Cardinal.aleph0_le_add_iff /-
 theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b := by
   simp only [← not_lt, add_lt_aleph_0_iff, not_and_or]
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
+-/
 
 #print Cardinal.nsmul_lt_aleph0_iff /-
 /-- See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
@@ -1681,11 +1869,14 @@ theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) :
 #align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
 -/
 
+#print Cardinal.mul_lt_aleph0 /-
 theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← Nat.cast_mul] <;> apply nat_lt_aleph_0
 #align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0
+-/
 
+#print Cardinal.mul_lt_aleph0_iff /-
 theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ :=
   by
   refine' ⟨fun h => _, _⟩
@@ -1699,14 +1890,18 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
   rintro (rfl | rfl | ⟨ha, hb⟩) <;>
     simp only [*, mul_lt_aleph_0, aleph_0_pos, MulZeroClass.zero_mul, MulZeroClass.mul_zero]
 #align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff
+-/
 
+#print Cardinal.aleph0_le_mul_iff /-
 /-- See also `cardinal.aleph_0_le_mul_iff`. -/
 theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) :=
   by
   let h := (@mul_lt_aleph0_iff a b).Not
   rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h 
 #align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff
+-/
 
+#print Cardinal.aleph0_le_mul_iff' /-
 /-- See also `cardinal.aleph_0_le_mul_iff'`. -/
 theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 :=
   by
@@ -1714,10 +1909,13 @@ theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 
   simp only [aleph_0_le_mul_iff, and_or_left, and_iff_right_of_imp this, @and_left_comm (a ≠ 0)]
   simp only [and_comm, or_comm]
 #align cardinal.aleph_0_le_mul_iff' Cardinal.aleph0_le_mul_iff'
+-/
 
+#print Cardinal.mul_lt_aleph0_iff_of_ne_zero /-
 theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠ 0) :
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
+-/
 
 #print Cardinal.power_lt_aleph0 /-
 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
@@ -1769,40 +1967,54 @@ theorem mk_denumerable (α : Type u) [Denumerable α] : (#α) = ℵ₀ :=
 #align cardinal.mk_denumerable Cardinal.mk_denumerable
 -/
 
+#print Cardinal.aleph0_add_aleph0 /-
 @[simp]
 theorem aleph0_add_aleph0 : ℵ₀ + ℵ₀ = ℵ₀ :=
   mk_denumerable _
 #align cardinal.aleph_0_add_aleph_0 Cardinal.aleph0_add_aleph0
+-/
 
+#print Cardinal.aleph0_mul_aleph0 /-
 theorem aleph0_mul_aleph0 : ℵ₀ * ℵ₀ = ℵ₀ :=
   mk_denumerable _
 #align cardinal.aleph_0_mul_aleph_0 Cardinal.aleph0_mul_aleph0
+-/
 
+#print Cardinal.nat_mul_aleph0 /-
 @[simp]
 theorem nat_mul_aleph0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ :=
   le_antisymm (lift_mk_fin n ▸ mk_le_aleph0) <|
     le_mul_of_one_le_left (zero_le _) <| by
       rwa [← Nat.cast_one, nat_cast_le, Nat.one_le_iff_ne_zero]
 #align cardinal.nat_mul_aleph_0 Cardinal.nat_mul_aleph0
+-/
 
+#print Cardinal.aleph0_mul_nat /-
 @[simp]
 theorem aleph0_mul_nat {n : ℕ} (hn : n ≠ 0) : ℵ₀ * n = ℵ₀ := by rw [mul_comm, nat_mul_aleph_0 hn]
 #align cardinal.aleph_0_mul_nat Cardinal.aleph0_mul_nat
+-/
 
+#print Cardinal.add_le_aleph0 /-
 @[simp]
 theorem add_le_aleph0 {c₁ c₂ : Cardinal} : c₁ + c₂ ≤ ℵ₀ ↔ c₁ ≤ ℵ₀ ∧ c₂ ≤ ℵ₀ :=
   ⟨fun h => ⟨le_self_add.trans h, le_add_self.trans h⟩, fun h =>
     aleph0_add_aleph0 ▸ add_le_add h.1 h.2⟩
 #align cardinal.add_le_aleph_0 Cardinal.add_le_aleph0
+-/
 
+#print Cardinal.aleph0_add_nat /-
 @[simp]
 theorem aleph0_add_nat (n : ℕ) : ℵ₀ + n = ℵ₀ :=
   (add_le_aleph0.2 ⟨le_rfl, (nat_lt_aleph0 n).le⟩).antisymm le_self_add
 #align cardinal.aleph_0_add_nat Cardinal.aleph0_add_nat
+-/
 
+#print Cardinal.nat_add_aleph0 /-
 @[simp]
 theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ := by rw [add_comm, aleph_0_add_nat]
 #align cardinal.nat_add_aleph_0 Cardinal.nat_add_aleph0
+-/
 
 #print Cardinal.toNat /-
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
@@ -1815,10 +2027,12 @@ def toNat : ZeroHom Cardinal ℕ :=
 #align cardinal.to_nat Cardinal.toNat
 -/
 
+#print Cardinal.toNat_apply_of_lt_aleph0 /-
 theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :
     c.toNat = Classical.choose (lt_aleph0.1 h) :=
   dif_pos h
 #align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0
+-/
 
 #print Cardinal.toNat_apply_of_aleph0_le /-
 theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0 :=
@@ -1826,13 +2040,17 @@ theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0
 #align cardinal.to_nat_apply_of_aleph_0_le Cardinal.toNat_apply_of_aleph0_le
 -/
 
+#print Cardinal.cast_toNat_of_lt_aleph0 /-
 theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑c.toNat = c := by
   rw [to_nat_apply_of_lt_aleph_0 h, ← Classical.choose_spec (lt_aleph_0.1 h)]
 #align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0
+-/
 
+#print Cardinal.cast_toNat_of_aleph0_le /-
 theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat = (0 : Cardinal) := by
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
+-/
 
 #print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
@@ -1862,17 +2080,21 @@ theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
 -/
 
+#print Cardinal.toNat_cast /-
 @[simp]
 theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
   by
   rw [to_nat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), ← nat_cast_inj]
   exact (Classical.choose_spec (lt_aleph_0.1 (nat_lt_aleph_0 n))).symm
 #align cardinal.to_nat_cast Cardinal.toNat_cast
+-/
 
+#print Cardinal.toNat_rightInverse /-
 /-- `to_nat` has a right-inverse: coercion. -/
 theorem toNat_rightInverse : Function.RightInverse (coe : ℕ → Cardinal) toNat :=
   toNat_cast
 #align cardinal.to_nat_right_inverse Cardinal.toNat_rightInverse
+-/
 
 #print Cardinal.toNat_surjective /-
 theorem toNat_surjective : Surjective toNat :=
@@ -1880,10 +2102,12 @@ theorem toNat_surjective : Surjective toNat :=
 #align cardinal.to_nat_surjective Cardinal.toNat_surjective
 -/
 
+#print Cardinal.exists_nat_eq_of_le_nat /-
 theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m :=
   let he := cast_toNat_of_lt_aleph0 (h.trans_lt <| nat_lt_aleph0 n)
   ⟨c.toNat, natCast_le.1 (he.trans_le h), he.symm⟩
 #align cardinal.exists_nat_eq_of_le_nat Cardinal.exists_nat_eq_of_le_nat
+-/
 
 #print Cardinal.mk_toNat_of_infinite /-
 @[simp]
@@ -1916,6 +2140,7 @@ theorem one_toNat : toNat 1 = 1 := by rw [← to_nat_cast 1, Nat.cast_one]
 #align cardinal.one_to_nat Cardinal.one_toNat
 -/
 
+#print Cardinal.toNat_eq_iff /-
 theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c = n :=
   ⟨fun h =>
     (cast_toNat_of_lt_aleph0
@@ -1923,6 +2148,7 @@ theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c
       (congr_arg coe h),
     fun h => (congr_arg toNat h).trans (toNat_cast n)⟩
 #align cardinal.to_nat_eq_iff Cardinal.toNat_eq_iff
+-/
 
 #print Cardinal.toNat_eq_one /-
 @[simp]
@@ -1956,6 +2182,7 @@ theorem toNat_congr {β : Type v} (e : α ≃ β) : (#α).toNat = (#β).toNat :=
 #align cardinal.to_nat_congr Cardinal.toNat_congr
 -/
 
+#print Cardinal.toNat_mul /-
 @[simp]
 theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
   by
@@ -1972,6 +2199,7 @@ theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
   · rw [to_nat_apply_of_aleph_0_le hx2, MulZeroClass.zero_mul, to_nat_apply_of_aleph_0_le]
     exact aleph_0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)
 #align cardinal.to_nat_mul Cardinal.toNat_mul
+-/
 
 #print Cardinal.toNatHom /-
 /-- `cardinal.to_nat` as a `monoid_with_zero_hom`. -/
@@ -1984,11 +2212,14 @@ def toNatHom : Cardinal →*₀ ℕ where
 #align cardinal.to_nat_hom Cardinal.toNatHom
 -/
 
+#print Cardinal.toNat_finset_prod /-
 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
   map_prod toNatHom _ _
 #align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prod
+-/
 
+#print Cardinal.toNat_add_of_lt_aleph0 /-
 @[simp]
 theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
     (lift.{v, u} a + lift.{u, v} b).toNat = a.toNat + b.toNat :=
@@ -1999,7 +2230,9 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
   rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_to_nat_of_lt_aleph_0 ha,
     cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0 (add_lt_aleph_0 ha hb)]
 #align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0
+-/
 
+#print Cardinal.toPartENat /-
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to `⊤`. -/
 def toPartENat : Cardinal →+ PartENat
@@ -2020,36 +2253,51 @@ def toPartENat : Cardinal →+ PartENat
       contrapose! hx
       apply le_self_add.trans_lt hx
 #align cardinal.to_part_enat Cardinal.toPartENat
+-/
 
+#print Cardinal.toPartENat_apply_of_lt_aleph0 /-
 theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPartENat = c.toNat :=
   if_pos h
 #align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0
+-/
 
+#print Cardinal.toPartENat_apply_of_aleph0_le /-
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPartENat = ⊤ :=
   if_neg h.not_lt
 #align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_le
+-/
 
+#print Cardinal.toPartENat_cast /-
 @[simp]
 theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
   rw [to_part_enat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), to_nat_cast]
 #align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
+-/
 
+#print Cardinal.mk_toPartENat_of_infinite /-
 @[simp]
 theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
   toPartENat_apply_of_aleph0_le (infinite_iff.1 h)
 #align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infinite
+-/
 
+#print Cardinal.aleph0_toPartENat /-
 @[simp]
 theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
   toPartENat_apply_of_aleph0_le le_rfl
 #align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
+-/
 
+#print Cardinal.toPartENat_surjective /-
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
+-/
 
+#print Cardinal.mk_toPartENat_eq_coe_card /-
 theorem mk_toPartENat_eq_coe_card [Fintype α] : (#α).toPartENat = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card
+-/
 
 #print Cardinal.mk_int /-
 theorem mk_int : (#ℤ) = ℵ₀ :=
@@ -2063,6 +2311,7 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 #align cardinal.mk_pnat Cardinal.mk_pNat
 -/
 
+#print Cardinal.sum_lt_prod /-
 /-- **König's theorem** -/
 theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
   lt_of_not_ge fun ⟨F⟩ =>
@@ -2085,6 +2334,7 @@ theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f
       let ⟨⟨i, a⟩, h⟩ := sG C
       hc i a (congr_fun h _)
 #align cardinal.sum_lt_prod Cardinal.sum_lt_prod
+-/
 
 #print Cardinal.mk_empty /-
 @[simp]
@@ -2253,24 +2503,32 @@ theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i)
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 -/
 
+#print Cardinal.mk_iUnion_eq_sum_mk /-
 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
   calc
     (#⋃ i, f i) = (#Σ i, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
+-/
 
+#print Cardinal.mk_iUnion_le /-
 theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
   mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
 #align cardinal.mk_Union_le Cardinal.mk_iUnion_le
+-/
 
+#print Cardinal.mk_sUnion_le /-
 theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
   rw [sUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
+-/
 
+#print Cardinal.mk_biUnion_le /-
 theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
     (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by rw [bUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
+-/
 
 #print Cardinal.finset_card_lt_aleph0 /-
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
@@ -2278,6 +2536,7 @@ theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
 -/
 
+#print Cardinal.mk_set_eq_nat_iff_finset /-
 theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
     (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n :=
   by
@@ -2288,11 +2547,15 @@ theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
   · rintro ⟨t, rfl, rfl⟩
     exact mk_coe_finset
 #align cardinal.mk_set_eq_nat_iff_finset Cardinal.mk_set_eq_nat_iff_finset
+-/
 
+#print Cardinal.mk_eq_nat_iff_finset /-
 theorem mk_eq_nat_iff_finset {n : ℕ} : (#α) = n ↔ ∃ t : Finset α, (t : Set α) = univ ∧ t.card = n :=
   by rw [← mk_univ, mk_set_eq_nat_iff_finset]
 #align cardinal.mk_eq_nat_iff_finset Cardinal.mk_eq_nat_iff_finset
+-/
 
+#print Cardinal.mk_eq_nat_iff_fintype /-
 theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fintype.card α h = n :=
   by
   rw [mk_eq_nat_iff_finset]
@@ -2302,31 +2565,42 @@ theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fin
   · rintro ⟨⟨t, ht⟩, hn⟩
     exact ⟨t, eq_univ_iff_forall.2 ht, hn⟩
 #align cardinal.mk_eq_nat_iff_fintype Cardinal.mk_eq_nat_iff_fintype
+-/
 
+#print Cardinal.mk_union_add_mk_inter /-
 theorem mk_union_add_mk_inter {α : Type u} {S T : Set α} :
     (#(S ∪ T : Set α)) + (#(S ∩ T : Set α)) = (#S) + (#T) :=
   Quot.sound ⟨Equiv.Set.unionSumInter S T⟩
 #align cardinal.mk_union_add_mk_inter Cardinal.mk_union_add_mk_inter
+-/
 
+#print Cardinal.mk_union_le /-
 /-- The cardinality of a union is at most the sum of the cardinalities
 of the two sets. -/
 theorem mk_union_le {α : Type u} (S T : Set α) : (#(S ∪ T : Set α)) ≤ (#S) + (#T) :=
   @mk_union_add_mk_inter α S T ▸ self_le_add_right (#(S ∪ T : Set α)) (#(S ∩ T : Set α))
 #align cardinal.mk_union_le Cardinal.mk_union_le
+-/
 
+#print Cardinal.mk_union_of_disjoint /-
 theorem mk_union_of_disjoint {α : Type u} {S T : Set α} (H : Disjoint S T) :
     (#(S ∪ T : Set α)) = (#S) + (#T) :=
   Quot.sound ⟨Equiv.Set.union H.le_bot⟩
 #align cardinal.mk_union_of_disjoint Cardinal.mk_union_of_disjoint
+-/
 
+#print Cardinal.mk_insert /-
 theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
     (#(insert a s : Set α)) = (#s) + 1 := by
   rw [← union_singleton, mk_union_of_disjoint, mk_singleton]; simpa
 #align cardinal.mk_insert Cardinal.mk_insert
+-/
 
+#print Cardinal.mk_sum_compl /-
 theorem mk_sum_compl {α} (s : Set α) : (#s) + (#(sᶜ : Set α)) = (#α) :=
   mk_congr (Equiv.Set.sumCompl s)
 #align cardinal.mk_sum_compl Cardinal.mk_sum_compl
+-/
 
 #print Cardinal.mk_le_mk_of_subset /-
 theorem mk_le_mk_of_subset {α} {s t : Set α} (h : s ⊆ t) : (#s) ≤ (#t) :=
@@ -2340,17 +2614,23 @@ theorem mk_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) : (#{ x //
 #align cardinal.mk_subtype_mono Cardinal.mk_subtype_mono
 -/
 
+#print Cardinal.le_mk_diff_add_mk /-
 theorem le_mk_diff_add_mk (S T : Set α) : (#S) ≤ (#(S \ T : Set α)) + (#T) :=
   (mk_le_mk_of_subset <| subset_diff_union _ _).trans <| mk_union_le _ _
 #align cardinal.le_mk_diff_add_mk Cardinal.le_mk_diff_add_mk
+-/
 
+#print Cardinal.mk_diff_add_mk /-
 theorem mk_diff_add_mk {S T : Set α} (h : T ⊆ S) : (#(S \ T : Set α)) + (#T) = (#S) :=
   (mk_union_of_disjoint <| disjoint_sdiff_self_left).symm.trans <| by rw [diff_union_of_subset h]
 #align cardinal.mk_diff_add_mk Cardinal.mk_diff_add_mk
+-/
 
+#print Cardinal.mk_union_le_aleph0 /-
 theorem mk_union_le_aleph0 {α} {P Q : Set α} : (#(P ∪ Q : Set α)) ≤ ℵ₀ ↔ (#P) ≤ ℵ₀ ∧ (#Q) ≤ ℵ₀ := by
   simp
 #align cardinal.mk_union_le_aleph_0 Cardinal.mk_union_le_aleph0
+-/
 
 #print Cardinal.mk_image_eq_lift /-
 theorem mk_image_eq_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α) (h : Injective f) :
@@ -2462,14 +2742,19 @@ theorem le_mk_iff_exists_subset {c : Cardinal} {α : Type u} {s : Set α} :
 #align cardinal.le_mk_iff_exists_subset Cardinal.le_mk_iff_exists_subset
 -/
 
+#print Cardinal.two_le_iff /-
 theorem two_le_iff : (2 : Cardinal) ≤ (#α) ↔ ∃ x y : α, x ≠ y := by
   rw [← Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]
 #align cardinal.two_le_iff Cardinal.two_le_iff
+-/
 
+#print Cardinal.two_le_iff' /-
 theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ (#α) ↔ ∃ y : α, y ≠ x := by
   rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
 #align cardinal.two_le_iff' Cardinal.two_le_iff'
+-/
 
+#print Cardinal.mk_eq_two_iff /-
 theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ :=
   by
   simp only [← @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]
@@ -2479,7 +2764,9 @@ theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α
   · rintro ⟨x, y, hne, h⟩
     exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩
 #align cardinal.mk_eq_two_iff Cardinal.mk_eq_two_iff
+-/
 
+#print Cardinal.mk_eq_two_iff' /-
 theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
   by
   rw [mk_eq_two_iff]; constructor
@@ -2490,7 +2777,9 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
   · rintro ⟨y, hne, hy⟩
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'
+-/
 
+#print Cardinal.exists_not_mem_of_length_lt /-
 theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length < (#α)) :
     ∃ z : α, z ∉ l := by
   contrapose! h
@@ -2500,7 +2789,9 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
     _ = l.to_finset.card := Cardinal.mk_coe_finset
     _ ≤ l.length := cardinal.nat_cast_le.mpr (List.toFinset_card_le l)
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
+-/
 
+#print Cardinal.three_le /-
 theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y :=
   by
   have : ↑(3 : ℕ) ≤ (#α); simpa using h
@@ -2508,6 +2799,7 @@ theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α,
   have := exists_not_mem_of_length_lt [x, y] this
   simpa [not_or] using this
 #align cardinal.three_le Cardinal.three_le
+-/
 
 #print Cardinal.powerlt /-
 /-- The function `a ^< b`, defined as the supremum of `a ^ c` for `c < b`. -/
@@ -2516,7 +2808,6 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 #align cardinal.powerlt Cardinal.powerlt
 -/
 
--- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
 #print Cardinal.le_powerlt /-
@@ -2549,18 +2840,24 @@ theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun b c => powerlt_l
 #align cardinal.powerlt_mono_left Cardinal.powerlt_mono_left
 -/
 
+#print Cardinal.powerlt_succ /-
 theorem powerlt_succ {a b : Cardinal} (h : a ≠ 0) : a ^< succ b = (a^b) :=
   (powerlt_le.2 fun c h' => power_le_power_left h <| le_of_lt_succ h').antisymm <|
     le_powerlt a (lt_succ b)
 #align cardinal.powerlt_succ Cardinal.powerlt_succ
+-/
 
+#print Cardinal.powerlt_min /-
 theorem powerlt_min {a b c : Cardinal} : a ^< min b c = min (a ^< b) (a ^< c) :=
   (powerlt_mono_left a).map_min
 #align cardinal.powerlt_min Cardinal.powerlt_min
+-/
 
+#print Cardinal.powerlt_max /-
 theorem powerlt_max {a b c : Cardinal} : a ^< max b c = max (a ^< b) (a ^< c) :=
   (powerlt_mono_left a).map_max
 #align cardinal.powerlt_max Cardinal.powerlt_max
+-/
 
 #print Cardinal.zero_powerlt /-
 theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 :=

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -892,7 +892,6 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   calc
     (#γ) + 1 = (#Option γ) := mk_option.symm
     _ ≤ (#β) := (f.option_elim b hb).cardinal_le
-    
 #align cardinal.add_one_le_succ Cardinal.add_one_le_succ
 
 #print Cardinal.IsLimit /-
@@ -953,7 +952,6 @@ theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
       calc
         (Σ i : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
         _ ≃ ULift ι × ULift α := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)
-        
 #align cardinal.sum_const Cardinal.sum_const
 
 theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = (#ι) * a := by simp
@@ -1734,7 +1732,6 @@ theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ None
     (#α) = 1 ↔ (#α) ≤ 1 ∧ 1 ≤ (#α) := le_antisymm_iff
     _ ↔ Subsingleton α ∧ Nonempty α :=
       le_one_iff_subsingleton.And (one_le_iff_ne_zero.trans mk_ne_zero_iff)
-    
 #align cardinal.eq_one_iff_unique Cardinal.eq_one_iff_unique
 -/
 
@@ -2149,7 +2146,6 @@ theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α)
   calc
     (#List α) = (#Σ n, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
     _ = sum fun n : ℕ => (#α) ^ℕ n := by simp
-    
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
 -/
 
@@ -2254,7 +2250,6 @@ theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i)
   calc
     (#⋃ i, f i) ≤ (#Σ i, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
     _ = sum fun i => #f i := mk_sigma _
-    
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 -/
 
@@ -2263,7 +2258,6 @@ theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
   calc
     (#⋃ i, f i) = (#Σ i, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
-    
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
 theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
@@ -2505,7 +2499,6 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
     _ ≤ (#l.to_finset) := (mk_le_mk_of_subset fun x _ => list.mem_to_finset.mpr (h x))
     _ = l.to_finset.card := Cardinal.mk_coe_finset
     _ ≤ l.length := cardinal.nat_cast_le.mpr (List.toFinset_card_le l)
-    
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
 theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y :=

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -866,10 +866,10 @@ theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
 
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
-  SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' }) fun a b =>
+  SuccOrder.ofSuccLeIff (fun c => sInf {c' | c < c'}) fun a b =>
     ⟨lt_of_lt_of_le <| csInf_mem <| exists_gt a, csInf_le'⟩
 
-theorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=
+theorem succ_def (c : Cardinal) : succ c = sInf {c' | c < c'} :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
 
@@ -1607,7 +1607,7 @@ alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 
 #print Cardinal.lt_aleph0_iff_subtype_finite /-
 @[simp]
-theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
+theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ {x | p x}.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
 -/
@@ -1638,7 +1638,7 @@ alias le_aleph_0_iff_set_countable ↔ _ _root_.set.countable.le_aleph_0
 #print Cardinal.le_aleph0_iff_subtype_countable /-
 @[simp]
 theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
-    (#{ x // p x }) ≤ ℵ₀ ↔ { x | p x }.Countable :=
+    (#{ x // p x }) ≤ ℵ₀ ↔ {x | p x}.Countable :=
   le_aleph0_iff_set_countable
 #align cardinal.le_aleph_0_iff_subtype_countable Cardinal.le_aleph0_iff_subtype_countable
 -/
@@ -2387,7 +2387,7 @@ theorem mk_subtype_of_equiv {α β : Type u} (p : β → Prop) (e : α ≃ β) :
 -/
 
 #print Cardinal.mk_sep /-
-theorem mk_sep (s : Set α) (t : α → Prop) : (#({ x ∈ s | t x } : Set α)) = (#{ x : s | t x.1 }) :=
+theorem mk_sep (s : Set α) (t : α → Prop) : (#({x ∈ s | t x} : Set α)) = (#{x : s | t x.1}) :=
   mk_congr (Equiv.Set.sep s t)
 #align cardinal.mk_sep Cardinal.mk_sep
 -/
@@ -2443,7 +2443,7 @@ theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h
 
 #print Cardinal.mk_subset_ge_of_subset_image_lift /-
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
-    {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) :=
+    {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({x ∈ s | f x ∈ t} : Set α)) :=
   by
   rw [image_eq_range] at h ; convert mk_preimage_of_subset_range_lift _ _ h using 1
   rw [mk_sep]; rfl
@@ -2452,7 +2452,7 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 
 #print Cardinal.mk_subset_ge_of_subset_image /-
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
-    (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) :=
+    (#t) ≤ (#({x ∈ s | f x ∈ t} : Set α)) :=
   by
   rw [image_eq_range] at h ; convert mk_preimage_of_subset_range _ _ h using 1
   rw [mk_sep]; rfl

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -528,7 +528,7 @@ theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   by
   refine' Fintype.induction_empty_option _ _ _ α
   · intro α β h e hα; letI := Fintype.ofEquiv β e.symm
-    rwa [mk_congr e, Fintype.card_congr e] at hα
+    rwa [mk_congr e, Fintype.card_congr e] at hα 
   · rfl
   · intro α h hα; simp [hα]; rfl
 #align cardinal.mk_fintype Cardinal.mk_fintype
@@ -887,7 +887,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
   cases' le_of_lt hlt with f
   have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
-  simp only [surjective, not_forall] at this
+  simp only [surjective, not_forall] at this 
   rcases this with ⟨b, hb⟩
   calc
     (#γ) + 1 = (#Option γ) := mk_option.symm
@@ -931,7 +931,7 @@ theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
 /-- The indexed sum of cardinals is the cardinality of the
   indexed disjoint union, i.e. sigma type. -/
 def sum {ι} (f : ι → Cardinal) : Cardinal :=
-  mk (Σi, (f i).out)
+  mk (Σ i, (f i).out)
 #align cardinal.sum Cardinal.sum
 -/
 
@@ -941,7 +941,7 @@ theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
 #align cardinal.le_sum Cardinal.le_sum
 
 @[simp]
-theorem mk_sigma {ι} (f : ι → Type _) : (#Σi, f i) = sum fun i => #f i :=
+theorem mk_sigma {ι} (f : ι → Type _) : (#Σ i, f i) = sum fun i => #f i :=
   mk_congr <| Equiv.sigmaCongrRight fun i => outMkEquiv.symm
 #align cardinal.mk_sigma Cardinal.mk_sigma
 
@@ -951,7 +951,7 @@ theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
   inductionOn a fun α =>
     mk_congr <|
       calc
-        (Σi : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
+        (Σ i : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
         _ ≃ ULift ι × ULift α := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)
         
 #align cardinal.sum_const Cardinal.sum_const
@@ -984,7 +984,8 @@ theorem lift_sum {ι : Type u} (f : ι → Cardinal.{v}) :
 
 theorem sum_le_sum {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : sum f ≤ sum g :=
   ⟨(Embedding.refl _).sigma_map fun i =>
-      Classical.choice <| by have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this⟩
+      Classical.choice <| by
+        have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this ⟩
 #align cardinal.sum_le_sum Cardinal.sum_le_sum
 
 theorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : α → β) (hf : ∀ b : β, (#f ⁻¹' {b}) ≤ c) :
@@ -1048,7 +1049,7 @@ theorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s
 
 #print Cardinal.bddAbove_image /-
 theorem bddAbove_image (f : Cardinal.{u} → Cardinal.{max u v}) {s : Set Cardinal.{u}}
-    (hs : BddAbove s) : BddAbove (f '' s) := by rw [bdd_above_iff_small] at hs⊢; exact small_lift _
+    (hs : BddAbove s) : BddAbove (f '' s) := by rw [bdd_above_iff_small] at hs ⊢; exact small_lift _
 #align cardinal.bdd_above_image Cardinal.bddAbove_image
 -/
 
@@ -1142,7 +1143,7 @@ theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) =
 
 theorem prod_le_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : prod f ≤ prod g :=
   ⟨Embedding.piCongrRight fun i =>
-      Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this⟩
+      Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this ⟩
 #align cardinal.prod_le_prod Cardinal.prod_le_prod
 
 @[simp]
@@ -1233,7 +1234,7 @@ theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
   le_antisymm
     (le_of_not_gt fun h => by
       rcases lt_lift_iff.1 h with ⟨b, e, h⟩
-      rw [lt_succ_iff, ← lift_le, e] at h
+      rw [lt_succ_iff, ← lift_le, e] at h 
       exact h.not_lt (lt_succ _))
     (succ_le_of_lt <| lift_lt.2 <| lt_succ a)
 #align cardinal.lift_succ Cardinal.lift_succ
@@ -1262,8 +1263,8 @@ theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSu
   apply ((le_csSup_iff' (bdd_above_image _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
   · by_contra h
     obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
-    simp_rw [lift_le] at h hc
-    rw [csSup_le_iff' hs] at h
+    simp_rw [lift_le] at h hc 
+    rw [csSup_le_iff' hs] at h 
     exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
   · rintro i ⟨j, hj, rfl⟩
     exact lift_le.2 (le_csSup hs hj)
@@ -1477,7 +1478,7 @@ theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n :=
   by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
-  rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn
+  rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn 
   cases' hn with f
   refine' (H <| finset.univ.map f).not_lt _
   rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin]
@@ -1487,7 +1488,7 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
 #print Cardinal.cantor' /-
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
-  rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
+  rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb 
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
 -/
@@ -1562,7 +1563,7 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
   by_contra' h'
   rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
   · exact h.ne_zero.irrefl
-  · rw [nat_succ] at h
+  · rw [nat_succ] at h 
     exact not_is_succ_limit_succ _ h.is_succ_limit
 #align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
 -/
@@ -1692,7 +1693,7 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
   refine' ⟨fun h => _, _⟩
   · by_cases ha : a = 0; · exact Or.inl ha
     right; by_cases hb : b = 0; · exact Or.inl hb
-    right; rw [← Ne, ← one_le_iff_ne_zero] at ha hb; constructor
+    right; rw [← Ne, ← one_le_iff_ne_zero] at ha hb ; constructor
     · rw [← mul_one a]
       refine' (mul_le_mul' le_rfl hb).trans_lt h
     · rw [← one_mul b]
@@ -1705,7 +1706,7 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
 theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) :=
   by
   let h := (@mul_lt_aleph0_iff a b).Not
-  rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h
+  rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h 
 #align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff
 
 /-- See also `cardinal.aleph_0_le_mul_iff'`. -/
@@ -2146,7 +2147,7 @@ theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
 #print Cardinal.mk_list_eq_sum_pow /-
 theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α) ^ℕ n :=
   calc
-    (#List α) = (#Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
+    (#List α) = (#Σ n, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
     _ = sum fun n : ℕ => (#α) ^ℕ n := by simp
     
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
@@ -2183,7 +2184,7 @@ theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅
   by
   constructor
   · intro h
-    rw [mk_eq_zero_iff] at h
+    rw [mk_eq_zero_iff] at h 
     exact eq_empty_iff_forall_not_mem.2 fun x hx => h.elim' ⟨x, hx⟩
   · rintro rfl; exact mk_emptyc _
 #align cardinal.mk_emptyc_iff Cardinal.mk_emptyCollection_iff
@@ -2251,7 +2252,7 @@ theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injectiv
 #print Cardinal.mk_iUnion_le_sum_mk /-
 theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
   calc
-    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
+    (#⋃ i, f i) ≤ (#Σ i, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
     _ = sum fun i => #f i := mk_sigma _
     
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
@@ -2260,7 +2261,7 @@ theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i)
 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
   calc
-    (#⋃ i, f i) = (#Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
+    (#⋃ i, f i) = (#Σ i, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
     
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
@@ -2444,7 +2445,7 @@ theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
     {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) :=
   by
-  rw [image_eq_range] at h; convert mk_preimage_of_subset_range_lift _ _ h using 1
+  rw [image_eq_range] at h ; convert mk_preimage_of_subset_range_lift _ _ h using 1
   rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image_lift Cardinal.mk_subset_ge_of_subset_image_lift
 -/
@@ -2453,7 +2454,7 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
     (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) :=
   by
-  rw [image_eq_range] at h; convert mk_preimage_of_subset_range _ _ h using 1
+  rw [image_eq_range] at h ; convert mk_preimage_of_subset_range _ _ h using 1
   rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image Cardinal.mk_subset_ge_of_subset_image
 -/
@@ -2489,9 +2490,9 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
   by
   rw [mk_eq_two_iff]; constructor
   · rintro ⟨a, b, hne, h⟩
-    simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
+    simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h 
     rcases h x with (rfl | rfl)
-    exacts[⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
+    exacts [⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
   · rintro ⟨y, hne, hy⟩
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -88,7 +88,7 @@ Cantor's theorem, König's theorem, Konig's theorem
 
 open Function Set Order
 
-open BigOperators Classical
+open scoped BigOperators Classical
 
 noncomputable section
 
@@ -409,10 +409,12 @@ theorem lift_inj {a b : Cardinal} : lift a = lift b ↔ a = b :=
 #align cardinal.lift_inj Cardinal.lift_inj
 -/
 
+#print Cardinal.lift_lt /-
 @[simp]
 theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
   liftOrderEmbedding.lt_iff_lt
 #align cardinal.lift_lt Cardinal.lift_lt
+-/
 
 theorem lift_strictMono : StrictMono lift := fun a b => lift_lt.2
 #align cardinal.lift_strict_mono Cardinal.lift_strictMono
@@ -804,9 +806,11 @@ instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := by infer_instance
 
+#print Cardinal.one_lt_iff_nontrivial /-
 theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
+-/
 
 theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 :=
   by
@@ -821,12 +825,15 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 -/
 
+#print Cardinal.power_pos /-
 theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
+-/
 
 end OrderProperties
 
+#print Cardinal.lt_wf /-
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -839,12 +846,15 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [f, mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
+-/
 
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
+#print Cardinal.wo /-
 instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 #align cardinal.wo Cardinal.wo
+-/
 
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
@@ -901,17 +911,21 @@ protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
 #align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
 -/
 
+#print Cardinal.IsLimit.isSuccLimit /-
 protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
   h.2
 #align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
+-/
 
 theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
   h.IsSuccLimit.succ_lt
 #align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
 
+#print Cardinal.isSuccLimit_zero /-
 theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
   isSuccLimit_bot
 #align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
+-/
 
 #print Cardinal.sum /-
 /-- The indexed sum of cardinals is the cardinality of the
@@ -1204,6 +1218,7 @@ theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.le_lift_iff Cardinal.le_lift_iff
 -/
 
+#print Cardinal.lt_lift_iff /-
 theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
   ⟨fun h =>
@@ -1211,6 +1226,7 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
     fun ⟨a', e, h⟩ => e ▸ lift_lt.2 h⟩
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
+-/
 
 @[simp]
 theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
@@ -1314,9 +1330,11 @@ theorem aleph0_ne_zero : ℵ₀ ≠ 0 :=
 #align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero
 -/
 
+#print Cardinal.aleph0_pos /-
 theorem aleph0_pos : 0 < ℵ₀ :=
   pos_iff_ne_zero.2 aleph0_ne_zero
 #align cardinal.aleph_0_pos Cardinal.aleph0_pos
+-/
 
 #print Cardinal.lift_aleph0 /-
 @[simp]
@@ -1339,15 +1357,19 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 -/
 
+#print Cardinal.aleph0_lt_lift /-
 @[simp]
 theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
+-/
 
+#print Cardinal.lift_lt_aleph0 /-
 @[simp]
 theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
+-/
 
 /-! ### Properties about the cast from `ℕ` -/
 
@@ -1462,14 +1484,18 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
+#print Cardinal.cantor' /-
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
+-/
 
+#print Cardinal.one_le_iff_pos /-
 theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by rw [← succ_zero, succ_le_iff]
 #align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos
+-/
 
 #print Cardinal.one_le_iff_ne_zero /-
 theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le_iff_pos, pos_iff_ne_zero]
@@ -1483,9 +1509,11 @@ theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
+#print Cardinal.one_lt_aleph0 /-
 @[simp]
 theorem one_lt_aleph0 : 1 < ℵ₀ := by simpa using nat_lt_aleph_0 1
 #align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0
+-/
 
 #print Cardinal.one_le_aleph0 /-
 theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
@@ -1512,6 +1540,7 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       exact (Nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
+#print Cardinal.isSuccLimit_aleph0 /-
 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
   isSuccLimit_of_succ_lt fun a ha =>
     by
@@ -1519,6 +1548,7 @@ theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
     rw [← nat_succ]
     apply nat_lt_aleph_0
 #align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
+-/
 
 #print Cardinal.isLimit_aleph0 /-
 theorem isLimit_aleph0 : IsLimit ℵ₀ :=
@@ -1546,30 +1576,40 @@ theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ F
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
+#print Cardinal.lt_aleph0_iff_finite /-
 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
+-/
 
+#print Cardinal.lt_aleph0_iff_fintype /-
 theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
+-/
 
+#print Cardinal.lt_aleph0_of_finite /-
 theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
+-/
 
+#print Cardinal.lt_aleph0_iff_set_finite /-
 @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
+-/
 
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
+#print Cardinal.lt_aleph0_iff_subtype_finite /-
 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
+-/
 
 #print Cardinal.mk_le_aleph0_iff /-
 theorem mk_le_aleph0_iff : (#α) ≤ ℵ₀ ↔ Countable α := by
@@ -1622,6 +1662,7 @@ theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a 
   simp only [← not_lt, add_lt_aleph_0_iff, not_and_or]
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
 
+#print Cardinal.nsmul_lt_aleph0_iff /-
 /-- See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
 theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ :=
   by
@@ -1632,11 +1673,14 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   · simp
   rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self_iff]
 #align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff
+-/
 
+#print Cardinal.nsmul_lt_aleph0_iff_of_ne_zero /-
 /-- See also `cardinal.nsmul_lt_aleph_0_iff` for a hypothesis-free version. -/
 theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ :=
   nsmul_lt_aleph0_iff.trans <| or_iff_right h
 #align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
+-/
 
 theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
@@ -1676,10 +1720,12 @@ theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
+#print Cardinal.power_lt_aleph0 /-
 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← nat_cast_pow] <;> apply nat_lt_aleph_0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
+-/
 
 #print Cardinal.eq_one_iff_unique /-
 theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
@@ -1790,25 +1836,33 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat =
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
+#print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
   rw [← nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
+-/
 
+#print Cardinal.toNat_lt_iff_lt_of_lt_aleph0 /-
 theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat < d.toNat ↔ c < d := by
   rw [← nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
+-/
 
+#print Cardinal.toNat_le_of_le_of_lt_aleph0 /-
 theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
     c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
 #align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
+-/
 
+#print Cardinal.toNat_lt_of_lt_of_lt_aleph0 /-
 theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
     c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
+-/
 
 @[simp]
 theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
@@ -2223,9 +2277,11 @@ theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
     (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by rw [bUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
+#print Cardinal.finset_card_lt_aleph0 /-
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
+-/
 
 theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
     (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n :=
@@ -2469,13 +2525,16 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 -- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
+#print Cardinal.le_powerlt /-
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
   apply @le_ciSup _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
+-/
 
+#print Cardinal.powerlt_le /-
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
   rw [powerlt, ciSup_le_iff']
@@ -2483,6 +2542,7 @@ theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) 
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.powerlt_le Cardinal.powerlt_le
+-/
 
 #print Cardinal.powerlt_le_powerlt_left /-
 theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a ^< c :=

bump to nightly-2023-05-28-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

ba5d8b97

Diff

@@ -141,24 +141,12 @@ theorem inductionOn {p : Cardinal → Prop} (c : Cardinal) (h : ∀ α, p (#α))
 #align cardinal.induction_on Cardinal.inductionOn
 -/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.induction_on₂ Cardinal.inductionOn₂ₓ'. -/
 @[elab_as_elim]
 theorem inductionOn₂ {p : Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
     (h : ∀ α β, p (#α) (#β)) : p c₁ c₂ :=
   Quotient.induction_on₂ c₁ c₂ h
 #align cardinal.induction_on₂ Cardinal.inductionOn₂
 
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 @[elab_as_elim]
 theorem inductionOn₃ {p : Cardinal → Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
     (c₃ : Cardinal) (h : ∀ α β γ, p (#α) (#β) (#γ)) : p c₁ c₂ c₃ :=
@@ -421,32 +409,14 @@ theorem lift_inj {a b : Cardinal} : lift a = lift b ↔ a = b :=
 #align cardinal.lift_inj Cardinal.lift_inj
 -/
 
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 @[simp]
 theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
   liftOrderEmbedding.lt_iff_lt
 #align cardinal.lift_lt Cardinal.lift_lt
 
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 theorem lift_strictMono : StrictMono lift := fun a b => lift_lt.2
 #align cardinal.lift_strict_mono Cardinal.lift_strictMono
 
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 theorem lift_monotone : Monotone lift :=
   lift_strictMono.Monotone
 #align cardinal.lift_monotone Cardinal.lift_monotone
@@ -463,12 +433,6 @@ theorem mk_eq_zero (α : Type u) [IsEmpty α] : (#α) = 0 :=
 #align cardinal.mk_eq_zero Cardinal.mk_eq_zero
 -/
 
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 @[simp]
 theorem lift_zero : lift 0 = 0 :=
   mk_congr (Equiv.equivPEmpty _)
@@ -535,12 +499,6 @@ alias mk_le_one_iff_set_subsingleton ↔ _ _root_.set.subsingleton.cardinal_mk_l
 instance : Add Cardinal.{u} :=
   ⟨map₂ Sum fun α β γ δ => Equiv.sumCongr⟩
 
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 theorem add_def (α β : Type u) : (#α) + (#β) = (#Sum α β) :=
   rfl
 #align cardinal.add_def Cardinal.add_def
@@ -548,45 +506,21 @@ theorem add_def (α β : Type u) : (#α) + (#β) = (#Sum α β) :=
 instance : NatCast Cardinal.{u} :=
   ⟨Nat.unaryCast⟩
 
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 @[simp]
 theorem mk_sum (α : Type u) (β : Type v) : (#Sum α β) = lift.{v, u} (#α) + lift.{u, v} (#β) :=
   mk_congr (Equiv.ulift.symm.sumCongr Equiv.ulift.symm)
 #align cardinal.mk_sum Cardinal.mk_sum
 
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 @[simp]
 theorem mk_option {α : Type u} : (#Option α) = (#α) + 1 :=
   (Equiv.optionEquivSumPUnit α).cardinal_eq
 #align cardinal.mk_option Cardinal.mk_option
 
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 @[simp]
 theorem mk_psum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
   (mk_congr (Equiv.psumEquivSum α β)).trans (mk_sum α β)
 #align cardinal.mk_psum Cardinal.mk_psum
 
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 @[simp]
 theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   by
@@ -600,22 +534,10 @@ theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
 instance : Mul Cardinal.{u} :=
   ⟨map₂ Prod fun α β γ δ => Equiv.prodCongr⟩
 
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 theorem mul_def (α β : Type u) : (#α) * (#β) = (#α × β) :=
   rfl
 #align cardinal.mul_def Cardinal.mul_def
 
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 @[simp]
 theorem mk_prod (α : Type u) (β : Type v) : (#α × β) = lift.{v, u} (#α) * lift.{u, v} (#β) :=
   mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)
@@ -668,12 +590,6 @@ theorem power_one {a : Cardinal} : (a^1) = a :=
 #align cardinal.power_one Cardinal.power_one
 -/
 
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 theorem power_add {a b c : Cardinal} : (a^b + c) = (a^b) * (a^c) :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumArrowEquivProdArrow β γ α
 #align cardinal.power_add Cardinal.power_add
@@ -699,22 +615,10 @@ instance : CommSemiring Cardinal.{u} where
   npow_zero := @power_zero
   npow_succ n c := show (c^n + 1) = c * (c^n) by rw [power_add, power_one, mul_comm']
 
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 theorem power_bit0 (a b : Cardinal) : (a^bit0 b) = (a^b) * (a^b) :=
   power_add
 #align cardinal.power_bit0 Cardinal.power_bit0
 
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 theorem power_bit1 (a b : Cardinal) : (a^bit1 b) = (a^b) * (a^b) * a := by
   rw [bit1, ← power_bit0, power_add, power_one]
 #align cardinal.power_bit1 Cardinal.power_bit1
@@ -726,22 +630,10 @@ theorem one_power {a : Cardinal} : (1^a) = 1 :=
 #align cardinal.one_power Cardinal.one_power
 -/
 
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 @[simp]
 theorem mk_bool : (#Bool) = 2 := by simp
 #align cardinal.mk_bool Cardinal.mk_bool
 
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 @[simp]
 theorem mk_Prop : (#Prop) = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
@@ -765,130 +657,58 @@ theorem power_ne_zero {a : Cardinal} (b) : a ≠ 0 → (a^b) ≠ 0 :=
 #align cardinal.power_ne_zero Cardinal.power_ne_zero
 -/
 
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 theorem mul_power {a b c : Cardinal} : (a * b^c) = (a^c) * (b^c) :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.arrowProdEquivProdArrow α β γ
 #align cardinal.mul_power Cardinal.mul_power
 
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 theorem power_mul {a b c : Cardinal} : (a^b * c) = ((a^b)^c) := by rw [mul_comm b c];
   exact induction_on₃ a b c fun α β γ => mk_congr <| Equiv.curry γ β α
 #align cardinal.power_mul Cardinal.power_mul
 
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 @[simp]
 theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : (a^(↑n : Cardinal.{u})) = a ^ℕ n :=
   rfl
 #align cardinal.pow_cast_right Cardinal.pow_cast_right
 
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 @[simp]
 theorem lift_one : lift 1 = 1 :=
   mk_congr <| Equiv.ulift.trans Equiv.punitEquivPUnit
 #align cardinal.lift_one Cardinal.lift_one
 
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 @[simp]
 theorem lift_add (a b) : lift (a + b) = lift a + lift b :=
   inductionOn₂ a b fun α β =>
     mk_congr <| Equiv.ulift.trans (Equiv.sumCongr Equiv.ulift Equiv.ulift).symm
 #align cardinal.lift_add Cardinal.lift_add
 
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 @[simp]
 theorem lift_mul (a b) : lift (a * b) = lift a * lift b :=
   inductionOn₂ a b fun α β =>
     mk_congr <| Equiv.ulift.trans (Equiv.prodCongr Equiv.ulift Equiv.ulift).symm
 #align cardinal.lift_mul Cardinal.lift_mul
 
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 @[simp]
 theorem lift_bit0 (a : Cardinal) : lift (bit0 a) = bit0 (lift a) :=
   lift_add a a
 #align cardinal.lift_bit0 Cardinal.lift_bit0
 
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 @[simp]
 theorem lift_bit1 (a : Cardinal) : lift (bit1 a) = bit1 (lift a) := by simp [bit1]
 #align cardinal.lift_bit1 Cardinal.lift_bit1
 
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 theorem lift_two : lift.{u, v} 2 = 2 := by simp
 #align cardinal.lift_two Cardinal.lift_two
 
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 @[simp]
 theorem mk_set {α : Type u} : (#Set α) = (2^#α) := by simp [Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
 
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 /-- A variant of `cardinal.mk_set` expressed in terms of a `set` instead of a `Type`. -/
 @[simp]
 theorem mk_powerset {α : Type u} (s : Set α) : (#↥(𝒫 s)) = (2^#↥s) :=
   (mk_congr (Equiv.Set.powerset s)).trans mk_set
 #align cardinal.mk_powerset Cardinal.mk_powerset
 
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 theorem lift_two_power (a) : lift (2^a) = (2^lift a) := by simp
 #align cardinal.lift_two_power Cardinal.lift_two_power
 
@@ -904,22 +724,10 @@ protected theorem zero_le : ∀ a : Cardinal, 0 ≤ a := by rintro ⟨α⟩ <;>
 private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d → a + c ≤ b + d := by
   rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.sum_map e₂⟩
 
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 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
 #align cardinal.add_covariant_class Cardinal.add_covariantClass
 
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 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
 #align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClass
@@ -981,12 +789,6 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 #align cardinal.self_le_power Cardinal.self_le_power
 -/
 
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 /-- **Cantor's theorem** -/
 theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
   by
@@ -1002,22 +804,10 @@ instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := by infer_instance
 
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 theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
 
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 theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 :=
   by
   by_cases ha : a = 0
@@ -1031,24 +821,12 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 -/
 
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 theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
 
 end OrderProperties
 
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 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -1065,24 +843,12 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
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 instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 #align cardinal.wo Cardinal.wo
 
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
 
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 @[simp]
 theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg not_nonempty_empty
@@ -1093,42 +859,18 @@ instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' }) fun a b =>
     ⟨lt_of_lt_of_le <| csInf_mem <| exists_gt a, csInf_le'⟩
 
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 theorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
 
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 theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
   bot_lt_succ
 #align cardinal.succ_pos Cardinal.succ_pos
 
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 theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
   (succ_pos _).ne'
 #align cardinal.succ_ne_zero Cardinal.succ_ne_zero
 
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 theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   by
   refine' (le_csInf_iff'' (exists_gt c)).2 fun b hlt => _
@@ -1159,32 +901,14 @@ protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
 #align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
 -/
 
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 protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
   h.2
 #align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
 
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 theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
   h.IsSuccLimit.succ_lt
 #align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
 
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 theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
   isSuccLimit_bot
 #align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
@@ -1197,34 +921,16 @@ def sum {ι} (f : ι → Cardinal) : Cardinal :=
 #align cardinal.sum Cardinal.sum
 -/
 
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 theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
   rw [← Quotient.out_eq (f i)] <;>
     exact ⟨⟨fun a => ⟨i, a⟩, fun a b h => eq_of_hEq <| by injection h⟩⟩
 #align cardinal.le_sum Cardinal.le_sum
 
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 @[simp]
 theorem mk_sigma {ι} (f : ι → Type _) : (#Σi, f i) = sum fun i => #f i :=
   mk_congr <| Equiv.sigmaCongrRight fun i => outMkEquiv.symm
 #align cardinal.mk_sigma Cardinal.mk_sigma
 
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 @[simp]
 theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
     (sum fun i : ι => a) = lift.{v} (#ι) * lift.{u} a :=
@@ -1236,33 +942,15 @@ theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
         
 #align cardinal.sum_const Cardinal.sum_const
 
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 theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = (#ι) * a := by simp
 #align cardinal.sum_const' Cardinal.sum_const'
 
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 @[simp]
 theorem sum_add_distrib {ι} (f g : ι → Cardinal) : sum (f + g) = sum f + sum g := by
   simpa only [mk_sigma, mk_sum, mk_out, lift_id] using
     mk_congr (Equiv.sigmaSumDistrib (Quotient.out ∘ f) (Quotient.out ∘ g))
 #align cardinal.sum_add_distrib Cardinal.sum_add_distrib
 
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 @[simp]
 theorem sum_add_distrib' {ι} (f g : ι → Cardinal) :
     (Cardinal.sum fun i => f i + g i) = sum f + sum g :=
@@ -1280,35 +968,17 @@ theorem lift_sum {ι : Type u} (f : ι → Cardinal.{v}) :
 #align cardinal.lift_sum Cardinal.lift_sum
 -/
 
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 theorem sum_le_sum {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : sum f ≤ sum g :=
   ⟨(Embedding.refl _).sigma_map fun i =>
       Classical.choice <| by have := H i <;> rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this⟩
 #align cardinal.sum_le_sum Cardinal.sum_le_sum
 
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 theorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : α → β) (hf : ∀ b : β, (#f ⁻¹' {b}) ≤ c) :
     (#α) ≤ (#β) * c := by
   simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using
     sum_le_sum _ _ hf
 #align cardinal.mk_le_mk_mul_of_mk_preimage_le Cardinal.mk_le_mk_mul_of_mk_preimage_le
 
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 theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v} {c : Cardinal}
     (f : α → β) (hf : ∀ b : β, lift.{v} (#f ⁻¹' {b}) ≤ c) : lift.{v} (#α) ≤ lift.{u} (#β) * c :=
   (mk_le_mk_mul_of_mk_preimage_le fun x : ULift.{v} α => ULift.up.{u} (f x.1)) <|
@@ -1375,44 +1045,20 @@ theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbo
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
 -/
 
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 theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
   ciSup_le' <| le_sum _
 #align cardinal.supr_le_sum Cardinal.iSup_le_sum
 
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 theorem sum_le_iSup_lift {ι : Type u} (f : ι → Cardinal.{max u v}) : sum f ≤ (#ι).lift * iSup f :=
   by
   rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
   exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)
 #align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift
 
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 theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f := by
   rw [← lift_id (#ι)]; exact sum_le_supr_lift f
 #align cardinal.sum_le_supr Cardinal.sum_le_iSup
 
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 theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
     Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) :=
   by
@@ -1420,12 +1066,6 @@ theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
   simp only [mk_sum, mk_out, lift_id, mk_sigma]
 #align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ
 
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 /-- A variant of `csupr_of_empty` but with `0` on the RHS for convenience -/
 @[simp]
 protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
@@ -1486,34 +1126,16 @@ theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) =
 #align cardinal.prod_const' Cardinal.prod_const'
 -/
 
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 theorem prod_le_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : prod f ≤ prod g :=
   ⟨Embedding.piCongrRight fun i =>
       Classical.choice <| by have := H i <;> rwa [← mk_out (f i), ← mk_out (g i)] at this⟩
 #align cardinal.prod_le_prod Cardinal.prod_le_prod
 
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 @[simp]
 theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 := by
   lift f to ι → Type u using fun _ => trivial; simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi]
 #align cardinal.prod_eq_zero Cardinal.prod_eq_zero
 
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 theorem prod_ne_zero {ι} (f : ι → Cardinal) : prod f ≠ 0 ↔ ∀ i, f i ≠ 0 := by simp [prod_eq_zero]
 #align cardinal.prod_ne_zero Cardinal.prod_ne_zero
 
@@ -1546,12 +1168,6 @@ theorem prod_eq_of_fintype {α : Type u} [Fintype α] (f : α → Cardinal.{v})
 #align cardinal.prod_eq_of_fintype Cardinal.prod_eq_of_fintype
 -/
 
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 @[simp]
 theorem lift_sInf (s : Set Cardinal) : lift (sInf s) = sInf (lift '' s) :=
   by
@@ -1560,12 +1176,6 @@ theorem lift_sInf (s : Set Cardinal) : lift (sInf s) = sInf (lift '' s) :=
   · exact lift_monotone.map_Inf hs
 #align cardinal.lift_Inf Cardinal.lift_sInf
 
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 @[simp]
 theorem lift_iInf {ι} (f : ι → Cardinal) : lift (iInf f) = ⨅ i, lift (f i) := by unfold iInf;
   convert lift_Inf (range f); rw [range_comp]
@@ -1594,12 +1204,6 @@ theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.le_lift_iff Cardinal.le_lift_iff
 -/
 
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 theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
   ⟨fun h =>
@@ -1608,12 +1212,6 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     fun ⟨a', e, h⟩ => e ▸ lift_lt.2 h⟩
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
 
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 @[simp]
 theorem lift_succ (a) : lift (succ a) = succ (lift a) :=
   le_antisymm
@@ -1632,34 +1230,16 @@ theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
 #align cardinal.lift_umax_eq Cardinal.lift_umax_eq
 -/
 
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 @[simp]
 theorem lift_min {a b : Cardinal} : lift (min a b) = min (lift a) (lift b) :=
   lift_monotone.map_min
 #align cardinal.lift_min Cardinal.lift_min
 
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 @[simp]
 theorem lift_max {a b : Cardinal} : lift (max a b) = max (lift a) (lift b) :=
   lift_monotone.map_max
 #align cardinal.lift_max Cardinal.lift_max
 
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 /-- The lift of a supremum is the supremum of the lifts. -/
 theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSup (lift.{u} '' s) :=
   by
@@ -1673,35 +1253,17 @@ theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSu
     exact lift_le.2 (le_csSup hs hj)
 #align cardinal.lift_Sup Cardinal.lift_sSup
 
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 /-- The lift of a supremum is the supremum of the lifts. -/
 theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
     lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by rw [iSup, iSup, lift_Sup hf, ← range_comp]
 #align cardinal.lift_supr Cardinal.lift_iSup
 
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 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
 theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
     (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by rw [lift_supr hf]; exact ciSup_le' w
 #align cardinal.lift_supr_le Cardinal.lift_iSup_le
 
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 @[simp]
 theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
     {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by rw [lift_supr hf];
@@ -1710,12 +1272,6 @@ theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove
 
 universe v' w'
 
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 /-- To prove an inequality between the lifts to a common universe of two different supremums,
 it suffices to show that the lift of each cardinal from the smaller supremum
 if bounded by the lift of some cardinal from the larger supremum.
@@ -1728,12 +1284,6 @@ theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardina
   exact ciSup_mono' (bdd_above_range_comp hf' _) fun i => ⟨_, h i⟩
 #align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup
 
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 /-- A variant of `lift_supr_le_lift_supr` with universes specialized via `w = v` and `w' = v'`.
 This is sometimes necessary to avoid universe unification issues. -/
 theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
@@ -1764,12 +1314,6 @@ theorem aleph0_ne_zero : ℵ₀ ≠ 0 :=
 #align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero
 -/
 
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 theorem aleph0_pos : 0 < ℵ₀ :=
   pos_iff_ne_zero.2 aleph0_ne_zero
 #align cardinal.aleph_0_pos Cardinal.aleph0_pos
@@ -1795,23 +1339,11 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 -/
 
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 @[simp]
 theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
 
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 @[simp]
 theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
   rw [← lift_aleph_0, lift_lt]
@@ -1820,116 +1352,50 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
 /-! ### Properties about the cast from `ℕ` -/
 
 
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 @[simp]
 theorem mk_fin (n : ℕ) : (#Fin n) = n := by simp
 #align cardinal.mk_fin Cardinal.mk_fin
 
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 @[simp]
 theorem lift_natCast (n : ℕ) : lift.{u} (n : Cardinal.{v}) = n := by induction n <;> simp [*]
 #align cardinal.lift_nat_cast Cardinal.lift_natCast
 
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 @[simp]
 theorem lift_eq_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a = n ↔ a = n :=
   lift_injective.eq_iff' (lift_natCast n)
 #align cardinal.lift_eq_nat_iff Cardinal.lift_eq_nat_iff
 
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 @[simp]
 theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a := by rw [← lift_natCast.{v} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
 
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 @[simp]
 theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
   simp only [← lift_nat_cast, lift_le]
 #align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
 
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 @[simp]
 theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) ≤ lift.{v} a ↔ (n : Cardinal) ≤ a := by simp only [← lift_nat_cast, lift_le]
 #align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
 
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 @[simp]
 theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
   simp only [← lift_nat_cast, lift_lt]
 #align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
 
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 @[simp]
 theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) < lift.{v} a ↔ (n : Cardinal) < a := by simp only [← lift_nat_cast, lift_lt]
 #align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
 
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 theorem lift_mk_fin (n : ℕ) : lift (#Fin n) = n := by simp
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
 
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 theorem mk_coe_finset {α : Type u} {s : Finset α} : (#s) = ↑(Finset.card s) := by simp
 #align cardinal.mk_coe_finset Cardinal.mk_coe_finset
 
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 theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
 
@@ -1947,45 +1413,21 @@ theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
 #align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype
 -/
 
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 theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ (#α) :=
   @mk_coe_finset _ s ▸ mk_set_le _
 #align cardinal.card_le_of_finset Cardinal.card_le_of_finset
 
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 @[simp, norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(pow m n) : Cardinal) = (m^n) := by
   induction n <;> simp [pow_succ', power_add, *]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 
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 @[simp, norm_cast]
 theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
   rw [← lift_mk_fin, ← lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le,
     Fintype.card_fin, Fintype.card_fin]
 #align cardinal.nat_cast_le Cardinal.natCast_le
 
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 @[simp, norm_cast]
 theorem natCast_lt {m n : ℕ} : (m : Cardinal) < n ↔ m < n := by simp [lt_iff_le_not_le, ← not_le]
 #align cardinal.nat_cast_lt Cardinal.natCast_lt
@@ -1993,53 +1435,23 @@ theorem natCast_lt {m n : ℕ} : (m : Cardinal) < n ↔ m < n := by simp [lt_iff
 instance : CharZero Cardinal :=
   ⟨StrictMono.injective fun m n => natCast_lt.2⟩
 
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 theorem natCast_inj {m n : ℕ} : (m : Cardinal) = n ↔ m = n :=
   Nat.cast_inj
 #align cardinal.nat_cast_inj Cardinal.natCast_inj
 
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 theorem natCast_injective : Injective (coe : ℕ → Cardinal) :=
   Nat.cast_injective
 #align cardinal.nat_cast_injective Cardinal.natCast_injective
 
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 @[simp, norm_cast]
 theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ n :=
   (add_one_le_succ _).antisymm (succ_le_of_lt <| natCast_lt.2 <| Nat.lt_succ_self _)
 #align cardinal.nat_succ Cardinal.nat_succ
 
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 @[simp]
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
 
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 theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n :=
   by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
@@ -2050,24 +1462,12 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
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 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
 
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 theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by rw [← succ_zero, succ_le_iff]
 #align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos
 
@@ -2076,12 +1476,6 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le
 #align cardinal.one_le_iff_ne_zero Cardinal.one_le_iff_ne_zero
 -/
 
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 theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
   succ_le_iff.1
     (by
@@ -2089,12 +1483,6 @@ theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
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 @[simp]
 theorem one_lt_aleph0 : 1 < ℵ₀ := by simpa using nat_lt_aleph_0 1
 #align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0
@@ -2105,12 +1493,6 @@ theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
 #align cardinal.one_le_aleph_0 Cardinal.one_le_aleph0
 -/
 
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 theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n :=
   ⟨fun h => by
     rcases lt_lift_iff.1 h with ⟨c, rfl, h'⟩
@@ -2123,12 +1505,6 @@ theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n :=
     exact ⟨Infinite.natEmbedding S⟩, fun ⟨n, e⟩ => e.symm ▸ nat_lt_aleph0 _⟩
 #align cardinal.lt_aleph_0 Cardinal.lt_aleph0
 
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 theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
   ⟨fun h n => (nat_lt_aleph0 _).le.trans h, fun h =>
     le_of_not_lt fun hn => by
@@ -2136,12 +1512,6 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       exact (Nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
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 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
   isSuccLimit_of_succ_lt fun a ha =>
     by
@@ -2167,83 +1537,35 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
 #align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
 -/
 
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 @[simp]
 theorem range_natCast : range (coe : ℕ → Cardinal) = Iio ℵ₀ :=
   ext fun x => by simp only [mem_Iio, mem_range, eq_comm, lt_aleph_0]
 #align cardinal.range_nat_cast Cardinal.range_natCast
 
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 theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ Fin n) := by
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
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 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
 
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 theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
 
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 theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
 
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 @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
 
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 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
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 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
@@ -2280,56 +1602,26 @@ theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
 #align cardinal.le_aleph_0_iff_subtype_countable Cardinal.le_aleph0_iff_subtype_countable
 -/
 
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 instance canLiftCardinalNat : CanLift Cardinal ℕ coe fun x => x < ℵ₀ :=
   ⟨fun x hx =>
     let ⟨n, hn⟩ := lt_aleph0.mp hx
     ⟨n, hn.symm⟩⟩
 #align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNat
 
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 theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a + b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← Nat.cast_add] <;> apply nat_lt_aleph_0
 #align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0
 
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 theorem add_lt_aleph0_iff {a b : Cardinal} : a + b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ :=
   ⟨fun h => ⟨(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h⟩, fun ⟨h1, h2⟩ =>
     add_lt_aleph0 h1 h2⟩
 #align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iff
 
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 theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b := by
   simp only [← not_lt, add_lt_aleph_0_iff, not_and_or]
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
 
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 /-- See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
 theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ :=
   by
@@ -2341,34 +1633,16 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self_iff]
 #align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff
 
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 /-- See also `cardinal.nsmul_lt_aleph_0_iff` for a hypothesis-free version. -/
 theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ :=
   nsmul_lt_aleph0_iff.trans <| or_iff_right h
 #align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
 
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 theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← Nat.cast_mul] <;> apply nat_lt_aleph_0
 #align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0
 
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 theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ :=
   by
   refine' ⟨fun h => _, _⟩
@@ -2383,12 +1657,6 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
     simp only [*, mul_lt_aleph_0, aleph_0_pos, MulZeroClass.zero_mul, MulZeroClass.mul_zero]
 #align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff
 
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 /-- See also `cardinal.aleph_0_le_mul_iff`. -/
 theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) :=
   by
@@ -2396,12 +1664,6 @@ theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b
   rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h
 #align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff
 
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 /-- See also `cardinal.aleph_0_le_mul_iff'`. -/
 theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 :=
   by
@@ -2410,22 +1672,10 @@ theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 
   simp only [and_comm, or_comm]
 #align cardinal.aleph_0_le_mul_iff' Cardinal.aleph0_le_mul_iff'
 
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 theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠ 0) :
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
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 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← nat_cast_pow] <;> apply nat_lt_aleph_0
@@ -2475,33 +1725,15 @@ theorem mk_denumerable (α : Type u) [Denumerable α] : (#α) = ℵ₀ :=
 #align cardinal.mk_denumerable Cardinal.mk_denumerable
 -/
 
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 @[simp]
 theorem aleph0_add_aleph0 : ℵ₀ + ℵ₀ = ℵ₀ :=
   mk_denumerable _
 #align cardinal.aleph_0_add_aleph_0 Cardinal.aleph0_add_aleph0
 
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 theorem aleph0_mul_aleph0 : ℵ₀ * ℵ₀ = ℵ₀ :=
   mk_denumerable _
 #align cardinal.aleph_0_mul_aleph_0 Cardinal.aleph0_mul_aleph0
 
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 @[simp]
 theorem nat_mul_aleph0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ :=
   le_antisymm (lift_mk_fin n ▸ mk_le_aleph0) <|
@@ -2509,45 +1741,21 @@ theorem nat_mul_aleph0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ :=
       rwa [← Nat.cast_one, nat_cast_le, Nat.one_le_iff_ne_zero]
 #align cardinal.nat_mul_aleph_0 Cardinal.nat_mul_aleph0
 
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 @[simp]
 theorem aleph0_mul_nat {n : ℕ} (hn : n ≠ 0) : ℵ₀ * n = ℵ₀ := by rw [mul_comm, nat_mul_aleph_0 hn]
 #align cardinal.aleph_0_mul_nat Cardinal.aleph0_mul_nat
 
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 @[simp]
 theorem add_le_aleph0 {c₁ c₂ : Cardinal} : c₁ + c₂ ≤ ℵ₀ ↔ c₁ ≤ ℵ₀ ∧ c₂ ≤ ℵ₀ :=
   ⟨fun h => ⟨le_self_add.trans h, le_add_self.trans h⟩, fun h =>
     aleph0_add_aleph0 ▸ add_le_add h.1 h.2⟩
 #align cardinal.add_le_aleph_0 Cardinal.add_le_aleph0
 
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 @[simp]
 theorem aleph0_add_nat (n : ℕ) : ℵ₀ + n = ℵ₀ :=
   (add_le_aleph0.2 ⟨le_rfl, (nat_lt_aleph0 n).le⟩).antisymm le_self_add
 #align cardinal.aleph_0_add_nat Cardinal.aleph0_add_nat
 
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 @[simp]
 theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ := by rw [add_comm, aleph_0_add_nat]
 #align cardinal.nat_add_aleph_0 Cardinal.nat_add_aleph0
@@ -2563,12 +1771,6 @@ def toNat : ZeroHom Cardinal ℕ :=
 #align cardinal.to_nat Cardinal.toNat
 -/
 
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 theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :
     c.toNat = Classical.choose (lt_aleph0.1 h) :=
   dif_pos h
@@ -2580,76 +1782,34 @@ theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0
 #align cardinal.to_nat_apply_of_aleph_0_le Cardinal.toNat_apply_of_aleph0_le
 -/
 
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 theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑c.toNat = c := by
   rw [to_nat_apply_of_lt_aleph_0 h, ← Classical.choose_spec (lt_aleph_0.1 h)]
 #align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0
 
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 theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat = (0 : Cardinal) := by
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
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 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
   rw [← nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
 
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 theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat < d.toNat ↔ c < d := by
   rw [← nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
 
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 theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
     c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
 #align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
 
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 theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
     c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
 
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 @[simp]
 theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
   by
@@ -2657,12 +1817,6 @@ theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
   exact (Classical.choose_spec (lt_aleph_0.1 (nat_lt_aleph_0 n))).symm
 #align cardinal.to_nat_cast Cardinal.toNat_cast
 
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 /-- `to_nat` has a right-inverse: coercion. -/
 theorem toNat_rightInverse : Function.RightInverse (coe : ℕ → Cardinal) toNat :=
   toNat_cast
@@ -2674,12 +1828,6 @@ theorem toNat_surjective : Surjective toNat :=
 #align cardinal.to_nat_surjective Cardinal.toNat_surjective
 -/
 
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 theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m :=
   let he := cast_toNat_of_lt_aleph0 (h.trans_lt <| nat_lt_aleph0 n)
   ⟨c.toNat, natCast_le.1 (he.trans_le h), he.symm⟩
@@ -2716,12 +1864,6 @@ theorem one_toNat : toNat 1 = 1 := by rw [← to_nat_cast 1, Nat.cast_one]
 #align cardinal.one_to_nat Cardinal.one_toNat
 -/
 
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 theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c = n :=
   ⟨fun h =>
     (cast_toNat_of_lt_aleph0
@@ -2762,12 +1904,6 @@ theorem toNat_congr {β : Type v} (e : α ≃ β) : (#α).toNat = (#β).toNat :=
 #align cardinal.to_nat_congr Cardinal.toNat_congr
 -/
 
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 @[simp]
 theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
   by
@@ -2796,23 +1932,11 @@ def toNatHom : Cardinal →*₀ ℕ where
 #align cardinal.to_nat_hom Cardinal.toNatHom
 -/
 
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 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
   map_prod toNatHom _ _
 #align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prod
 
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 @[simp]
 theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
     (lift.{v, u} a + lift.{u, v} b).toNat = a.toNat + b.toNat :=
@@ -2824,12 +1948,6 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
     cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0 (add_lt_aleph_0 ha hb)]
 #align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0
 
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 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to `⊤`. -/
 def toPartENat : Cardinal →+ PartENat
@@ -2851,75 +1969,33 @@ def toPartENat : Cardinal →+ PartENat
       apply le_self_add.trans_lt hx
 #align cardinal.to_part_enat Cardinal.toPartENat
 
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 theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPartENat = c.toNat :=
   if_pos h
 #align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_leₓ'. -/
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPartENat = ⊤ :=
   if_neg h.not_lt
 #align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_le
 
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 @[simp]
 theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
   rw [to_part_enat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), to_nat_cast]
 #align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
 
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 @[simp]
 theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
   toPartENat_apply_of_aleph0_le (infinite_iff.1 h)
 #align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infinite
 
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 @[simp]
 theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
   toPartENat_apply_of_aleph0_le le_rfl
 #align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
 
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 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 
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 theorem mk_toPartENat_eq_coe_card [Fintype α] : (#α).toPartENat = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card
 
@@ -2935,12 +2011,6 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 #align cardinal.mk_pnat Cardinal.mk_pNat
 -/
 
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 /-- **König's theorem** -/
 theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
   lt_of_not_ge fun ⟨F⟩ =>
@@ -3133,12 +2203,6 @@ theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i)
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 -/
 
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 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
   calc
@@ -3147,52 +2211,22 @@ theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
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 theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
   mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
 #align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
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 theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
   rw [sUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
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 theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
     (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by rw [bUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
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 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
 
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 theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
     (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n :=
   by
@@ -3204,22 +2238,10 @@ theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
     exact mk_coe_finset
 #align cardinal.mk_set_eq_nat_iff_finset Cardinal.mk_set_eq_nat_iff_finset
 
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 theorem mk_eq_nat_iff_finset {n : ℕ} : (#α) = n ↔ ∃ t : Finset α, (t : Set α) = univ ∧ t.card = n :=
   by rw [← mk_univ, mk_set_eq_nat_iff_finset]
 #align cardinal.mk_eq_nat_iff_finset Cardinal.mk_eq_nat_iff_finset
 
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 theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fintype.card α h = n :=
   by
   rw [mk_eq_nat_iff_finset]
@@ -3230,57 +2252,27 @@ theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fin
     exact ⟨t, eq_univ_iff_forall.2 ht, hn⟩
 #align cardinal.mk_eq_nat_iff_fintype Cardinal.mk_eq_nat_iff_fintype
 
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 theorem mk_union_add_mk_inter {α : Type u} {S T : Set α} :
     (#(S ∪ T : Set α)) + (#(S ∩ T : Set α)) = (#S) + (#T) :=
   Quot.sound ⟨Equiv.Set.unionSumInter S T⟩
 #align cardinal.mk_union_add_mk_inter Cardinal.mk_union_add_mk_inter
 
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 /-- The cardinality of a union is at most the sum of the cardinalities
 of the two sets. -/
 theorem mk_union_le {α : Type u} (S T : Set α) : (#(S ∪ T : Set α)) ≤ (#S) + (#T) :=
   @mk_union_add_mk_inter α S T ▸ self_le_add_right (#(S ∪ T : Set α)) (#(S ∩ T : Set α))
 #align cardinal.mk_union_le Cardinal.mk_union_le
 
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 theorem mk_union_of_disjoint {α : Type u} {S T : Set α} (H : Disjoint S T) :
     (#(S ∪ T : Set α)) = (#S) + (#T) :=
   Quot.sound ⟨Equiv.Set.union H.le_bot⟩
 #align cardinal.mk_union_of_disjoint Cardinal.mk_union_of_disjoint
 
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 theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
     (#(insert a s : Set α)) = (#s) + 1 := by
   rw [← union_singleton, mk_union_of_disjoint, mk_singleton]; simpa
 #align cardinal.mk_insert Cardinal.mk_insert
 
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 theorem mk_sum_compl {α} (s : Set α) : (#s) + (#(sᶜ : Set α)) = (#α) :=
   mk_congr (Equiv.Set.sumCompl s)
 #align cardinal.mk_sum_compl Cardinal.mk_sum_compl
@@ -3297,32 +2289,14 @@ theorem mk_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) : (#{ x //
 #align cardinal.mk_subtype_mono Cardinal.mk_subtype_mono
 -/
 
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 theorem le_mk_diff_add_mk (S T : Set α) : (#S) ≤ (#(S \ T : Set α)) + (#T) :=
   (mk_le_mk_of_subset <| subset_diff_union _ _).trans <| mk_union_le _ _
 #align cardinal.le_mk_diff_add_mk Cardinal.le_mk_diff_add_mk
 
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 theorem mk_diff_add_mk {S T : Set α} (h : T ⊆ S) : (#(S \ T : Set α)) + (#T) = (#S) :=
   (mk_union_of_disjoint <| disjoint_sdiff_self_left).symm.trans <| by rw [diff_union_of_subset h]
 #align cardinal.mk_diff_add_mk Cardinal.mk_diff_add_mk
 
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 theorem mk_union_le_aleph0 {α} {P Q : Set α} : (#(P ∪ Q : Set α)) ≤ ℵ₀ ↔ (#P) ≤ ℵ₀ ∧ (#Q) ≤ ℵ₀ := by
   simp
 #align cardinal.mk_union_le_aleph_0 Cardinal.mk_union_le_aleph0
@@ -3437,32 +2411,14 @@ theorem le_mk_iff_exists_subset {c : Cardinal} {α : Type u} {s : Set α} :
 #align cardinal.le_mk_iff_exists_subset Cardinal.le_mk_iff_exists_subset
 -/
 
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 theorem two_le_iff : (2 : Cardinal) ≤ (#α) ↔ ∃ x y : α, x ≠ y := by
   rw [← Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]
 #align cardinal.two_le_iff Cardinal.two_le_iff
 
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 theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ (#α) ↔ ∃ y : α, y ≠ x := by
   rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
 #align cardinal.two_le_iff' Cardinal.two_le_iff'
 
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 theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ :=
   by
   simp only [← @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]
@@ -3473,12 +2429,6 @@ theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α
     exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩
 #align cardinal.mk_eq_two_iff Cardinal.mk_eq_two_iff
 
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 theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
   by
   rw [mk_eq_two_iff]; constructor
@@ -3490,12 +2440,6 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'
 
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 theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length < (#α)) :
     ∃ z : α, z ∉ l := by
   contrapose! h
@@ -3507,12 +2451,6 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
     
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
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 theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y :=
   by
   have : ↑(3 : ℕ) ≤ (#α); simpa using h
@@ -3531,12 +2469,6 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 -- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.le_powerlt Cardinal.le_powerltₓ'. -/
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
   apply @le_ciSup _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
@@ -3544,12 +2476,6 @@ theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
 
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 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
   rw [powerlt, ciSup_le_iff']
@@ -3569,33 +2495,15 @@ theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun b c => powerlt_l
 #align cardinal.powerlt_mono_left Cardinal.powerlt_mono_left
 -/
 
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 theorem powerlt_succ {a b : Cardinal} (h : a ≠ 0) : a ^< succ b = (a^b) :=
   (powerlt_le.2 fun c h' => power_le_power_left h <| le_of_lt_succ h').antisymm <|
     le_powerlt a (lt_succ b)
 #align cardinal.powerlt_succ Cardinal.powerlt_succ
 
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 theorem powerlt_min {a b c : Cardinal} : a ^< min b c = min (a ^< b) (a ^< c) :=
   (powerlt_mono_left a).map_min
 #align cardinal.powerlt_min Cardinal.powerlt_min
 
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 theorem powerlt_max {a b c : Cardinal} : a ^< max b c = max (a ^< b) (a ^< c) :=
   (powerlt_mono_left a).map_max
 #align cardinal.powerlt_max Cardinal.powerlt_max

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -299,16 +299,12 @@ instance : PartialOrder Cardinal.{u} where
   le := (· ≤ ·)
   le_refl := by rintro ⟨α⟩ <;> exact ⟨embedding.refl _⟩
   le_trans := by rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.trans e₂⟩
-  le_antisymm := by
-    rintro ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩
-    exact Quotient.sound (e₁.antisymm e₂)
+  le_antisymm := by rintro ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩; exact Quotient.sound (e₁.antisymm e₂)
 
 instance : LinearOrder Cardinal.{u} :=
   {
     Cardinal.partialOrder.{u} with
-    le_total := by
-      rintro ⟨α⟩ ⟨β⟩
-      apply embedding.total
+    le_total := by rintro ⟨α⟩ ⟨β⟩; apply embedding.total
     decidableLe := Classical.decRel _ }
 
 #print Cardinal.le_def /-
@@ -355,11 +351,8 @@ theorem mk_set_le (s : Set α) : (#s) ≤ (#α) :=
 -/
 
 #print Cardinal.out_embedding /-
-theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.out) :=
-  by
-  trans _
-  rw [← Quotient.out_eq c, ← Quotient.out_eq c']
-  rfl
+theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.out) := by trans _;
+  rw [← Quotient.out_eq c, ← Quotient.out_eq c']; rfl
 #align cardinal.out_embedding Cardinal.out_embedding
 -/
 
@@ -403,9 +396,7 @@ theorem lift_mk_eq' {α : Type u} {β : Type v} : lift.{v} (#α) = lift.{u} (#β
 #print Cardinal.lift_le /-
 @[simp]
 theorem lift_le {a b : Cardinal} : lift a ≤ lift b ↔ a ≤ b :=
-  inductionOn₂ a b fun α β => by
-    rw [← lift_umax]
-    exact lift_mk_le
+  inductionOn₂ a b fun α β => by rw [← lift_umax]; exact lift_mk_le
 #align cardinal.lift_le Cardinal.lift_le
 -/
 
@@ -600,13 +591,10 @@ Case conversion may be inaccurate. Consider using '#align cardinal.mk_fintype Ca
 theorem mk_fintype (α : Type u) [Fintype α] : (#α) = Fintype.card α :=
   by
   refine' Fintype.induction_empty_option _ _ _ α
-  · intro α β h e hα
-    letI := Fintype.ofEquiv β e.symm
+  · intro α β h e hα; letI := Fintype.ofEquiv β e.symm
     rwa [mk_congr e, Fintype.card_congr e] at hα
   · rfl
-  · intro α h hα
-    simp [hα]
-    rfl
+  · intro α h hα; simp [hα]; rfl
 #align cardinal.mk_fintype Cardinal.mk_fintype
 
 instance : Mul Cardinal.{u} :=
@@ -793,9 +781,7 @@ lean 3 declaration is
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) b c)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b) c)
 Case conversion may be inaccurate. Consider using '#align cardinal.power_mul Cardinal.power_mulₓ'. -/
-theorem power_mul {a b c : Cardinal} : (a^b * c) = ((a^b)^c) :=
-  by
-  rw [mul_comm b c]
+theorem power_mul {a b c : Cardinal} : (a^b * c) = ((a^b)^c) := by rw [mul_comm b c];
   exact induction_on₃ a b c fun α β γ => mk_congr <| Equiv.curry γ β α
 #align cardinal.power_mul Cardinal.power_mul
 
@@ -972,12 +958,8 @@ instance : CommMonoidWithZero Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
 
 #print Cardinal.zero_power_le /-
-theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 :=
-  by
-  by_cases h : c = 0
-  rw [h, power_zero]
-  rw [zero_power h]
-  apply zero_le
+theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 := by by_cases h : c = 0;
+  rw [h, power_zero]; rw [zero_power h]; apply zero_le
 #align cardinal.zero_power_le Cardinal.zero_power_le
 -/
 
@@ -995,8 +977,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
   by
   rcases eq_or_ne a 0 with (rfl | ha)
   · exact zero_le _
-  · convert power_le_power_left ha hb
-    exact power_one.symm
+  · convert power_le_power_left ha hb; exact power_one.symm
 #align cardinal.self_le_power Cardinal.self_le_power
 -/
 
@@ -1334,10 +1315,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
     ULift.forall.2 fun b =>
       (mk_congr <|
             (Equiv.ulift.image _).trans
-              (Equiv.trans
-                (by
-                  rw [Equiv.image_eq_preimage]
-                  simp [Set.preimage])
+              (Equiv.trans (by rw [Equiv.image_eq_preimage]; simp [Set.preimage])
                 Equiv.ulift.symm)).trans_le
         (hf b)
 #align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
@@ -1346,9 +1324,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
     inputs, is always bounded above. -/
 theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove (Set.range f) :=
-  ⟨_, by
-    rintro a ⟨i, rfl⟩
-    exact le_sum f i⟩
+  ⟨_, by rintro a ⟨i, rfl⟩; exact le_sum f i⟩
 #align cardinal.bdd_above_range Cardinal.bddAbove_range
 -/
 
@@ -1376,8 +1352,7 @@ theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :
     refine' ⟨_, fun hx => ⟨e ⟨x, hx⟩, _⟩⟩
     · rintro ⟨a, rfl⟩
       exact (e.symm a).Prop
-    · simp_rw [Subtype.val_eq_coe, Equiv.symm_apply_apply]
-      rfl⟩
+    · simp_rw [Subtype.val_eq_coe, Equiv.symm_apply_apply]; rfl⟩
 #align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_small
 -/
 
@@ -1389,18 +1364,13 @@ theorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s
 
 #print Cardinal.bddAbove_image /-
 theorem bddAbove_image (f : Cardinal.{u} → Cardinal.{max u v}) {s : Set Cardinal.{u}}
-    (hs : BddAbove s) : BddAbove (f '' s) :=
-  by
-  rw [bdd_above_iff_small] at hs⊢
-  exact small_lift _
+    (hs : BddAbove s) : BddAbove (f '' s) := by rw [bdd_above_iff_small] at hs⊢; exact small_lift _
 #align cardinal.bdd_above_image Cardinal.bddAbove_image
 -/
 
 #print Cardinal.bddAbove_range_comp /-
 theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbove (range f))
-    (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) :=
-  by
-  rw [range_comp]
+    (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) := by rw [range_comp];
   exact bdd_above_image g hf
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
 -/
@@ -1433,10 +1403,8 @@ lean 3 declaration is
 but is expected to have type
   forall {ι : Type.{u1}} (f : ι -> Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.sum.{u1, u1} ι f) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} ι) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f))
 Case conversion may be inaccurate. Consider using '#align cardinal.sum_le_supr Cardinal.sum_le_iSupₓ'. -/
-theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f :=
-  by
-  rw [← lift_id (#ι)]
-  exact sum_le_supr_lift f
+theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f := by
+  rw [← lift_id (#ι)]; exact sum_le_supr_lift f
 #align cardinal.sum_le_supr Cardinal.sum_le_iSup
 
 /- warning: cardinal.sum_nat_eq_add_sum_succ -> Cardinal.sum_nat_eq_add_sum_succ is a dubious translation:
@@ -1536,10 +1504,8 @@ but is expected to have type
   forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}), Iff (Eq.{max (succ (succ u2)) (succ (succ u1))} Cardinal.{max u1 u2} (Cardinal.prod.{u1, u2} ι f) (OfNat.ofNat.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} 0 (Zero.toOfNat0.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} Cardinal.instZeroCardinal.{max u2 u1}))) (Exists.{succ u1} ι (fun (i : ι) => Eq.{succ (succ u2)} Cardinal.{u2} (f i) (OfNat.ofNat.{succ u2} Cardinal.{u2} 0 (Zero.toOfNat0.{succ u2} Cardinal.{u2} Cardinal.instZeroCardinal.{u2}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.prod_eq_zero Cardinal.prod_eq_zeroₓ'. -/
 @[simp]
-theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 :=
-  by
-  lift f to ι → Type u using fun _ => trivial
-  simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi]
+theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 := by
+  lift f to ι → Type u using fun _ => trivial; simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi]
 #align cardinal.prod_eq_zero Cardinal.prod_eq_zero
 
 /- warning: cardinal.prod_ne_zero -> Cardinal.prod_ne_zero is a dubious translation:
@@ -1601,11 +1567,8 @@ but is expected to have type
   forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u3}), Eq.{max (succ (succ u2)) (succ (succ u3))} Cardinal.{max u3 u2} (Cardinal.lift.{u2, u3} (iInf.{succ u3, u1} Cardinal.{u3} (ConditionallyCompleteLattice.toInfSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (iInf.{max (succ u2) (succ u3), u1} Cardinal.{max u3 u2} (ConditionallyCompleteLattice.toInfSet.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u3}))) ι (fun (i : ι) => Cardinal.lift.{u2, u3} (f i)))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_infi Cardinal.lift_iInfₓ'. -/
 @[simp]
-theorem lift_iInf {ι} (f : ι → Cardinal) : lift (iInf f) = ⨅ i, lift (f i) :=
-  by
-  unfold iInf
-  convert lift_Inf (range f)
-  rw [range_comp]
+theorem lift_iInf {ι} (f : ι → Cardinal) : lift (iInf f) = ⨅ i, lift (f i) := by unfold iInf;
+  convert lift_Inf (range f); rw [range_comp]
 #align cardinal.lift_infi Cardinal.lift_iInf
 
 #print Cardinal.lift_down /-
@@ -1730,10 +1693,7 @@ Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le
 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
 theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
-    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t :=
-  by
-  rw [lift_supr hf]
-  exact ciSup_le' w
+    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by rw [lift_supr hf]; exact ciSup_le' w
 #align cardinal.lift_supr_le Cardinal.lift_iSup_le
 
 /- warning: cardinal.lift_supr_le_iff -> Cardinal.lift_iSup_le_iff is a dubious translation:
@@ -1744,9 +1704,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iffₓ'. -/
 @[simp]
 theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
-    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t :=
-  by
-  rw [lift_supr hf]
+    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by rw [lift_supr hf];
   exact ciSup_le_iff' (bdd_above_range_comp hf _)
 #align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff
 
@@ -2414,14 +2372,9 @@ Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_
 theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ :=
   by
   refine' ⟨fun h => _, _⟩
-  · by_cases ha : a = 0
-    · exact Or.inl ha
-    right
-    by_cases hb : b = 0
-    · exact Or.inl hb
-    right
-    rw [← Ne, ← one_le_iff_ne_zero] at ha hb
-    constructor
+  · by_cases ha : a = 0; · exact Or.inl ha
+    right; by_cases hb : b = 0; · exact Or.inl hb
+    right; rw [← Ne, ← one_le_iff_ne_zero] at ha hb; constructor
     · rw [← mul_one a]
       refine' (mul_le_mul' le_rfl hb).trans_lt h
     · rw [← one_mul b]
@@ -2510,10 +2463,8 @@ theorem mk_eq_aleph0 (α : Type _) [Countable α] [Infinite α] : (#α) = ℵ₀
 
 #print Cardinal.denumerable_iff /-
 theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ (#α) = ℵ₀ :=
-  ⟨fun ⟨h⟩ => mk_congr ((@Denumerable.eqv α h).trans Equiv.ulift.symm), fun h =>
-    by
-    cases' Quotient.exact h with f
-    exact ⟨Denumerable.mk' <| f.trans Equiv.ulift⟩⟩
+  ⟨fun ⟨h⟩ => mk_congr ((@Denumerable.eqv α h).trans Equiv.ulift.symm), fun h => by
+    cases' Quotient.exact h with f; exact ⟨Denumerable.mk' <| f.trans Equiv.ulift⟩⟩
 #align cardinal.denumerable_iff Cardinal.denumerable_iff
 -/
 
@@ -2826,8 +2777,7 @@ theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
   · rw [MulZeroClass.mul_zero, zero_to_nat, MulZeroClass.mul_zero]
   cases' lt_or_le x ℵ₀ with hx2 hx2
   · cases' lt_or_le y ℵ₀ with hy2 hy2
-    · lift x to ℕ using hx2
-      lift y to ℕ using hy2
+    · lift x to ℕ using hx2; lift y to ℕ using hy2
       rw [← Nat.cast_mul, to_nat_cast, to_nat_cast, to_nat_cast]
     · rw [to_nat_apply_of_aleph_0_le hy2, MulZeroClass.mul_zero, to_nat_apply_of_aleph_0_le]
       exact aleph_0_le_mul_iff'.2 (Or.inl ⟨hx1, hy2⟩)
@@ -3111,8 +3061,7 @@ theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅
   · intro h
     rw [mk_eq_zero_iff] at h
     exact eq_empty_iff_forall_not_mem.2 fun x hx => h.elim' ⟨x, hx⟩
-  · rintro rfl
-    exact mk_emptyc _
+  · rintro rfl; exact mk_emptyc _
 #align cardinal.mk_emptyc_iff Cardinal.mk_emptyCollection_iff
 -/
 
@@ -3214,10 +3163,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} (A : Set.{u1} (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.sUnion.{u1} α A))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} (Set.{u1} α) A)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} (Set.{u1} α) A) (fun (s : Set.Elem.{u1} (Set.{u1} α) A) => Cardinal.mk.{u1} (Set.Elem.{u1} α (Subtype.val.{succ u1} (Set.{u1} α) (fun (x : Set.{u1} α) => Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) x A) s)))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_sUnion_le Cardinal.mk_sUnion_leₓ'. -/
-theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s :=
-  by
-  rw [sUnion_eq_Union]
-  apply mk_Union_le
+theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
+  rw [sUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
 /- warning: cardinal.mk_bUnion_le -> Cardinal.mk_biUnion_le is a dubious translation:
@@ -3227,10 +3174,7 @@ but is expected to have type
   forall {ι : Type.{u1}} {α : Type.{u1}} (A : ι -> (Set.{u1} α)) (s : Set.{u1} ι), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.iUnion.{u1, succ u1} α ι (fun (x : ι) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) => A x))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} ι s)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} ι s) (fun (x : Set.Elem.{u1} ι s) => Cardinal.mk.{u1} (Set.Elem.{u1} α (A (Subtype.val.{succ u1} ι (fun (x : ι) => Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) x))))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_bUnion_le Cardinal.mk_biUnion_leₓ'. -/
 theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
-    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 :=
-  by
-  rw [bUnion_eq_Union]
-  apply mk_Union_le
+    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by rw [bUnion_eq_Union]; apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
 /- warning: cardinal.finset_card_lt_aleph_0 -> Cardinal.finset_card_lt_aleph0 is a dubious translation:
@@ -3327,10 +3271,8 @@ but is expected to have type
   forall {α : Type.{u1}} {s : Set.{u1} α} {a : α}, (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s)) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Insert.insert.{u1, u1} α (Set.{u1} α) (Set.instInsertSet.{u1} α) a s))) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} α s)) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_insert Cardinal.mk_insertₓ'. -/
 theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
-    (#(insert a s : Set α)) = (#s) + 1 :=
-  by
-  rw [← union_singleton, mk_union_of_disjoint, mk_singleton]
-  simpa
+    (#(insert a s : Set α)) = (#s) + 1 := by
+  rw [← union_singleton, mk_union_of_disjoint, mk_singleton]; simpa
 #align cardinal.mk_insert Cardinal.mk_insert
 
 /- warning: cardinal.mk_sum_compl -> Cardinal.mk_sum_compl is a dubious translation:
@@ -3434,9 +3376,7 @@ theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α →
   by
   rw [lift_mk_le.{v, u, 0}]
   refine' ⟨⟨_, _⟩⟩
-  · rintro ⟨y, hy⟩
-    rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩
-    exact ⟨x, hy⟩
+  · rintro ⟨y, hy⟩; rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩; exact ⟨x, hy⟩
   rintro ⟨y, hy⟩ ⟨y', hy'⟩; dsimp
   rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩
   rcases Classical.subtype_of_exists (h hy') with ⟨x', rfl⟩
@@ -3474,10 +3414,8 @@ theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
     {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) :=
   by
-  rw [image_eq_range] at h
-  convert mk_preimage_of_subset_range_lift _ _ h using 1
-  rw [mk_sep]
-  rfl
+  rw [image_eq_range] at h; convert mk_preimage_of_subset_range_lift _ _ h using 1
+  rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image_lift Cardinal.mk_subset_ge_of_subset_image_lift
 -/
 
@@ -3485,10 +3423,8 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
     (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) :=
   by
-  rw [image_eq_range] at h
-  convert mk_preimage_of_subset_range _ _ h using 1
-  rw [mk_sep]
-  rfl
+  rw [image_eq_range] at h; convert mk_preimage_of_subset_range _ _ h using 1
+  rw [mk_sep]; rfl
 #align cardinal.mk_subset_ge_of_subset_image Cardinal.mk_subset_ge_of_subset_image
 -/

bump to nightly-2023-05-28-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6c6011cf

Diff

@@ -635,7 +635,6 @@ theorem mk_prod (α : Type u) (β : Type v) : (#α × β) = lift.{v, u} (#α) *
 
 private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
   inductionOn₂ a b fun α β => mk_congr <| Equiv.prodComm α β
-#align cardinal.mul_comm' cardinal.mul_comm'
 
 /-- The cardinal exponential. `#α ^ #β` is the cardinal of `β → α`. -/
 instance : Pow Cardinal.{u} Cardinal.{u} :=
@@ -918,7 +917,6 @@ protected theorem zero_le : ∀ a : Cardinal, 0 ≤ a := by rintro ⟨α⟩ <;>
 
 private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d → a + c ≤ b + d := by
   rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨δ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.sum_map e₂⟩
-#align cardinal.add_le_add' cardinal.add_le_add'
 
 /- warning: cardinal.add_covariant_class -> Cardinal.add_covariantClass is a dubious translation:
 lean 3 declaration is

bump to nightly-2023-05-16-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

9cb92e8c

Diff

@@ -430,12 +430,16 @@ theorem lift_inj {a b : Cardinal} : lift a = lift b ↔ a = b :=
 #align cardinal.lift_inj Cardinal.lift_inj
 -/
 
-#print Cardinal.lift_lt /-
+/- warning: cardinal.lift_lt -> Cardinal.lift_lt is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a b)
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a b)
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt Cardinal.lift_ltₓ'. -/
 @[simp]
 theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
   liftOrderEmbedding.lt_iff_lt
 #align cardinal.lift_lt Cardinal.lift_lt
--/
 
 /- warning: cardinal.lift_strict_mono -> Cardinal.lift_strictMono is a dubious translation:
 lean 3 declaration is
@@ -920,7 +924,7 @@ private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d →
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1})) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5063 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5063) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5076 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5078 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5076 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5078)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5065 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5067 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5065 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5067) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5080 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5082 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5080 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5082)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_covariant_class Cardinal.add_covariantClassₓ'. -/
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
@@ -930,7 +934,7 @@ instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}))) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5130 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5130)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5143 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5145 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5143 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5145)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5132 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5134 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5132 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5134)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5147 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5149 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5147 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5149)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClassₓ'. -/
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
@@ -1000,7 +1004,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 
 /- warning: cardinal.cantor -> Cardinal.cantor is a dubious translation:
 lean 3 declaration is
-  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
+  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
 but is expected to have type
   forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (instOfNat.{succ u1} Cardinal.{u1} 2 Cardinal.instNatCastCardinal.{u1} (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) a)
 Case conversion may be inaccurate. Consider using '#align cardinal.cantor Cardinal.cantorₓ'. -/
@@ -1019,11 +1023,15 @@ instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := by infer_instance
 
-#print Cardinal.one_lt_iff_nontrivial /-
+/- warning: cardinal.one_lt_iff_nontrivial -> Cardinal.one_lt_iff_nontrivial is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) (Cardinal.mk.{u1} α)) (Nontrivial.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) (Cardinal.mk.{u1} α)) (Nontrivial.{u1} α)
+Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivialₓ'. -/
 theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
--/
 
 /- warning: cardinal.power_le_max_power_one -> Cardinal.power_le_max_power_one is a dubious translation:
 lean 3 declaration is
@@ -1044,15 +1052,24 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 -/
 
-#print Cardinal.power_pos /-
+/- warning: cardinal.power_pos -> Cardinal.power_pos is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} (b : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) a) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
+but is expected to have type
+  forall {a : Cardinal.{u1}} (b : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) a) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b))
+Case conversion may be inaccurate. Consider using '#align cardinal.power_pos Cardinal.power_posₓ'. -/
 theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
--/
 
 end OrderProperties
 
-#print Cardinal.lt_wf /-
+/- warning: cardinal.lt_wf -> Cardinal.lt_wf is a dubious translation:
+lean 3 declaration is
+  WellFounded.{succ (succ u1)} Cardinal.{u1} (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})))
+but is expected to have type
+  WellFounded.{succ (succ u1)} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6252 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6254 : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6252 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6254)
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_wf Cardinal.lt_wfₓ'. -/
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -1065,15 +1082,18 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [f, mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
--/
 
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
-#print Cardinal.wo /-
+/- warning: cardinal.wo -> Cardinal.wo is a dubious translation:
+lean 3 declaration is
+  IsWellOrder.{succ u1} Cardinal.{u1} (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})))
+but is expected to have type
+  IsWellOrder.{succ u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6508 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6510 : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6508 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6510)
+Case conversion may be inaccurate. Consider using '#align cardinal.wo Cardinal.woₓ'. -/
 instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 #align cardinal.wo Cardinal.wo
--/
 
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
@@ -1096,7 +1116,7 @@ instance : SuccOrder Cardinal :=
 
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 but is expected to have type
   forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
@@ -1106,7 +1126,7 @@ theorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=
 
 /- warning: cardinal.succ_pos -> Cardinal.succ_pos is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c)
+  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c)
 but is expected to have type
   forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_pos Cardinal.succ_posₓ'. -/
@@ -1160,15 +1180,19 @@ protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
 #align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
 -/
 
-#print Cardinal.IsLimit.isSuccLimit /-
+/- warning: cardinal.is_limit.is_succ_limit -> Cardinal.IsLimit.isSuccLimit is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c)
+but is expected to have type
+  forall {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c)
+Case conversion may be inaccurate. Consider using '#align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimitₓ'. -/
 protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
   h.2
 #align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
--/
 
 /- warning: cardinal.is_limit.succ_lt -> Cardinal.IsLimit.succ_lt is a dubious translation:
 lean 3 declaration is
-  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} x) c)
+  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} x) c)
 but is expected to have type
   forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} x) c)
 Case conversion may be inaccurate. Consider using '#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_ltₓ'. -/
@@ -1176,11 +1200,15 @@ theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
   h.IsSuccLimit.succ_lt
 #align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
 
-#print Cardinal.isSuccLimit_zero /-
+/- warning: cardinal.is_succ_limit_zero -> Cardinal.isSuccLimit_zero is a dubious translation:
+lean 3 declaration is
+  Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
+but is expected to have type
+  Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zeroₓ'. -/
 theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
   isSuccLimit_bot
 #align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
--/
 
 #print Cardinal.sum /-
 /-- The indexed sum of cardinals is the cardinality of the
@@ -1605,7 +1633,12 @@ theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.le_lift_iff Cardinal.le_lift_iff
 -/
 
-#print Cardinal.lt_lift_iff /-
+/- warning: cardinal.lt_lift_iff -> Cardinal.lt_lift_iff is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{max u1 u2}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) b (Cardinal.lift.{u2, u1} a)) (Exists.{succ (succ u1)} Cardinal.{u1} (fun (a' : Cardinal.{u1}) => And (Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} a') b) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a' a)))
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{max u1 u2}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) b (Cardinal.lift.{u2, u1} a)) (Exists.{succ (succ u1)} Cardinal.{u1} (fun (a' : Cardinal.{u1}) => And (Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} a') b) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a' a)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_lift_iff Cardinal.lt_lift_iffₓ'. -/
 theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
   ⟨fun h =>
@@ -1613,7 +1646,6 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
     fun ⟨a', e, h⟩ => e ▸ lift_lt.2 h⟩
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
--/
 
 /- warning: cardinal.lift_succ -> Cardinal.lift_succ is a dubious translation:
 lean 3 declaration is
@@ -1776,11 +1808,15 @@ theorem aleph0_ne_zero : ℵ₀ ≠ 0 :=
 #align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero
 -/
 
-#print Cardinal.aleph0_pos /-
+/- warning: cardinal.aleph_0_pos -> Cardinal.aleph0_pos is a dubious translation:
+lean 3 declaration is
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) Cardinal.aleph0.{u1}
+but is expected to have type
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_pos Cardinal.aleph0_posₓ'. -/
 theorem aleph0_pos : 0 < ℵ₀ :=
   pos_iff_ne_zero.2 aleph0_ne_zero
 #align cardinal.aleph_0_pos Cardinal.aleph0_pos
--/
 
 #print Cardinal.lift_aleph0 /-
 @[simp]
@@ -1803,19 +1839,27 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 -/
 
-#print Cardinal.aleph0_lt_lift /-
+/- warning: cardinal.aleph_0_lt_lift -> Cardinal.aleph0_lt_lift is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) Cardinal.aleph0.{max u1 u2} (Cardinal.lift.{u2, u1} c)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) Cardinal.aleph0.{u1} c)
+but is expected to have type
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) Cardinal.aleph0.{max u1 u2} (Cardinal.lift.{u2, u1} c)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) Cardinal.aleph0.{u1} c)
+Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_liftₓ'. -/
 @[simp]
 theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
--/
 
-#print Cardinal.lift_lt_aleph0 /-
+/- warning: cardinal.lift_lt_aleph_0 -> Cardinal.lift_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} c) Cardinal.aleph0.{max u1 u2}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1})
+but is expected to have type
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} c) Cardinal.aleph0.{max u1 u2}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1})
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0ₓ'. -/
 @[simp]
 theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
--/
 
 /-! ### Properties about the cast from `ℕ` -/
 
@@ -1886,7 +1930,7 @@ theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
 
 /- warning: cardinal.lift_lt_nat_iff -> Cardinal.lift_lt_nat_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
+  forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
 but is expected to have type
   forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Nat.cast.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instNatCastCardinal.{max u1 u2} n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iffₓ'. -/
@@ -1897,7 +1941,7 @@ theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n
 
 /- warning: cardinal.nat_lt_lift_iff -> Cardinal.nat_lt_lift_iff is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) a)
+  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) a)
 but is expected to have type
   forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Nat.cast.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instNatCastCardinal.{max u1 u2} n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n) a)
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iffₓ'. -/
@@ -1982,7 +2026,7 @@ theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
 
 /- warning: cardinal.nat_cast_lt -> Cardinal.natCast_lt is a dubious translation:
 lean 3 declaration is
-  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
+  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
 but is expected to have type
   forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} m) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (LT.lt.{0} Nat instLTNat m n)
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_cast_lt Cardinal.natCast_ltₓ'. -/
@@ -2050,18 +2094,26 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
-#print Cardinal.cantor' /-
+/- warning: cardinal.cantor' -> Cardinal.cantor' is a dubious translation:
+lean 3 declaration is
+  forall (a : Cardinal.{u1}) {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) b) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) b a))
+but is expected to have type
+  forall (a : Cardinal.{u1}) {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) b) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) b a))
+Case conversion may be inaccurate. Consider using '#align cardinal.cantor' Cardinal.cantor'ₓ'. -/
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
--/
 
-#print Cardinal.one_le_iff_pos /-
+/- warning: cardinal.one_le_iff_pos -> Cardinal.one_le_iff_pos is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) c) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) c)
+but is expected to have type
+  forall {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) c) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) c)
+Case conversion may be inaccurate. Consider using '#align cardinal.one_le_iff_pos Cardinal.one_le_iff_posₓ'. -/
 theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by rw [← succ_zero, succ_le_iff]
 #align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos
--/
 
 #print Cardinal.one_le_iff_ne_zero /-
 theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le_iff_pos, pos_iff_ne_zero]
@@ -2070,7 +2122,7 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le
 
 /- warning: cardinal.nat_lt_aleph_0 -> Cardinal.nat_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
+  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
 but is expected to have type
   forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n) Cardinal.aleph0.{u1}
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0ₓ'. -/
@@ -2081,11 +2133,15 @@ theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
-#print Cardinal.one_lt_aleph0 /-
+/- warning: cardinal.one_lt_aleph_0 -> Cardinal.one_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) Cardinal.aleph0.{u1}
+but is expected to have type
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0ₓ'. -/
 @[simp]
 theorem one_lt_aleph0 : 1 < ℵ₀ := by simpa using nat_lt_aleph_0 1
 #align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0
--/
 
 #print Cardinal.one_le_aleph0 /-
 theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
@@ -2095,7 +2151,7 @@ theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
 
 /- warning: cardinal.lt_aleph_0 -> Cardinal.lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)))
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)))
 but is expected to have type
   forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)))
 Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0 Cardinal.lt_aleph0ₓ'. -/
@@ -2124,7 +2180,12 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       exact (Nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
-#print Cardinal.isSuccLimit_aleph0 /-
+/- warning: cardinal.is_succ_limit_aleph_0 -> Cardinal.isSuccLimit_aleph0 is a dubious translation:
+lean 3 declaration is
+  Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) Cardinal.aleph0.{u1}
+but is expected to have type
+  Order.IsSuccLimit.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0ₓ'. -/
 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
   isSuccLimit_of_succ_lt fun a ha =>
     by
@@ -2132,7 +2193,6 @@ theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
     rw [← nat_succ]
     apply nat_lt_aleph_0
 #align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
--/
 
 #print Cardinal.isLimit_aleph0 /-
 theorem isLimit_aleph0 : IsLimit ℵ₀ :=
@@ -2172,40 +2232,66 @@ theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ F
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
-#print Cardinal.lt_aleph0_iff_finite /-
+/- warning: cardinal.lt_aleph_0_iff_finite -> Cardinal.lt_aleph0_iff_finite is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}) (Finite.{succ u1} α)
+but is expected to have type
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}) (Finite.{succ u1} α)
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finiteₓ'. -/
 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
--/
 
-#print Cardinal.lt_aleph0_iff_fintype /-
+/- warning: cardinal.lt_aleph_0_iff_fintype -> Cardinal.lt_aleph0_iff_fintype is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}) (Nonempty.{succ u1} (Fintype.{u1} α))
+but is expected to have type
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}) (Nonempty.{succ u1} (Fintype.{u1} α))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintypeₓ'. -/
 theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
--/
 
-#print Cardinal.lt_aleph0_of_finite /-
+/- warning: cardinal.lt_aleph_0_of_finite -> Cardinal.lt_aleph0_of_finite is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : Finite.{succ u1} α], LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : Finite.{succ u1} α], LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finiteₓ'. -/
 theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
--/
 
-#print Cardinal.lt_aleph0_iff_set_finite /-
+/- warning: cardinal.lt_aleph_0_iff_set_finite -> Cardinal.lt_aleph0_iff_set_finite is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {S : Set.{u1} α}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α S)
+but is expected to have type
+  forall {α : Type.{u1}} {S : Set.{u1} α}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α S)) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α S)
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finiteₓ'. -/
 @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
--/
 
+/- warning: set.finite.lt_aleph_0 -> Set.Finite.lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {S : Set.{u1} α}, (Set.Finite.{u1} α S) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) Cardinal.aleph0.{u1})
+but is expected to have type
+  forall {α : Type.{u1}} {S : Set.{u1} α}, (Set.Finite.{u1} α S) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α S)) Cardinal.aleph0.{u1})
+Case conversion may be inaccurate. Consider using '#align set.finite.lt_aleph_0 Set.Finite.lt_aleph0ₓ'. -/
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
-#print Cardinal.lt_aleph0_iff_subtype_finite /-
+/- warning: cardinal.lt_aleph_0_iff_subtype_finite -> Cardinal.lt_aleph0_iff_subtype_finite is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {p : α -> Prop}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Subtype.{succ u1} α (fun (x : α) => p x))) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α (setOf.{u1} α (fun (x : α) => p x)))
+but is expected to have type
+  forall {α : Type.{u1}} {p : α -> Prop}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Subtype.{succ u1} α (fun (x : α) => p x))) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α (setOf.{u1} α (fun (x : α) => p x)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finiteₓ'. -/
 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
--/
 
 #print Cardinal.mk_le_aleph0_iff /-
 theorem mk_le_aleph0_iff : (#α) ≤ ℵ₀ ↔ Countable α := by
@@ -2240,7 +2326,7 @@ theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
 
 /- warning: cardinal.can_lift_cardinal_nat -> Cardinal.canLiftCardinalNat is a dubious translation:
 lean 3 declaration is
-  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
+  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
 but is expected to have type
   CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1}) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNatₓ'. -/
@@ -2252,7 +2338,7 @@ instance canLiftCardinalNat : CanLift Cardinal ℕ coe fun x => x < ℵ₀ :=
 
 /- warning: cardinal.add_lt_aleph_0 -> Cardinal.add_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0ₓ'. -/
@@ -2263,7 +2349,7 @@ theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a +
 
 /- warning: cardinal.add_lt_aleph_0_iff -> Cardinal.add_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iffₓ'. -/
@@ -2282,7 +2368,12 @@ theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a 
   simp only [← not_lt, add_lt_aleph_0_iff, not_and_or]
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
 
-#print Cardinal.nsmul_lt_aleph0_iff /-
+/- warning: cardinal.nsmul_lt_aleph_0_iff -> Cardinal.nsmul_lt_aleph0_iff is a dubious translation:
+lean 3 declaration is
+  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+but is expected to have type
+  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HSMul.hSMul.{0, succ u1, succ u1} Nat Cardinal.{u1} Cardinal.{u1} (instHSMul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) n a) Cardinal.aleph0.{u1}) (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iffₓ'. -/
 /-- See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
 theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ :=
   by
@@ -2293,18 +2384,21 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   · simp
   rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self_iff]
 #align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff
--/
 
-#print Cardinal.nsmul_lt_aleph0_iff_of_ne_zero /-
+/- warning: cardinal.nsmul_lt_aleph_0_iff_of_ne_zero -> Cardinal.nsmul_lt_aleph0_iff_of_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall {n : Nat} {a : Cardinal.{u1}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+but is expected to have type
+  forall {n : Nat} {a : Cardinal.{u1}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HSMul.hSMul.{0, succ u1, succ u1} Nat Cardinal.{u1} Cardinal.{u1} (instHSMul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) n a) Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zeroₓ'. -/
 /-- See also `cardinal.nsmul_lt_aleph_0_iff` for a hypothesis-free version. -/
 theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ :=
   nsmul_lt_aleph0_iff.trans <| or_iff_right h
 #align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
--/
 
 /- warning: cardinal.mul_lt_aleph_0 -> Cardinal.mul_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0ₓ'. -/
@@ -2315,7 +2409,7 @@ theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a *
 
 /- warning: cardinal.mul_lt_aleph_0_iff -> Cardinal.mul_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iffₓ'. -/
@@ -2367,7 +2461,7 @@ theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 
 
 /- warning: cardinal.mul_lt_aleph_0_iff_of_ne_zero -> Cardinal.mul_lt_aleph0_iff_of_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zeroₓ'. -/
@@ -2375,12 +2469,16 @@ theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
-#print Cardinal.power_lt_aleph0 /-
+/- warning: cardinal.power_lt_aleph_0 -> Cardinal.power_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b) Cardinal.aleph0.{u1})
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b) Cardinal.aleph0.{u1})
+Case conversion may be inaccurate. Consider using '#align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0ₓ'. -/
 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← nat_cast_pow] <;> apply nat_lt_aleph_0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
--/
 
 #print Cardinal.eq_one_iff_unique /-
 theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
@@ -2518,7 +2616,7 @@ def toNat : ZeroHom Cardinal ℕ :=
 
 /- warning: cardinal.to_nat_apply_of_lt_aleph_0 -> Cardinal.toNat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0ₓ'. -/
@@ -2535,7 +2633,7 @@ theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0
 
 /- warning: cardinal.cast_to_nat_of_lt_aleph_0 -> Cardinal.cast_toNat_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0ₓ'. -/
@@ -2553,33 +2651,49 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat =
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
-#print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_le_iff_le_of_lt_aleph_0 -> Cardinal.toNat_le_iff_le_of_lt_aleph0 is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0ₓ'. -/
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
   rw [← nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
--/
 
-#print Cardinal.toNat_lt_iff_lt_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 -> Cardinal.toNat_lt_iff_lt_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0ₓ'. -/
 theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat < d.toNat ↔ c < d := by
   rw [← nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
--/
 
-#print Cardinal.toNat_le_of_le_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_le_of_le_of_lt_aleph_0 -> Cardinal.toNat_le_of_le_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) d Cardinal.aleph0.{u1}) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} c d) -> (LE.le.{0} Nat Nat.hasLe (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d))
+but is expected to have type
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) d Cardinal.aleph0.{u1}) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} c d) -> (LE.le.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) c) instLENat (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} d))
+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0ₓ'. -/
 theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
     c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
 #align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
--/
 
-#print Cardinal.toNat_lt_of_lt_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_lt_of_lt_of_lt_aleph_0 -> Cardinal.toNat_lt_of_lt_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) d Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c d) -> (LT.lt.{0} Nat Nat.hasLt (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0ₓ'. -/
 theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
     c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
--/
 
 /- warning: cardinal.to_nat_cast -> Cardinal.toNat_cast is a dubious translation:
 lean 3 declaration is
@@ -2747,7 +2861,7 @@ theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
 
 /- warning: cardinal.to_nat_add_of_lt_aleph_0 -> Cardinal.toNat_add_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toHasLt.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) b Cardinal.aleph0.{u2}) -> (Eq.{1} Nat (coeFn.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) (fun (_x : ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) => Cardinal.{max u1 u2} -> Nat) (ZeroHom.hasCoeToFun.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHAdd.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasAdd.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} a) (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} b)))
 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0ₓ'. -/
@@ -2791,7 +2905,7 @@ def toPartENat : Cardinal →+ PartENat
 
 /- warning: cardinal.to_part_enat_apply_of_lt_aleph_0 -> Cardinal.toPartENat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
 but is expected to have type
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} 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PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ 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(OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0ₓ'. -/
@@ -2875,7 +2989,7 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 
 /- warning: cardinal.sum_lt_prod -> Cardinal.sum_lt_prod is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toHasLt.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toHasLt.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
 but is expected to have type
   forall {ι : Type.{u2}} (f : ι -> Cardinal.{u1}) (g : ι -> Cardinal.{u1}), (forall (i : ι), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (f i) (g i)) -> (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u2, u1} ι f) (Cardinal.prod.{u2, u1} ι g))
 Case conversion may be inaccurate. Consider using '#align cardinal.sum_lt_prod Cardinal.sum_lt_prodₓ'. -/
@@ -3121,11 +3235,15 @@ theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
   apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
-#print Cardinal.finset_card_lt_aleph0 /-
+/- warning: cardinal.finset_card_lt_aleph_0 -> Cardinal.finset_card_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} (s : Finset.{u1} α), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} α) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (Finset.Set.hasCoeT.{u1} α))) s))) Cardinal.aleph0.{u1}
+but is expected to have type
+  forall {α : Type.{u1}} (s : Finset.{u1} α), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α (Finset.toSet.{u1} α s))) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0ₓ'. -/
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
--/
 
 /- warning: cardinal.mk_set_eq_nat_iff_finset -> Cardinal.mk_set_eq_nat_iff_finset is a dubious translation:
 lean 3 declaration is
@@ -3440,7 +3558,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
 
 /- warning: cardinal.exists_not_mem_of_length_lt -> Cardinal.exists_not_mem_of_length_lt is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
+  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
 but is expected to have type
   forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.mem.{u1, u1} α (List.{u1} α) (List.instMembershipList.{u1} α) z l)))
 Case conversion may be inaccurate. Consider using '#align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_ltₓ'. -/
@@ -3479,16 +3597,25 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 -- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
-#print Cardinal.le_powerlt /-
+/- warning: cardinal.le_powerlt -> Cardinal.le_powerlt is a dubious translation:
+lean 3 declaration is
+  forall {b : Cardinal.{u1}} {c : Cardinal.{u1}} (a : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a c) (Cardinal.powerlt.{u1} a b))
+but is expected to have type
+  forall {b : Cardinal.{u1}} {c : Cardinal.{u1}} (a : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a c) (Cardinal.powerlt.{u1} a b))
+Case conversion may be inaccurate. Consider using '#align cardinal.le_powerlt Cardinal.le_powerltₓ'. -/
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
   apply @le_ciSup _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
--/
 
-#print Cardinal.powerlt_le /-
+/- warning: cardinal.powerlt_le -> Cardinal.powerlt_le is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.powerlt.{u1} a b) c) (forall (x : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toHasLt.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a x) c))
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.powerlt.{u1} a b) c) (forall (x : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a x) c))
+Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_le Cardinal.powerlt_leₓ'. -/
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
   rw [powerlt, ciSup_le_iff']
@@ -3496,7 +3623,6 @@ theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) 
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.powerlt_le Cardinal.powerlt_le
--/
 
 #print Cardinal.powerlt_le_powerlt_left /-
 theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a ^< c :=

bump to nightly-2023-05-14-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

d31da313

Diff

@@ -1078,29 +1078,29 @@ instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
 
-/- warning: cardinal.Inf_empty -> Cardinal.infₛ_empty is a dubious translation:
+/- warning: cardinal.Inf_empty -> Cardinal.sInf_empty is a dubious translation:
 lean 3 declaration is
-  Eq.{succ (succ u1)} Cardinal.{u1} (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (EmptyCollection.emptyCollection.{succ u1} (Set.{succ u1} Cardinal.{u1}) (Set.hasEmptyc.{succ u1} Cardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
+  Eq.{succ (succ u1)} Cardinal.{u1} (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (EmptyCollection.emptyCollection.{succ u1} (Set.{succ u1} Cardinal.{u1}) (Set.hasEmptyc.{succ u1} Cardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
 but is expected to have type
-  Eq.{succ (succ u1)} Cardinal.{u1} (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (EmptyCollection.emptyCollection.{succ u1} (Set.{succ u1} Cardinal.{u1}) (Set.instEmptyCollectionSet.{succ u1} Cardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
-Case conversion may be inaccurate. Consider using '#align cardinal.Inf_empty Cardinal.infₛ_emptyₓ'. -/
+  Eq.{succ (succ u1)} Cardinal.{u1} (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (EmptyCollection.emptyCollection.{succ u1} (Set.{succ u1} Cardinal.{u1}) (Set.instEmptyCollectionSet.{succ u1} Cardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.Inf_empty Cardinal.sInf_emptyₓ'. -/
 @[simp]
-theorem infₛ_empty : infₛ (∅ : Set Cardinal.{u}) = 0 :=
+theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg not_nonempty_empty
-#align cardinal.Inf_empty Cardinal.infₛ_empty
+#align cardinal.Inf_empty Cardinal.sInf_empty
 
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
-  SuccOrder.ofSuccLeIff (fun c => infₛ { c' | c < c' }) fun a b =>
-    ⟨lt_of_lt_of_le <| cinfₛ_mem <| exists_gt a, cinfₛ_le'⟩
+  SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' }) fun a b =>
+    ⟨lt_of_lt_of_le <| csInf_mem <| exists_gt a, csInf_le'⟩
 
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 but is expected to have type
-  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
-theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
+theorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
 
@@ -1132,7 +1132,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align cardinal.add_one_le_succ Cardinal.add_one_le_succₓ'. -/
 theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
   by
-  refine' (le_cinfₛ_iff'' (exists_gt c)).2 fun b hlt => _
+  refine' (le_csInf_iff'' (exists_gt c)).2 fun b hlt => _
   rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
   cases' le_of_lt hlt with f
   have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
@@ -1379,39 +1379,39 @@ theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbo
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
 -/
 
-/- warning: cardinal.supr_le_sum -> Cardinal.supᵢ_le_sum is a dubious translation:
+/- warning: cardinal.supr_le_sum -> Cardinal.iSup_le_sum is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (supᵢ.{succ (max u1 u2), succ u1} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasSup.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) ι f) (Cardinal.sum.{u1, max u1 u2} ι f)
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (iSup.{succ (max u1 u2), succ u1} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasSup.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) ι f) (Cardinal.sum.{u1, max u1 u2} ι f)
 but is expected to have type
-  forall {ι : Type.{u2}} (f : ι -> Cardinal.{max u2 u1}), LE.le.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.instLECardinal.{max u2 u1} (supᵢ.{succ (max u2 u1), succ u2} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toSupSet.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) ι f) (Cardinal.sum.{u2, max u2 u1} ι f)
-Case conversion may be inaccurate. Consider using '#align cardinal.supr_le_sum Cardinal.supᵢ_le_sumₓ'. -/
-theorem supᵢ_le_sum {ι} (f : ι → Cardinal) : supᵢ f ≤ sum f :=
-  csupᵢ_le' <| le_sum _
-#align cardinal.supr_le_sum Cardinal.supᵢ_le_sum
+  forall {ι : Type.{u2}} (f : ι -> Cardinal.{max u2 u1}), LE.le.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.instLECardinal.{max u2 u1} (iSup.{succ (max u2 u1), succ u2} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toSupSet.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) ι f) (Cardinal.sum.{u2, max u2 u1} ι f)
+Case conversion may be inaccurate. Consider using '#align cardinal.supr_le_sum Cardinal.iSup_le_sumₓ'. -/
+theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
+  ciSup_le' <| le_sum _
+#align cardinal.supr_le_sum Cardinal.iSup_le_sum
 
-/- warning: cardinal.sum_le_supr_lift -> Cardinal.sum_le_supᵢ_lift is a dubious translation:
+/- warning: cardinal.sum_le_supr_lift -> Cardinal.sum_le_iSup_lift is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.sum.{u1, max u1 u2} ι f) (HMul.hMul.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHMul.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasMul.{max u1 u2}) (Cardinal.lift.{u2, u1} (Cardinal.mk.{u1} ι)) (supᵢ.{succ (max u1 u2), succ u1} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasSup.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) ι f))
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.sum.{u1, max u1 u2} ι f) (HMul.hMul.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHMul.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasMul.{max u1 u2}) (Cardinal.lift.{u2, u1} (Cardinal.mk.{u1} ι)) (iSup.{succ (max u1 u2), succ u1} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasSup.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) ι f))
 but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align cardinal.sum_le_supr_lift Cardinal.sum_le_supᵢ_liftₓ'. -/
-theorem sum_le_supᵢ_lift {ι : Type u} (f : ι → Cardinal.{max u v}) : sum f ≤ (#ι).lift * supᵢ f :=
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.sum.{u1, max u1 u2} ι f) (HMul.hMul.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHMul.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instMulCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} (Cardinal.mk.{u1} ι)) (iSup.{max (succ u1) (succ u2), succ u1} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toSupSet.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u2}))) ι f))
+Case conversion may be inaccurate. Consider using '#align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_liftₓ'. -/
+theorem sum_le_iSup_lift {ι : Type u} (f : ι → Cardinal.{max u v}) : sum f ≤ (#ι).lift * iSup f :=
   by
-  rw [← (supᵢ f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
-  exact sum_le_sum _ _ (le_csupᵢ <| bddAbove_range.{u, v} f)
-#align cardinal.sum_le_supr_lift Cardinal.sum_le_supᵢ_lift
+  rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
+  exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)
+#align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift
 
-/- warning: cardinal.sum_le_supr -> Cardinal.sum_le_supᵢ is a dubious translation:
+/- warning: cardinal.sum_le_supr -> Cardinal.sum_le_iSup is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.sum.{u1, u1} ι f) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} ι) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f))
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.sum.{u1, u1} ι f) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} ι) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f))
 but is expected to have type
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.sum.{u1, u1} ι f) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} ι) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f))
-Case conversion may be inaccurate. Consider using '#align cardinal.sum_le_supr Cardinal.sum_le_supᵢₓ'. -/
-theorem sum_le_supᵢ {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * supᵢ f :=
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.sum.{u1, u1} ι f) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} ι) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f))
+Case conversion may be inaccurate. Consider using '#align cardinal.sum_le_supr Cardinal.sum_le_iSupₓ'. -/
+theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f :=
   by
   rw [← lift_id (#ι)]
   exact sum_le_supr_lift f
-#align cardinal.sum_le_supr Cardinal.sum_le_supᵢ
+#align cardinal.sum_le_supr Cardinal.sum_le_iSup
 
 /- warning: cardinal.sum_nat_eq_add_sum_succ -> Cardinal.sum_nat_eq_add_sum_succ is a dubious translation:
 lean 3 declaration is
@@ -1426,17 +1426,17 @@ theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
   simp only [mk_sum, mk_out, lift_id, mk_sigma]
 #align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ
 
-/- warning: cardinal.supr_of_empty -> Cardinal.supᵢ_of_empty is a dubious translation:
+/- warning: cardinal.supr_of_empty -> Cardinal.iSup_of_empty is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u2}) [_inst_1 : IsEmpty.{u1} ι], Eq.{succ (succ u2)} Cardinal.{u2} (supᵢ.{succ u2, u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f) (OfNat.ofNat.{succ u2} Cardinal.{u2} 0 (OfNat.mk.{succ u2} Cardinal.{u2} 0 (Zero.zero.{succ u2} Cardinal.{u2} Cardinal.hasZero.{u2})))
+  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u2}) [_inst_1 : IsEmpty.{u1} ι], Eq.{succ (succ u2)} Cardinal.{u2} (iSup.{succ u2, u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f) (OfNat.ofNat.{succ u2} Cardinal.{u2} 0 (OfNat.mk.{succ u2} Cardinal.{u2} 0 (Zero.zero.{succ u2} Cardinal.{u2} Cardinal.hasZero.{u2})))
 but is expected to have type
-  forall {ι : Sort.{u2}} (f : ι -> Cardinal.{u1}) [_inst_1 : IsEmpty.{u2} ι], Eq.{succ (succ u1)} Cardinal.{u1} (supᵢ.{succ u1, u2} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
-Case conversion may be inaccurate. Consider using '#align cardinal.supr_of_empty Cardinal.supᵢ_of_emptyₓ'. -/
+  forall {ι : Sort.{u2}} (f : ι -> Cardinal.{u1}) [_inst_1 : IsEmpty.{u2} ι], Eq.{succ (succ u1)} Cardinal.{u1} (iSup.{succ u1, u2} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.supr_of_empty Cardinal.iSup_of_emptyₓ'. -/
 /-- A variant of `csupr_of_empty` but with `0` on the RHS for convenience -/
 @[simp]
-protected theorem supᵢ_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : supᵢ f = 0 :=
-  csupᵢ_of_empty f
-#align cardinal.supr_of_empty Cardinal.supᵢ_of_empty
+protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
+  ciSup_of_empty f
+#align cardinal.supr_of_empty Cardinal.iSup_of_empty
 
 #print Cardinal.lift_mk_shrink /-
 @[simp]
@@ -1554,33 +1554,33 @@ theorem prod_eq_of_fintype {α : Type u} [Fintype α] (f : α → Cardinal.{v})
 #align cardinal.prod_eq_of_fintype Cardinal.prod_eq_of_fintype
 -/
 
-/- warning: cardinal.lift_Inf -> Cardinal.lift_infₛ is a dubious translation:
+/- warning: cardinal.lift_Inf -> Cardinal.lift_sInf is a dubious translation:
 lean 3 declaration is
-  forall (s : Set.{succ u1} Cardinal.{u1}), Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) s)) (InfSet.infₛ.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) (Set.image.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s))
+  forall (s : Set.{succ u1} Cardinal.{u1}), Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (InfSet.sInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) s)) (InfSet.sInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toHasInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2}))) (Set.image.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s))
 but is expected to have type
-  forall (s : Set.{succ u2} Cardinal.{u2}), Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (InfSet.infₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toInfSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) s)) (InfSet.infₛ.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toInfSet.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u2}))) (Set.image.{succ u2, max (succ u1) (succ u2)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_Inf Cardinal.lift_infₛₓ'. -/
+  forall (s : Set.{succ u2} Cardinal.{u2}), Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (InfSet.sInf.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toInfSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) s)) (InfSet.sInf.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toInfSet.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u2}))) (Set.image.{succ u2, max (succ u1) (succ u2)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_Inf Cardinal.lift_sInfₓ'. -/
 @[simp]
-theorem lift_infₛ (s : Set Cardinal) : lift (infₛ s) = infₛ (lift '' s) :=
+theorem lift_sInf (s : Set Cardinal) : lift (sInf s) = sInf (lift '' s) :=
   by
   rcases eq_empty_or_nonempty s with (rfl | hs)
   · simp
   · exact lift_monotone.map_Inf hs
-#align cardinal.lift_Inf Cardinal.lift_infₛ
+#align cardinal.lift_Inf Cardinal.lift_sInf
 
-/- warning: cardinal.lift_infi -> Cardinal.lift_infᵢ is a dubious translation:
+/- warning: cardinal.lift_infi -> Cardinal.lift_iInf is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u2}), Eq.{succ (succ (max u2 u3))} Cardinal.{max u2 u3} (Cardinal.lift.{u3, u2} (infᵢ.{succ u2, u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasInf.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (infᵢ.{succ (max u2 u3), u1} Cardinal.{max u2 u3} (ConditionallyCompleteLattice.toHasInf.{succ (max u2 u3)} Cardinal.{max u2 u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u3)} Cardinal.{max u2 u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u3)} Cardinal.{max u2 u3} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u3}))) ι (fun (i : ι) => Cardinal.lift.{u3, u2} (f i)))
+  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u2}), Eq.{succ (succ (max u2 u3))} Cardinal.{max u2 u3} (Cardinal.lift.{u3, u2} (iInf.{succ u2, u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasInf.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (iInf.{succ (max u2 u3), u1} Cardinal.{max u2 u3} (ConditionallyCompleteLattice.toHasInf.{succ (max u2 u3)} Cardinal.{max u2 u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u3)} Cardinal.{max u2 u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u3)} Cardinal.{max u2 u3} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u3}))) ι (fun (i : ι) => Cardinal.lift.{u3, u2} (f i)))
 but is expected to have type
-  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u3}), Eq.{max (succ (succ u2)) (succ (succ u3))} Cardinal.{max u3 u2} (Cardinal.lift.{u2, u3} (infᵢ.{succ u3, u1} Cardinal.{u3} (ConditionallyCompleteLattice.toInfSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (infᵢ.{max (succ u2) (succ u3), u1} Cardinal.{max u3 u2} (ConditionallyCompleteLattice.toInfSet.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u3}))) ι (fun (i : ι) => Cardinal.lift.{u2, u3} (f i)))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_infi Cardinal.lift_infᵢₓ'. -/
+  forall {ι : Sort.{u1}} (f : ι -> Cardinal.{u3}), Eq.{max (succ (succ u2)) (succ (succ u3))} Cardinal.{max u3 u2} (Cardinal.lift.{u2, u3} (iInf.{succ u3, u1} Cardinal.{u3} (ConditionallyCompleteLattice.toInfSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (iInf.{max (succ u2) (succ u3), u1} Cardinal.{max u3 u2} (ConditionallyCompleteLattice.toInfSet.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u3)} Cardinal.{max u3 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u3}))) ι (fun (i : ι) => Cardinal.lift.{u2, u3} (f i)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_infi Cardinal.lift_iInfₓ'. -/
 @[simp]
-theorem lift_infᵢ {ι} (f : ι → Cardinal) : lift (infᵢ f) = ⨅ i, lift (f i) :=
+theorem lift_iInf {ι} (f : ι → Cardinal) : lift (iInf f) = ⨅ i, lift (f i) :=
   by
-  unfold infᵢ
+  unfold iInf
   convert lift_Inf (range f)
   rw [range_comp]
-#align cardinal.lift_infi Cardinal.lift_infᵢ
+#align cardinal.lift_infi Cardinal.lift_iInf
 
 #print Cardinal.lift_down /-
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} : b ≤ lift a → ∃ a', lift a' = b :=
@@ -1661,98 +1661,98 @@ theorem lift_max {a b : Cardinal} : lift (max a b) = max (lift a) (lift b) :=
   lift_monotone.map_max
 #align cardinal.lift_max Cardinal.lift_max
 
-/- warning: cardinal.lift_Sup -> Cardinal.lift_supₛ is a dubious translation:
+/- warning: cardinal.lift_Sup -> Cardinal.lift_sSup is a dubious translation:
 lean 3 declaration is
-  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.supₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.supₛ.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
+  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.sSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.sSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
 but is expected to have type
-  forall {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) -> (Eq.{max (succ (succ u2)) (succ (succ u1))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (SupSet.supₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) s)) (SupSet.supₛ.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toSupSet.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) (Set.image.{succ u1, max (succ u2) (succ u1)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s)))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_Sup Cardinal.lift_supₛₓ'. -/
+  forall {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) -> (Eq.{max (succ (succ u2)) (succ (succ u1))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (SupSet.sSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) s)) (SupSet.sSup.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toSupSet.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) (Set.image.{succ u1, max (succ u2) (succ u1)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_Sup Cardinal.lift_sSupₓ'. -/
 /-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (supₛ s) = supₛ (lift.{u} '' s) :=
+theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (sSup s) = sSup (lift.{u} '' s) :=
   by
-  apply ((le_csupₛ_iff' (bdd_above_image _ hs)).2 fun c hc => _).antisymm (csupₛ_le' _)
+  apply ((le_csSup_iff' (bdd_above_image _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
   · by_contra h
     obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
     simp_rw [lift_le] at h hc
-    rw [csupₛ_le_iff' hs] at h
+    rw [csSup_le_iff' hs] at h
     exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
   · rintro i ⟨j, hj, rfl⟩
-    exact lift_le.2 (le_csupₛ hs hj)
-#align cardinal.lift_Sup Cardinal.lift_supₛ
+    exact lift_le.2 (le_csSup hs hj)
+#align cardinal.lift_Sup Cardinal.lift_sSup
 
-/- warning: cardinal.lift_supr -> Cardinal.lift_supᵢ is a dubious translation:
+/- warning: cardinal.lift_supr -> Cardinal.lift_iSup is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (supᵢ.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (iSup.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
 but is expected to have type
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{max (succ (succ u1)) (succ (succ u3))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (supᵢ.{max (succ u1) (succ u3), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toSupSet.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u3}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr Cardinal.lift_supᵢₓ'. -/
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{max (succ (succ u1)) (succ (succ u3))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (iSup.{max (succ u1) (succ u3), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toSupSet.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u3}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr Cardinal.lift_iSupₓ'. -/
 /-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_supᵢ {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
-    lift.{u} (supᵢ f) = ⨆ i, lift.{u} (f i) := by rw [supᵢ, supᵢ, lift_Sup hf, ← range_comp]
-#align cardinal.lift_supr Cardinal.lift_supᵢ
+theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
+    lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by rw [iSup, iSup, lift_Sup hf, ← range_comp]
+#align cardinal.lift_supr Cardinal.lift_iSup
 
-/- warning: cardinal.lift_supr_le -> Cardinal.lift_supᵢ_le is a dubious translation:
+/- warning: cardinal.lift_supr_le -> Cardinal.lift_iSup_le is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
 but is expected to have type
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u1 u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t)
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le Cardinal.lift_supᵢ_leₓ'. -/
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u1 u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t)
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le Cardinal.lift_iSup_leₓ'. -/
 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
-theorem lift_supᵢ_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
-    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (supᵢ f) ≤ t :=
+theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
+    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t :=
   by
   rw [lift_supr hf]
-  exact csupᵢ_le' w
-#align cardinal.lift_supr_le Cardinal.lift_supᵢ_le
+  exact ciSup_le' w
+#align cardinal.lift_supr_le Cardinal.lift_iSup_le
 
-/- warning: cardinal.lift_supr_le_iff -> Cardinal.lift_supᵢ_le_iff is a dubious translation:
+/- warning: cardinal.lift_supr_le_iff -> Cardinal.lift_iSup_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
 but is expected to have type
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u1 u3}}, Iff (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t) (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_iff Cardinal.lift_supᵢ_le_iffₓ'. -/
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u1 u3}}, Iff (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (iSup.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t) (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iffₓ'. -/
 @[simp]
-theorem lift_supᵢ_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
-    {t : Cardinal} : lift.{u} (supᵢ f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t :=
+theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
+    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t :=
   by
   rw [lift_supr hf]
-  exact csupᵢ_le_iff' (bdd_above_range_comp hf _)
-#align cardinal.lift_supr_le_iff Cardinal.lift_supᵢ_le_iff
+  exact ciSup_le_iff' (bdd_above_range_comp hf _)
+#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff
 
 universe v' w'
 
-/- warning: cardinal.lift_supr_le_lift_supr -> Cardinal.lift_supᵢ_le_lift_supᵢ is a dubious translation:
+/- warning: cardinal.lift_supr_le_lift_supr -> Cardinal.lift_iSup_le_lift_iSup is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (iSup.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (iSup.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
 but is expected to have type
-  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toSupSet.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u4}))) ι' f'))))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢₓ'. -/
+  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (iSup.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (iSup.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toSupSet.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u4}))) ι' f'))))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSupₓ'. -/
 /-- To prove an inequality between the lifts to a common universe of two different supremums,
 it suffices to show that the lift of each cardinal from the smaller supremum
 if bounded by the lift of some cardinal from the larger supremum.
 -/
-theorem lift_supᵢ_le_lift_supᵢ {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
+theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
     {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
-    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (supᵢ f) ≤ lift.{w} (supᵢ f') :=
+    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (iSup f) ≤ lift.{w} (iSup f') :=
   by
   rw [lift_supr hf, lift_supr hf']
-  exact csupᵢ_mono' (bdd_above_range_comp hf' _) fun i => ⟨_, h i⟩
-#align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢ
+  exact ciSup_mono' (bdd_above_range_comp hf' _) fun i => ⟨_, h i⟩
+#align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup
 
-/- warning: cardinal.lift_supr_le_lift_supr' -> Cardinal.lift_supᵢ_le_lift_supᵢ' is a dubious translation:
+/- warning: cardinal.lift_supr_le_lift_supr' -> Cardinal.lift_iSup_le_lift_iSup' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (iSup.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
 but is expected to have type
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι' f'))))
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_supᵢ_le_lift_supᵢ'ₓ'. -/
+  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (iSup.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι' f'))))
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'ₓ'. -/
 /-- A variant of `lift_supr_le_lift_supr` with universes specialized via `w = v` and `w' = v'`.
 This is sometimes necessary to avoid universe unification issues. -/
-theorem lift_supᵢ_le_lift_supᵢ' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
+theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
     {f' : ι' → Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : ι → ι')
-    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (supᵢ f) ≤ lift.{v} (supᵢ f') :=
-  lift_supᵢ_le_lift_supᵢ hf hf' h
-#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_supᵢ_le_lift_supᵢ'
+    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') :=
+  lift_iSup_le_lift_iSup hf hf' h
+#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'
 
 #print Cardinal.aleph0 /-
 /-- `ℵ₀` is the smallest infinite cardinal. -/
@@ -3063,63 +3063,63 @@ theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injectiv
 #align cardinal.mk_image_eq Cardinal.mk_image_eq
 -/
 
-#print Cardinal.mk_unionᵢ_le_sum_mk /-
-theorem mk_unionᵢ_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
+#print Cardinal.mk_iUnion_le_sum_mk /-
+theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
   calc
-    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToUnionᵢ_surjective f)
+    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
     _ = sum fun i => #f i := mk_sigma _
     
-#align cardinal.mk_Union_le_sum_mk Cardinal.mk_unionᵢ_le_sum_mk
+#align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 -/
 
-/- warning: cardinal.mk_Union_eq_sum_mk -> Cardinal.mk_unionᵢ_eq_sum_mk is a dubious translation:
+/- warning: cardinal.mk_Union_eq_sum_mk -> Cardinal.mk_iUnion_eq_sum_mk is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {ι : Type.{u1}} {f : ι -> (Set.{u1} α)}, (forall (i : ι) (j : ι), (Ne.{succ u1} ι i j) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (f i) (f j))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.unionᵢ.{u1, succ u1} α ι (fun (i : ι) => f i)))) (Cardinal.sum.{u1, u1} ι (fun (i : ι) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (f i)))))
+  forall {α : Type.{u1}} {ι : Type.{u1}} {f : ι -> (Set.{u1} α)}, (forall (i : ι) (j : ι), (Ne.{succ u1} ι i j) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (f i) (f j))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.iUnion.{u1, succ u1} α ι (fun (i : ι) => f i)))) (Cardinal.sum.{u1, u1} ι (fun (i : ι) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (f i)))))
 but is expected to have type
-  forall {α : Type.{u1}} {ι : Type.{u1}} {f : ι -> (Set.{u1} α)}, (forall (i : ι) (j : ι), (Ne.{succ u1} ι i j) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (f i) (f j))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.unionᵢ.{u1, succ u1} α ι (fun (i : ι) => f i)))) (Cardinal.sum.{u1, u1} ι (fun (i : ι) => Cardinal.mk.{u1} (Set.Elem.{u1} α (f i)))))
-Case conversion may be inaccurate. Consider using '#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_unionᵢ_eq_sum_mkₓ'. -/
-theorem mk_unionᵢ_eq_sum_mk {α ι : Type u} {f : ι → Set α}
+  forall {α : Type.{u1}} {ι : Type.{u1}} {f : ι -> (Set.{u1} α)}, (forall (i : ι) (j : ι), (Ne.{succ u1} ι i j) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (f i) (f j))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.iUnion.{u1, succ u1} α ι (fun (i : ι) => f i)))) (Cardinal.sum.{u1, u1} ι (fun (i : ι) => Cardinal.mk.{u1} (Set.Elem.{u1} α (f i)))))
+Case conversion may be inaccurate. Consider using '#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mkₓ'. -/
+theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
   calc
     (#⋃ i, f i) = (#Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
     
-#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_unionᵢ_eq_sum_mk
+#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
-/- warning: cardinal.mk_Union_le -> Cardinal.mk_unionᵢ_le is a dubious translation:
+/- warning: cardinal.mk_Union_le -> Cardinal.mk_iUnion_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {ι : Type.{u1}} (f : ι -> (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.unionᵢ.{u1, succ u1} α ι (fun (i : ι) => f i)))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} ι) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι (fun (i : ι) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (f i)))))
+  forall {α : Type.{u1}} {ι : Type.{u1}} (f : ι -> (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.iUnion.{u1, succ u1} α ι (fun (i : ι) => f i)))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} ι) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι (fun (i : ι) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (f i)))))
 but is expected to have type
-  forall {α : Type.{u1}} {ι : Type.{u1}} (f : ι -> (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.unionᵢ.{u1, succ u1} α ι (fun (i : ι) => f i)))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} ι) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι (fun (i : ι) => Cardinal.mk.{u1} (Set.Elem.{u1} α (f i)))))
-Case conversion may be inaccurate. Consider using '#align cardinal.mk_Union_le Cardinal.mk_unionᵢ_leₓ'. -/
-theorem mk_unionᵢ_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
-  mk_unionᵢ_le_sum_mk.trans (sum_le_supᵢ _)
-#align cardinal.mk_Union_le Cardinal.mk_unionᵢ_le
+  forall {α : Type.{u1}} {ι : Type.{u1}} (f : ι -> (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.iUnion.{u1, succ u1} α ι (fun (i : ι) => f i)))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} ι) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι (fun (i : ι) => Cardinal.mk.{u1} (Set.Elem.{u1} α (f i)))))
+Case conversion may be inaccurate. Consider using '#align cardinal.mk_Union_le Cardinal.mk_iUnion_leₓ'. -/
+theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
+  mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
+#align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
-/- warning: cardinal.mk_sUnion_le -> Cardinal.mk_unionₛ_le is a dubious translation:
+/- warning: cardinal.mk_sUnion_le -> Cardinal.mk_sUnion_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (A : Set.{u1} (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.unionₛ.{u1} α A))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A)) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (fun (s : coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) Type.{u1} (coeSortTrans.{succ (succ u1), succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (coeBaseAux.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (Set.{u1} α) (coeSubtype.{succ u1} (Set.{u1} α) (fun (x : Set.{u1} α) => Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) x A)))) s))))
+  forall {α : Type.{u1}} (A : Set.{u1} (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.sUnion.{u1} α A))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (fun (s : coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) Type.{u1} (coeSortTrans.{succ (succ u1), succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (coeBaseAux.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} (Set.{u1} α)) Type.{u1} (Set.hasCoeToSort.{u1} (Set.{u1} α)) A) (Set.{u1} α) (coeSubtype.{succ u1} (Set.{u1} α) (fun (x : Set.{u1} α) => Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) x A)))) s))))
 but is expected to have type
-  forall {α : Type.{u1}} (A : Set.{u1} (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.unionₛ.{u1} α A))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} (Set.{u1} α) A)) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} (Set.{u1} α) A) (fun (s : Set.Elem.{u1} (Set.{u1} α) A) => Cardinal.mk.{u1} (Set.Elem.{u1} α (Subtype.val.{succ u1} (Set.{u1} α) (fun (x : Set.{u1} α) => Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) x A) s)))))
-Case conversion may be inaccurate. Consider using '#align cardinal.mk_sUnion_le Cardinal.mk_unionₛ_leₓ'. -/
-theorem mk_unionₛ_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s :=
+  forall {α : Type.{u1}} (A : Set.{u1} (Set.{u1} α)), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.sUnion.{u1} α A))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} (Set.{u1} α) A)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} (Set.{u1} α) A) (fun (s : Set.Elem.{u1} (Set.{u1} α) A) => Cardinal.mk.{u1} (Set.Elem.{u1} α (Subtype.val.{succ u1} (Set.{u1} α) (fun (x : Set.{u1} α) => Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) x A) s)))))
+Case conversion may be inaccurate. Consider using '#align cardinal.mk_sUnion_le Cardinal.mk_sUnion_leₓ'. -/
+theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s :=
   by
   rw [sUnion_eq_Union]
   apply mk_Union_le
-#align cardinal.mk_sUnion_le Cardinal.mk_unionₛ_le
+#align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
-/- warning: cardinal.mk_bUnion_le -> Cardinal.mk_bunionᵢ_le is a dubious translation:
+/- warning: cardinal.mk_bUnion_le -> Cardinal.mk_biUnion_le is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {α : Type.{u1}} (A : ι -> (Set.{u1} α)) (s : Set.{u1} ι), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.unionᵢ.{u1, succ u1} α ι (fun (x : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) => A x))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s)) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s) (fun (x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (A (Subtype.val.{succ u1} ι (fun (x : ι) => Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) x))))))
+  forall {ι : Type.{u1}} {α : Type.{u1}} (A : ι -> (Set.{u1} α)) (s : Set.{u1} ι), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.iUnion.{u1, succ u1} α ι (fun (x : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) (fun (H : Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) => A x))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s) (fun (x : coeSort.{succ u1, succ (succ u1)} (Set.{u1} ι) Type.{u1} (Set.hasCoeToSort.{u1} ι) s) => Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (A (Subtype.val.{succ u1} ι (fun (x : ι) => Membership.Mem.{u1, u1} ι (Set.{u1} ι) (Set.hasMem.{u1} ι) x s) x))))))
 but is expected to have type
-  forall {ι : Type.{u1}} {α : Type.{u1}} (A : ι -> (Set.{u1} α)) (s : Set.{u1} ι), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.unionᵢ.{u1, succ u1} α ι (fun (x : ι) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) => A x))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} ι s)) (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} ι s) (fun (x : Set.Elem.{u1} ι s) => Cardinal.mk.{u1} (Set.Elem.{u1} α (A (Subtype.val.{succ u1} ι (fun (x : ι) => Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) x))))))
-Case conversion may be inaccurate. Consider using '#align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_leₓ'. -/
-theorem mk_bunionᵢ_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
+  forall {ι : Type.{u1}} {α : Type.{u1}} (A : ι -> (Set.{u1} α)) (s : Set.{u1} ι), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.mk.{u1} (Set.Elem.{u1} α (Set.iUnion.{u1, succ u1} α ι (fun (x : ι) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) => A x))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) (Cardinal.mk.{u1} (Set.Elem.{u1} ι s)) (iSup.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (Set.Elem.{u1} ι s) (fun (x : Set.Elem.{u1} ι s) => Cardinal.mk.{u1} (Set.Elem.{u1} α (A (Subtype.val.{succ u1} ι (fun (x : ι) => Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) x s) x))))))
+Case conversion may be inaccurate. Consider using '#align cardinal.mk_bUnion_le Cardinal.mk_biUnion_leₓ'. -/
+theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
     (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 :=
   by
   rw [bUnion_eq_Union]
   apply mk_Union_le
-#align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_le
+#align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
 #print Cardinal.finset_card_lt_aleph0 /-
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
@@ -3482,7 +3482,7 @@ infixl:80 " ^< " => powerlt
 #print Cardinal.le_powerlt /-
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
-  apply @le_csupᵢ _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
+  apply @le_ciSup _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
@@ -3491,7 +3491,7 @@ theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
 #print Cardinal.powerlt_le /-
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
-  rw [powerlt, csupᵢ_le_iff']
+  rw [powerlt, ciSup_le_iff']
   · simp
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
@@ -3553,7 +3553,7 @@ theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 :=
 @[simp]
 theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 :=
   by
-  convert Cardinal.supᵢ_of_empty _
+  convert Cardinal.iSup_of_empty _
   exact Subtype.isEmpty_of_false fun x => (Cardinal.zero_le _).not_lt
 #align cardinal.powerlt_zero Cardinal.powerlt_zero
 -/

bump to nightly-2023-04-17-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

4784256b

Diff

@@ -575,16 +575,16 @@ theorem mk_option {α : Type u} : (#Option α) = (#α) + 1 :=
   (Equiv.optionEquivSumPUnit α).cardinal_eq
 #align cardinal.mk_option Cardinal.mk_option
 
-/- warning: cardinal.mk_psum -> Cardinal.mk_pSum is a dubious translation:
+/- warning: cardinal.mk_psum -> Cardinal.mk_psum is a dubious translation:
 lean 3 declaration is
   forall (α : Type.{u1}) (β : Type.{u2}), Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.mk.{max u1 u2} (PSum.{succ u1, succ u2} α β)) (HAdd.hAdd.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHAdd.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasAdd.{max u1 u2}) (Cardinal.lift.{u2, u1} (Cardinal.mk.{u1} α)) (Cardinal.lift.{u1, u2} (Cardinal.mk.{u2} β)))
 but is expected to have type
   forall (α : Type.{u1}) (β : Type.{u2}), Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u1 u2} (Cardinal.mk.{max u1 u2} (PSum.{succ u1, succ u2} α β)) (HAdd.hAdd.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u2 u1} Cardinal.{max u1 u2} (instHAdd.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instAddCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} (Cardinal.mk.{u1} α)) (Cardinal.lift.{u1, u2} (Cardinal.mk.{u2} β)))
-Case conversion may be inaccurate. Consider using '#align cardinal.mk_psum Cardinal.mk_pSumₓ'. -/
+Case conversion may be inaccurate. Consider using '#align cardinal.mk_psum Cardinal.mk_psumₓ'. -/
 @[simp]
-theorem mk_pSum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
+theorem mk_psum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
   (mk_congr (Equiv.psumEquivSum α β)).trans (mk_sum α β)
-#align cardinal.mk_psum Cardinal.mk_pSum
+#align cardinal.mk_psum Cardinal.mk_psum
 
 /- warning: cardinal.mk_fintype -> Cardinal.mk_fintype is a dubious translation:
 lean 3 declaration is
@@ -920,7 +920,7 @@ private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d →
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1})) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5036 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5038 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5036 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5038) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5051 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5053 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5051 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5053)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5063 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5063) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5076 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5078 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5076 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5078)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_covariant_class Cardinal.add_covariantClassₓ'. -/
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
@@ -930,7 +930,7 @@ instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}))) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5103 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5105 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5103 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5105)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5118 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5120 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5118 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5120)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5130 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5130)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5143 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5145 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5143 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5145)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClassₓ'. -/
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
@@ -2909,23 +2909,23 @@ theorem mk_empty : (#Empty) = 0 :=
 #align cardinal.mk_empty Cardinal.mk_empty
 -/
 
-#print Cardinal.mk_pEmpty /-
+#print Cardinal.mk_pempty /-
 @[simp]
-theorem mk_pEmpty : (#PEmpty) = 0 :=
+theorem mk_pempty : (#PEmpty) = 0 :=
   mk_eq_zero _
-#align cardinal.mk_pempty Cardinal.mk_pEmpty
+#align cardinal.mk_pempty Cardinal.mk_pempty
 -/
 
-#print Cardinal.mk_pUnit /-
+#print Cardinal.mk_punit /-
 @[simp]
-theorem mk_pUnit : (#PUnit) = 1 :=
+theorem mk_punit : (#PUnit) = 1 :=
   mk_eq_one PUnit
-#align cardinal.mk_punit Cardinal.mk_pUnit
+#align cardinal.mk_punit Cardinal.mk_punit
 -/
 
 #print Cardinal.mk_unit /-
 theorem mk_unit : (#Unit) = 1 :=
-  mk_pUnit
+  mk_punit
 #align cardinal.mk_unit Cardinal.mk_unit
 -/
 
@@ -2936,18 +2936,18 @@ theorem mk_singleton {α : Type u} (x : α) : (#({x} : Set α)) = 1 :=
 #align cardinal.mk_singleton Cardinal.mk_singleton
 -/
 
-#print Cardinal.mk_pLift_true /-
+#print Cardinal.mk_plift_true /-
 @[simp]
-theorem mk_pLift_true : (#PLift True) = 1 :=
+theorem mk_plift_true : (#PLift True) = 1 :=
   mk_eq_one _
-#align cardinal.mk_plift_true Cardinal.mk_pLift_true
+#align cardinal.mk_plift_true Cardinal.mk_plift_true
 -/
 
-#print Cardinal.mk_pLift_false /-
+#print Cardinal.mk_plift_false /-
 @[simp]
-theorem mk_pLift_false : (#PLift False) = 0 :=
+theorem mk_plift_false : (#PLift False) = 0 :=
   mk_eq_zero _
-#align cardinal.mk_plift_false Cardinal.mk_pLift_false
+#align cardinal.mk_plift_false Cardinal.mk_plift_false
 -/
 
 #print Cardinal.mk_vector /-

bump to nightly-2023-04-14-22

mathlib commit https://github.com/leanprover-community/mathlib/commit/7ebf83ed9c262adbf983ef64d5e8c2ae94b625f4

60fdc414

Diff

@@ -281,16 +281,12 @@ theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=
 #align cardinal.lift_uzero Cardinal.lift_uzero
 -/
 
-/- warning: cardinal.lift_lift -> Cardinal.lift_lift is a dubious translation:
-lean 3 declaration is
-  forall (a : Cardinal.{u3}), Eq.{succ (succ (max (max u3 u1) u2))} Cardinal.{max (max u3 u1) u2} (Cardinal.lift.{u2, max u3 u1} (Cardinal.lift.{u1, u3} a)) (Cardinal.lift.{max u1 u2, u3} a)
-but is expected to have type
-  forall (a : Cardinal.{u1}), Eq.{max (max (succ (succ u2)) (succ (succ u3))) (succ (succ u1))} Cardinal.{max (max u2 u1) u3} (Cardinal.lift.{u3, max u2 u1} (Cardinal.lift.{u2, u1} a)) (Cardinal.lift.{max u2 u3, u1} a)
-Case conversion may be inaccurate. Consider using '#align cardinal.lift_lift Cardinal.lift_liftₓ'. -/
+#print Cardinal.lift_lift /-
 @[simp]
 theorem lift_lift (a : Cardinal) : lift.{w} (lift.{v} a) = lift.{max v w} a :=
   inductionOn a fun α => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
 #align cardinal.lift_lift Cardinal.lift_lift
+-/
 
 /-- We define the order on cardinal numbers by `#α ≤ #β` if and only if
   there exists an embedding (injective function) from α to β. -/

bump to nightly-2023-04-13-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/347636a7a80595d55bedf6e6fbd996a3c39da69a

b6e2bcea

Diff

@@ -924,7 +924,7 @@ private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d →
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1})) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5044 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5046 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5044 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5046) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5059 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5059 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5036 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5038 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5036 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5038) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5051 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5053 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5051 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5053)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_covariant_class Cardinal.add_covariantClassₓ'. -/
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
@@ -934,7 +934,7 @@ instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}))) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5111 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5113 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5111 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5113)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5126 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5126 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5103 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5105 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5103 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5105)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5118 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5120 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5118 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5120)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClassₓ'. -/
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
@@ -2742,7 +2742,7 @@ def toNatHom : Cardinal →*₀ ℕ where
 lean 3 declaration is
   forall {α : Type.{u1}} (s : Finset.{u1} α) (f : α -> Cardinal.{u2}), Eq.{1} Nat (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} (Finset.prod.{succ u2, u1} Cardinal.{u2} α (CommSemiring.toCommMonoid.{succ u2} Cardinal.{u2} Cardinal.commSemiring.{u2}) s (fun (i : α) => f i))) (Finset.prod.{0, u1} Nat α Nat.commMonoid s (fun (i : α) => coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} (f i)))
 but is expected to have type
-  forall {α : Type.{u2}} (s : Finset.{u2} α) (f : α -> Cardinal.{u1}), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.commMonoid.{u1} s (fun (i : α) => f i))) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.commMonoid.{u1} s (fun (i : α) => f i))) (Finset.prod.{0, u2} Nat α Nat.commMonoid s (fun (i : α) => FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (f i)))
+  forall {α : Type.{u2}} (s : Finset.{u2} α) (f : α -> Cardinal.{u1}), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.instCommMonoidCardinal.{u1} s (fun (i : α) => f i))) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.instCommMonoidCardinal.{u1} s (fun (i : α) => f i))) (Finset.prod.{0, u2} Nat α Nat.commMonoid s (fun (i : α) => FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (f i)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prodₓ'. -/
 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=

bump to nightly-2023-04-13-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/347636a7a80595d55bedf6e6fbd996a3c39da69a

bd75793a

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit e05ead7993520a432bec94ac504842d90707ad63
+! leanprover-community/mathlib commit 9bb28972724354ac0574e2b318be896ec252025f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -299,17 +299,21 @@ instance : LE Cardinal.{u} :=
     Quotient.liftOn₂ q₁ q₂ (fun α β => Nonempty <| α ↪ β) fun α β γ δ ⟨e₁⟩ ⟨e₂⟩ =>
       propext ⟨fun ⟨e⟩ => ⟨e.congr e₁ e₂⟩, fun ⟨e⟩ => ⟨e.congr e₁.symm e₂.symm⟩⟩⟩
 
-instance : LinearOrder Cardinal.{u} where
+instance : PartialOrder Cardinal.{u} where
   le := (· ≤ ·)
   le_refl := by rintro ⟨α⟩ <;> exact ⟨embedding.refl _⟩
   le_trans := by rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.trans e₂⟩
   le_antisymm := by
     rintro ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩
     exact Quotient.sound (e₁.antisymm e₂)
-  le_total := by
-    rintro ⟨α⟩ ⟨β⟩
-    apply embedding.total
-  decidableLe := Classical.decRel _
+
+instance : LinearOrder Cardinal.{u} :=
+  {
+    Cardinal.partialOrder.{u} with
+    le_total := by
+      rintro ⟨α⟩ ⟨β⟩
+      apply embedding.total
+    decidableLe := Classical.decRel _ }
 
 #print Cardinal.le_def /-
 theorem le_def (α β : Type u) : (#α) ≤ (#β) ↔ Nonempty (α ↪ β) :=
@@ -439,7 +443,7 @@ theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
 
 /- warning: cardinal.lift_strict_mono -> Cardinal.lift_strictMono is a dubious translation:
 lean 3 declaration is
-  StrictMono.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (SemilatticeInf.toPartialOrder.{succ u1} Cardinal.{u1} (Lattice.toSemilatticeInf.{succ u1} Cardinal.{u1} (LinearOrder.toLattice.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1})))) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (SemilatticeInf.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (Lattice.toSemilatticeInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (LinearOrder.toLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2})))) Cardinal.lift.{u2, u1}
+  StrictMono.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.lift.{u2, u1}
 but is expected to have type
   StrictMono.{succ u2, max (succ u2) (succ u1)} Cardinal.{u2} Cardinal.{max u2 u1} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (PartialOrder.toPreorder.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.partialOrder.{max u2 u1}) Cardinal.lift.{u1, u2}
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_strict_mono Cardinal.lift_strictMonoₓ'. -/
@@ -448,7 +452,7 @@ theorem lift_strictMono : StrictMono lift := fun a b => lift_lt.2
 
 /- warning: cardinal.lift_monotone -> Cardinal.lift_monotone is a dubious translation:
 lean 3 declaration is
-  Monotone.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (SemilatticeInf.toPartialOrder.{succ u1} Cardinal.{u1} (Lattice.toSemilatticeInf.{succ u1} Cardinal.{u1} (LinearOrder.toLattice.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1})))) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (SemilatticeInf.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (Lattice.toSemilatticeInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (LinearOrder.toLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2})))) Cardinal.lift.{u2, u1}
+  Monotone.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.lift.{u2, u1}
 but is expected to have type
   Monotone.{succ u2, max (succ u2) (succ u1)} Cardinal.{u2} Cardinal.{max u2 u1} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (PartialOrder.toPreorder.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.partialOrder.{max u2 u1}) Cardinal.lift.{u1, u2}
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_monotone Cardinal.lift_monotoneₓ'. -/
@@ -938,7 +942,7 @@ instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· +
 
 instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
   { Cardinal.commSemiring,
-    Cardinal.linearOrder with
+    Cardinal.partialOrder with
     bot := 0
     bot_le := Cardinal.zero_le
     add_le_add_left := fun a b => add_le_add_left
@@ -955,12 +959,20 @@ instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
 instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : CanonicallyOrderedAddMonoid Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring with }
+
 instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=
   { Cardinal.commSemiring,
     Cardinal.linearOrder with
     mul_le_mul_left := @mul_le_mul_left' _ _ _ _
     zero_le_one := zero_le _ }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : CommMonoidWithZero Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring with }
+
 #print Cardinal.zero_power_le /-
 theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 :=
   by
@@ -992,7 +1004,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 
 /- warning: cardinal.cantor -> Cardinal.cantor is a dubious translation:
 lean 3 declaration is
-  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
+  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
 but is expected to have type
   forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (instOfNat.{succ u1} Cardinal.{u1} 2 Cardinal.instNatCastCardinal.{u1} (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) a)
 Case conversion may be inaccurate. Consider using '#align cardinal.cantor Cardinal.cantorₓ'. -/
@@ -1088,7 +1100,7 @@ instance : SuccOrder Cardinal :=
 
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c c')))
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 but is expected to have type
   forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
@@ -1098,7 +1110,7 @@ theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
 
 /- warning: cardinal.succ_pos -> Cardinal.succ_pos is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
+  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c)
 but is expected to have type
   forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_pos Cardinal.succ_posₓ'. -/
@@ -1108,7 +1120,7 @@ theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
 
 /- warning: cardinal.succ_ne_zero -> Cardinal.succ_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
+  forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
 but is expected to have type
   forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_ne_zero Cardinal.succ_ne_zeroₓ'. -/
@@ -1118,7 +1130,7 @@ theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
 
 /- warning: cardinal.add_one_le_succ -> Cardinal.add_one_le_succ is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
+  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c)
 but is expected to have type
   forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_one_le_succ Cardinal.add_one_le_succₓ'. -/
@@ -1160,7 +1172,7 @@ protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
 
 /- warning: cardinal.is_limit.succ_lt -> Cardinal.IsLimit.succ_lt is a dubious translation:
 lean 3 declaration is
-  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} x) c)
+  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} x) c)
 but is expected to have type
   forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} x) c)
 Case conversion may be inaccurate. Consider using '#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_ltₓ'. -/
@@ -1609,7 +1621,7 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 
 /- warning: cardinal.lift_succ -> Cardinal.lift_succ is a dubious translation:
 lean 3 declaration is
-  forall (a : Cardinal.{u1}), Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} a)) (Order.succ.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2}))))) Cardinal.succOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a))
+  forall (a : Cardinal.{u1}), Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} a)) (Order.succ.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.succOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a))
 but is expected to have type
   forall (a : Cardinal.{u1}), Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} a)) (Order.succ.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_succ Cardinal.lift_succₓ'. -/
@@ -1655,7 +1667,7 @@ theorem lift_max {a b : Cardinal} : lift (max a b) = max (lift a) (lift b) :=
 
 /- warning: cardinal.lift_Sup -> Cardinal.lift_supₛ is a dubious translation:
 lean 3 declaration is
-  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.supₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.supₛ.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
+  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.supₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.supₛ.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
 but is expected to have type
   forall {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) -> (Eq.{max (succ (succ u2)) (succ (succ u1))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (SupSet.supₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) s)) (SupSet.supₛ.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toSupSet.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) (Set.image.{succ u1, max (succ u2) (succ u1)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s)))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_Sup Cardinal.lift_supₛₓ'. -/
@@ -1674,7 +1686,7 @@ theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (supₛ s) =
 
 /- warning: cardinal.lift_supr -> Cardinal.lift_supᵢ is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (supᵢ.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (supᵢ.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{max (succ (succ u1)) (succ (succ u3))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (supᵢ.{max (succ u1) (succ u3), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toSupSet.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u3}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr Cardinal.lift_supᵢₓ'. -/
@@ -1685,7 +1697,7 @@ theorem lift_supᵢ {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (rang
 
 /- warning: cardinal.lift_supr_le -> Cardinal.lift_supᵢ_le is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u1 u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t)
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le Cardinal.lift_supᵢ_leₓ'. -/
@@ -1700,7 +1712,7 @@ theorem lift_supᵢ_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (h
 
 /- warning: cardinal.lift_supr_le_iff -> Cardinal.lift_supᵢ_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u1 u3}}, Iff (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t) (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_iff Cardinal.lift_supᵢ_le_iffₓ'. -/
@@ -1716,7 +1728,7 @@ universe v' w'
 
 /- warning: cardinal.lift_supr_le_lift_supr -> Cardinal.lift_supᵢ_le_lift_supᵢ is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} (OrderedAddCommMonoid.toPartialOrder.{succ u4} Cardinal.{u4} (OrderedSemiring.toOrderedAddCommMonoid.{succ u4} Cardinal.{u4} (OrderedCommSemiring.toOrderedSemiring.{succ u4} Cardinal.{u4} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u4} Cardinal.{u4} Cardinal.canonicallyOrderedCommSemiring.{u4}))))) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
 but is expected to have type
   forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toSupSet.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u4}))) ι' f'))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢₓ'. -/
@@ -1734,7 +1746,7 @@ theorem lift_supᵢ_le_lift_supᵢ {ι : Type v} {ι' : Type v'} {f : ι → Car
 
 /- warning: cardinal.lift_supr_le_lift_supr' -> Cardinal.lift_supᵢ_le_lift_supᵢ' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
 but is expected to have type
   forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι' f'))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_supᵢ_le_lift_supᵢ'ₓ'. -/
@@ -1878,7 +1890,7 @@ theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
 
 /- warning: cardinal.lift_lt_nat_iff -> Cardinal.lift_lt_nat_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2})))))) (Cardinal.lift.{u2, u1} a) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
+  forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
 but is expected to have type
   forall {a : Cardinal.{u1}} {n : Nat}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Nat.cast.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instNatCastCardinal.{max u1 u2} n)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iffₓ'. -/
@@ -1889,7 +1901,7 @@ theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n
 
 /- warning: cardinal.nat_lt_lift_iff -> Cardinal.nat_lt_lift_iff is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2})))))) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) a)
+  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) ((fun (a : Type) (b : Type.{succ (max u1 u2)}) [self : HasLiftT.{1, succ (succ (max u1 u2))} a b] => self.0) Nat Cardinal.{max u1 u2} (HasLiftT.mk.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (CoeTCₓ.coe.{1, succ (succ (max u1 u2))} Nat Cardinal.{max u1 u2} (Nat.castCoe.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasNatCast.{max u1 u2}))) n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) a)
 but is expected to have type
   forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Nat.cast.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instNatCastCardinal.{max u1 u2} n) (Cardinal.lift.{u2, u1} a)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n) a)
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iffₓ'. -/
@@ -1974,7 +1986,7 @@ theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
 
 /- warning: cardinal.nat_cast_lt -> Cardinal.natCast_lt is a dubious translation:
 lean 3 declaration is
-  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
+  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
 but is expected to have type
   forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} m) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (LT.lt.{0} Nat instLTNat m n)
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_cast_lt Cardinal.natCast_ltₓ'. -/
@@ -2007,7 +2019,7 @@ theorem natCast_injective : Injective (coe : ℕ → Cardinal) :=
 
 /- warning: cardinal.nat_succ -> Cardinal.nat_succ is a dubious translation:
 lean 3 declaration is
-  forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
+  forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
 but is expected to have type
   forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_succ Cardinal.nat_succₓ'. -/
@@ -2018,7 +2030,7 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ n :=
 
 /- warning: cardinal.succ_zero -> Cardinal.succ_zero is a dubious translation:
 lean 3 declaration is
-  Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))
+  Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))
 but is expected to have type
   Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_zero Cardinal.succ_zeroₓ'. -/
@@ -2062,7 +2074,7 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le
 
 /- warning: cardinal.nat_lt_aleph_0 -> Cardinal.nat_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
+  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
 but is expected to have type
   forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n) Cardinal.aleph0.{u1}
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0ₓ'. -/
@@ -2087,7 +2099,7 @@ theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
 
 /- warning: cardinal.lt_aleph_0 -> Cardinal.lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)))
+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)))
 but is expected to have type
   forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)))
 Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0 Cardinal.lt_aleph0ₓ'. -/
@@ -2145,7 +2157,7 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
 
 /- warning: cardinal.range_nat_cast -> Cardinal.range_natCast is a dubious translation:
 lean 3 declaration is
-  Eq.{succ (succ u1)} (Set.{succ u1} Cardinal.{u1}) (Set.range.{succ u1, 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))))) (Set.Iio.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.aleph0.{u1})
+  Eq.{succ (succ u1)} (Set.{succ u1} Cardinal.{u1}) (Set.range.{succ u1, 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))))) (Set.Iio.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.aleph0.{u1})
 but is expected to have type
   Eq.{succ (succ u1)} (Set.{succ u1} Cardinal.{u1}) (Set.range.{succ u1, 1} Cardinal.{u1} Nat (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1})) (Set.Iio.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.range_nat_cast Cardinal.range_natCastₓ'. -/
@@ -2232,7 +2244,7 @@ theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
 
 /- warning: cardinal.can_lift_cardinal_nat -> Cardinal.canLiftCardinalNat is a dubious translation:
 lean 3 declaration is
-  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) x Cardinal.aleph0.{u1})
+  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
 but is expected to have type
   CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1}) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNatₓ'. -/
@@ -2244,7 +2256,7 @@ instance canLiftCardinalNat : CanLift Cardinal ℕ coe fun x => x < ℵ₀ :=
 
 /- warning: cardinal.add_lt_aleph_0 -> Cardinal.add_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0ₓ'. -/
@@ -2255,7 +2267,7 @@ theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a +
 
 /- warning: cardinal.add_lt_aleph_0_iff -> Cardinal.add_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iffₓ'. -/
@@ -2296,7 +2308,7 @@ theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) :
 
 /- warning: cardinal.mul_lt_aleph_0 -> Cardinal.mul_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0ₓ'. -/
@@ -2307,7 +2319,7 @@ theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a *
 
 /- warning: cardinal.mul_lt_aleph_0_iff -> Cardinal.mul_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}))))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iffₓ'. -/
@@ -2359,7 +2371,7 @@ theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 
 
 /- warning: cardinal.mul_lt_aleph_0_iff_of_ne_zero -> Cardinal.mul_lt_aleph0_iff_of_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1})))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zeroₓ'. -/
@@ -2510,7 +2522,7 @@ def toNat : ZeroHom Cardinal ℕ :=
 
 /- warning: cardinal.to_nat_apply_of_lt_aleph_0 -> Cardinal.toNat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}} (h : LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}), Eq.{1} Nat (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (Classical.choose.{1} Nat (fun (x : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) x)) (Iff.mp (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))) (Cardinal.lt_aleph0.{u1} c) h))
+  forall {c : Cardinal.{u1}} (h : LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}), Eq.{1} Nat (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (Classical.choose.{1} Nat (fun (x : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) x)) (Iff.mp (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))) (Cardinal.lt_aleph0.{u1} c) h))
 but is expected to have type
   forall {c : Cardinal.{u1}} (h : LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c) (Classical.choose.{1} Nat (fun (x : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} x)) (Iff.mp (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))) (Cardinal.lt_aleph0.{u1} c) h))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0ₓ'. -/
@@ -2527,7 +2539,7 @@ theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0
 
 /- warning: cardinal.cast_to_nat_of_lt_aleph_0 -> Cardinal.cast_toNat_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)) c)
+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)) c)
 but is expected to have type
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)) c)
 Case conversion may be inaccurate. Consider using '#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0ₓ'. -/
@@ -2739,7 +2751,7 @@ theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
 
 /- warning: cardinal.to_nat_add_of_lt_aleph_0 -> Cardinal.toNat_add_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2})))))) b Cardinal.aleph0.{u2}) -> (Eq.{1} Nat (coeFn.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) (fun (_x : ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) => Cardinal.{max u1 u2} -> Nat) (ZeroHom.hasCoeToFun.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHAdd.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasAdd.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} a) (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} b)))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) b Cardinal.aleph0.{u2}) -> (Eq.{1} Nat (coeFn.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) (fun (_x : ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) => Cardinal.{max u1 u2} -> Nat) (ZeroHom.hasCoeToFun.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHAdd.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasAdd.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} a) (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} b)))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) b Cardinal.aleph0.{u2}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{max u1 u2}) => Nat) (HAdd.hAdd.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u2 u1} Cardinal.{max u1 u2} (instHAdd.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instAddCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (FunLike.coe.{succ (succ (max u1 u2)), succ (succ (max u1 u2)), 1} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{max u1 u2} (fun (_x : Cardinal.{max u1 u2}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{max u1 u2}) => Nat) _x) (ZeroHomClass.toFunLike.{succ (max u1 u2), succ (max u1 u2), 0} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u2 u1} Cardinal.{max u1 u2} (instHAdd.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instAddCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u2}) => Nat) b) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) (instHAdd.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) instAddNat) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} a) (FunLike.coe.{succ (succ u2), succ (succ u2), 1} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u2} (fun (_x : Cardinal.{u2}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u2}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u2, succ u2, 0} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u2} b)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0ₓ'. -/
@@ -2783,7 +2795,7 @@ def toPartENat : Cardinal →+ PartENat
 
 /- warning: cardinal.to_part_enat_apply_of_lt_aleph_0 -> Cardinal.toPartENat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
 but is expected to have type
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0ₓ'. -/
@@ -2867,7 +2879,7 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 
 /- warning: cardinal.sum_lt_prod -> Cardinal.sum_lt_prod is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2})))))) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2})))))) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
 but is expected to have type
   forall {ι : Type.{u2}} (f : ι -> Cardinal.{u1}) (g : ι -> Cardinal.{u1}), (forall (i : ι), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (f i) (g i)) -> (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u2, u1} ι f) (Cardinal.prod.{u2, u1} ι g))
 Case conversion may be inaccurate. Consider using '#align cardinal.sum_lt_prod Cardinal.sum_lt_prodₓ'. -/
@@ -3432,7 +3444,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
 
 /- warning: cardinal.exists_not_mem_of_length_lt -> Cardinal.exists_not_mem_of_length_lt is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
+  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
 but is expected to have type
   forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.mem.{u1, u1} α (List.{u1} α) (List.instMembershipList.{u1} α) z l)))
 Case conversion may be inaccurate. Consider using '#align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_ltₓ'. -/
@@ -3503,7 +3515,7 @@ theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun b c => powerlt_l
 
 /- warning: cardinal.powerlt_succ -> Cardinal.powerlt_succ is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b))
 Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_succ Cardinal.powerlt_succₓ'. -/

e05ead79

bump to nightly-2023-04-10-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/e05ead7993520a432bec94ac504842d90707ad63

e7234864

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 9dba31df156d9d65b9d78db449542ca73d147c68
+! leanprover-community/mathlib commit e05ead7993520a432bec94ac504842d90707ad63
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1086,6 +1086,56 @@ instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => infₛ { c' | c < c' }) fun a b =>
     ⟨lt_of_lt_of_le <| cinfₛ_mem <| exists_gt a, cinfₛ_le'⟩
 
+/- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
+lean 3 declaration is
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c c')))
+but is expected to have type
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
+Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
+theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
+  rfl
+#align cardinal.succ_def Cardinal.succ_def
+
+/- warning: cardinal.succ_pos -> Cardinal.succ_pos is a dubious translation:
+lean 3 declaration is
+  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
+but is expected to have type
+  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
+Case conversion may be inaccurate. Consider using '#align cardinal.succ_pos Cardinal.succ_posₓ'. -/
+theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
+  bot_lt_succ
+#align cardinal.succ_pos Cardinal.succ_pos
+
+/- warning: cardinal.succ_ne_zero -> Cardinal.succ_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
+but is expected to have type
+  forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))
+Case conversion may be inaccurate. Consider using '#align cardinal.succ_ne_zero Cardinal.succ_ne_zeroₓ'. -/
+theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
+  (succ_pos _).ne'
+#align cardinal.succ_ne_zero Cardinal.succ_ne_zero
+
+/- warning: cardinal.add_one_le_succ -> Cardinal.add_one_le_succ is a dubious translation:
+lean 3 declaration is
+  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
+but is expected to have type
+  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
+Case conversion may be inaccurate. Consider using '#align cardinal.add_one_le_succ Cardinal.add_one_le_succₓ'. -/
+theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
+  by
+  refine' (le_cinfₛ_iff'' (exists_gt c)).2 fun b hlt => _
+  rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
+  cases' le_of_lt hlt with f
+  have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
+  simp only [surjective, not_forall] at this
+  rcases this with ⟨b, hb⟩
+  calc
+    (#γ) + 1 = (#Option γ) := mk_option.symm
+    _ ≤ (#β) := (f.option_elim b hb).cardinal_le
+    
+#align cardinal.add_one_le_succ Cardinal.add_one_le_succ
+
 #print Cardinal.IsLimit /-
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
   cardinal by this definition, but `0` isn't.
@@ -1124,56 +1174,6 @@ theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
 #align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
 -/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
-theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
-  rfl
-#align cardinal.succ_def Cardinal.succ_def
-
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-Case conversion may be inaccurate. Consider using '#align cardinal.add_one_le_succ Cardinal.add_one_le_succₓ'. -/
-theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
-  by
-  refine' (le_cinfₛ_iff'' (exists_gt c)).2 fun b hlt => _
-  rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
-  cases' le_of_lt hlt with f
-  have : ¬surjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)
-  simp only [surjective, not_forall] at this
-  rcases this with ⟨b, hb⟩
-  calc
-    (#γ) + 1 = (#Option γ) := mk_option.symm
-    _ ≤ (#β) := (f.option_elim b hb).cardinal_le
-    
-#align cardinal.add_one_le_succ Cardinal.add_one_le_succ
-
-/- warning: cardinal.succ_pos -> Cardinal.succ_pos is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cardinal.succ_pos Cardinal.succ_posₓ'. -/
-theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
-  bot_lt_succ
-#align cardinal.succ_pos Cardinal.succ_pos
-
-/- warning: cardinal.succ_ne_zero -> Cardinal.succ_ne_zero is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cardinal.succ_ne_zero Cardinal.succ_ne_zeroₓ'. -/
-theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
-  (succ_pos _).ne'
-#align cardinal.succ_ne_zero Cardinal.succ_ne_zero
-
 #print Cardinal.sum /-
 /-- The indexed sum of cardinals is the cardinality of the
   indexed disjoint union, i.e. sigma type. -/
@@ -2116,6 +2116,33 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       exact (Nat.lt_succ_self _).not_le (nat_cast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
+#print Cardinal.isSuccLimit_aleph0 /-
+theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
+  isSuccLimit_of_succ_lt fun a ha =>
+    by
+    rcases lt_aleph_0.1 ha with ⟨n, rfl⟩
+    rw [← nat_succ]
+    apply nat_lt_aleph_0
+#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
+-/
+
+#print Cardinal.isLimit_aleph0 /-
+theorem isLimit_aleph0 : IsLimit ℵ₀ :=
+  ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
+#align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
+-/
+
+#print Cardinal.IsLimit.aleph0_le /-
+theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
+  by
+  by_contra' h'
+  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
+  · exact h.ne_zero.irrefl
+  · rw [nat_succ] at h
+    exact not_is_succ_limit_succ _ h.is_succ_limit
+#align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
+-/
+
 /- warning: cardinal.range_nat_cast -> Cardinal.range_natCast is a dubious translation:
 lean 3 declaration is
   Eq.{succ (succ u1)} (Set.{succ u1} Cardinal.{u1}) (Set.range.{succ u1, 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))))) (Set.Iio.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.aleph0.{u1})
@@ -2165,33 +2192,6 @@ theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
-#print Cardinal.isSuccLimit_aleph0 /-
-theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
-  isSuccLimit_of_succ_lt fun a ha =>
-    by
-    rcases lt_aleph_0.1 ha with ⟨n, rfl⟩
-    rw [← nat_succ]
-    apply nat_lt_aleph_0
-#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
--/
-
-#print Cardinal.isLimit_aleph0 /-
-theorem isLimit_aleph0 : IsLimit ℵ₀ :=
-  ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
-#align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
--/
-
-#print Cardinal.IsLimit.aleph0_le /-
-theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
-  by
-  by_contra' h'
-  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
-  · exact h.ne_zero.irrefl
-  · rw [nat_succ] at h
-    exact not_is_succ_limit_succ _ h.is_succ_limit
-#align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
--/
-
 #print Cardinal.lt_aleph0_iff_subtype_finite /-
 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=

bump to nightly-2023-04-10-00

mathlib commit https://github.com/leanprover-community/mathlib/commit/284fdd2962e67d2932fa3a79ce19fcf92d38e228

542f5cb1

Diff

@@ -430,16 +430,12 @@ theorem lift_inj {a b : Cardinal} : lift a = lift b ↔ a = b :=
 #align cardinal.lift_inj Cardinal.lift_inj
 -/
 
-/- warning: cardinal.lift_lt -> Cardinal.lift_lt is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt Cardinal.lift_ltₓ'. -/
+#print Cardinal.lift_lt /-
 @[simp]
 theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
   liftOrderEmbedding.lt_iff_lt
 #align cardinal.lift_lt Cardinal.lift_lt
+-/
 
 /- warning: cardinal.lift_strict_mono -> Cardinal.lift_strictMono is a dubious translation:
 lean 3 declaration is
@@ -924,7 +920,7 @@ private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d →
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1})) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4896 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4898 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4896 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4898) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4911 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4913 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4911 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4913)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5044 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5046 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5044 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5046) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5059 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5059 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5061)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_covariant_class Cardinal.add_covariantClassₓ'. -/
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
@@ -934,7 +930,7 @@ instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}))) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
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+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5111 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5113 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5111 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5113)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5126 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5126 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5128)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClassₓ'. -/
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
@@ -1015,15 +1011,11 @@ instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := by infer_instance
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivialₓ'. -/
+#print Cardinal.one_lt_iff_nontrivial /-
 theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
+-/
 
 /- warning: cardinal.power_le_max_power_one -> Cardinal.power_le_max_power_one is a dubious translation:
 lean 3 declaration is
@@ -1044,24 +1036,15 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 -/
 
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+#print Cardinal.power_pos /-
 theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
+-/
 
 end OrderProperties
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.lt_wf Cardinal.lt_wfₓ'. -/
+#print Cardinal.lt_wf /-
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -1074,18 +1057,15 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [f, mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
+-/
 
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.wo Cardinal.woₓ'. -/
+#print Cardinal.wo /-
 instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 #align cardinal.wo Cardinal.wo
+-/
 
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
@@ -1122,9 +1102,11 @@ protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
 #align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
 -/
 
+#print Cardinal.IsLimit.isSuccLimit /-
 protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
   h.2
 #align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
+-/
 
 /- warning: cardinal.is_limit.succ_lt -> Cardinal.IsLimit.succ_lt is a dubious translation:
 lean 3 declaration is
@@ -1136,9 +1118,11 @@ theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
   h.IsSuccLimit.succ_lt
 #align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
 
+#print Cardinal.isSuccLimit_zero /-
 theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
   isSuccLimit_bot
 #align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
+-/
 
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
@@ -1324,12 +1308,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
         (hf b)
 #align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
 
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+#print Cardinal.bddAbove_range /-
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
     inputs, is always bounded above. -/
 theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove (Set.range f) :=
@@ -1337,6 +1316,7 @@ theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove
     rintro a ⟨i, rfl⟩
     exact le_sum f i⟩
 #align cardinal.bdd_above_range Cardinal.bddAbove_range
+-/
 
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) :=
   by
@@ -1348,12 +1328,7 @@ instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) :=
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=
   small_subset Iio_subset_Iic_self
 
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+#print Cardinal.bddAbove_iff_small /-
 /-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to an usual ZFC set. -/
 theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :=
   ⟨fun ⟨a, ha⟩ => @small_subset _ (Iic a) s (fun x h => ha h) _,
@@ -1370,42 +1345,31 @@ theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :
     · simp_rw [Subtype.val_eq_coe, Equiv.symm_apply_apply]
       rfl⟩
 #align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_small
+-/
 
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+#print Cardinal.bddAbove_of_small /-
 theorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s :=
   bddAbove_iff_small.2 h
 #align cardinal.bdd_above_of_small Cardinal.bddAbove_of_small
+-/
 
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+#print Cardinal.bddAbove_image /-
 theorem bddAbove_image (f : Cardinal.{u} → Cardinal.{max u v}) {s : Set Cardinal.{u}}
     (hs : BddAbove s) : BddAbove (f '' s) :=
   by
   rw [bdd_above_iff_small] at hs⊢
   exact small_lift _
 #align cardinal.bdd_above_image Cardinal.bddAbove_image
+-/
 
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+#print Cardinal.bddAbove_range_comp /-
 theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbove (range f))
     (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) :=
   by
   rw [range_comp]
   exact bdd_above_image g hf
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
+-/
 
 /- warning: cardinal.supr_le_sum -> Cardinal.supᵢ_le_sum is a dubious translation:
 lean 3 declaration is
@@ -1633,12 +1597,7 @@ theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.le_lift_iff Cardinal.le_lift_iff
 -/
 
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+#print Cardinal.lt_lift_iff /-
 theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
   ⟨fun h =>
@@ -1646,6 +1605,7 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
     fun ⟨a', e, h⟩ => e ▸ lift_lt.2 h⟩
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
+-/
 
 /- warning: cardinal.lift_succ -> Cardinal.lift_succ is a dubious translation:
 lean 3 declaration is
@@ -1808,15 +1768,11 @@ theorem aleph0_ne_zero : ℵ₀ ≠ 0 :=
 #align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero
 -/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_pos Cardinal.aleph0_posₓ'. -/
+#print Cardinal.aleph0_pos /-
 theorem aleph0_pos : 0 < ℵ₀ :=
   pos_iff_ne_zero.2 aleph0_ne_zero
 #align cardinal.aleph_0_pos Cardinal.aleph0_pos
+-/
 
 #print Cardinal.lift_aleph0 /-
 @[simp]
@@ -1839,15 +1795,19 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 -/
 
+#print Cardinal.aleph0_lt_lift /-
 @[simp]
 theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
+-/
 
+#print Cardinal.lift_lt_aleph0 /-
 @[simp]
 theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
   rw [← lift_aleph_0, lift_lt]
 #align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
+-/
 
 /-! ### Properties about the cast from `ℕ` -/
 
@@ -1894,21 +1854,45 @@ theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a := by rw [← lift_natCast.{v} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
 
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 @[simp]
 theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
   simp only [← lift_nat_cast, lift_le]
 #align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
 
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 @[simp]
 theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) ≤ lift.{v} a ↔ (n : Cardinal) ≤ a := by simp only [← lift_nat_cast, lift_le]
 #align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
 
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 @[simp]
 theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
   simp only [← lift_nat_cast, lift_lt]
 #align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
 
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 @[simp]
 theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) < lift.{v} a ↔ (n : Cardinal) < a := by simp only [← lift_nat_cast, lift_lt]
@@ -1936,32 +1920,24 @@ theorem mk_coe_finset {α : Type u} {s : Finset α} : (#s) = ↑(Finset.card s)
 lean 3 declaration is
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 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintypeₓ'. -/
 theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintypeₓ'. -/
+#print Cardinal.mk_finsupp_lift_of_fintype /-
 @[simp]
 theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zero β] :
     (#α →₀ β) = lift.{u} (#β) ^ℕ Fintype.card α := by
   simpa using (@Finsupp.equivFunOnFinite α β _ _).cardinal_eq
 #align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintype
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintypeₓ'. -/
+#print Cardinal.mk_finsupp_of_fintype /-
 theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
     (#α →₀ β) = (#β) ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype
+-/
 
 /- warning: cardinal.card_le_of_finset -> Cardinal.card_le_of_finset is a dubious translation:
 lean 3 declaration is
@@ -2066,26 +2042,18 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.cantor' Cardinal.cantor'ₓ'. -/
+#print Cardinal.cantor' /-
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.one_le_iff_pos Cardinal.one_le_iff_posₓ'. -/
+#print Cardinal.one_le_iff_pos /-
 theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by rw [← succ_zero, succ_le_iff]
 #align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos
+-/
 
 #print Cardinal.one_le_iff_ne_zero /-
 theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le_iff_pos, pos_iff_ne_zero]
@@ -2105,15 +2073,11 @@ theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0ₓ'. -/
+#print Cardinal.one_lt_aleph0 /-
 @[simp]
 theorem one_lt_aleph0 : 1 < ℵ₀ := by simpa using nat_lt_aleph_0 1
 #align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0
+-/
 
 #print Cardinal.one_le_aleph0 /-
 theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
@@ -2173,56 +2137,35 @@ theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ F
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finiteₓ'. -/
+#print Cardinal.lt_aleph0_iff_finite /-
 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintypeₓ'. -/
+#print Cardinal.lt_aleph0_iff_fintype /-
 theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finiteₓ'. -/
+#print Cardinal.lt_aleph0_of_finite /-
 theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finiteₓ'. -/
+#print Cardinal.lt_aleph0_iff_set_finite /-
 @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
+-/
 
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-Case conversion may be inaccurate. Consider using '#align set.finite.lt_aleph_0 Set.Finite.lt_aleph0ₓ'. -/
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
+#print Cardinal.isSuccLimit_aleph0 /-
 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
   isSuccLimit_of_succ_lt fun a ha =>
     by
@@ -2230,6 +2173,7 @@ theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
     rw [← nat_succ]
     apply nat_lt_aleph_0
 #align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
+-/
 
 #print Cardinal.isLimit_aleph0 /-
 theorem isLimit_aleph0 : IsLimit ℵ₀ :=
@@ -2248,16 +2192,12 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
 #align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
 -/
 
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+#print Cardinal.lt_aleph0_iff_subtype_finite /-
 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
+-/
 
 #print Cardinal.mk_le_aleph0_iff /-
 theorem mk_le_aleph0_iff : (#α) ≤ ℵ₀ ↔ Countable α := by
@@ -2334,12 +2274,7 @@ theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a 
   simp only [← not_lt, add_lt_aleph_0_iff, not_and_or]
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iffₓ'. -/
+#print Cardinal.nsmul_lt_aleph0_iff /-
 /-- See also `cardinal.nsmul_lt_aleph_0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
 theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ :=
   by
@@ -2350,17 +2285,14 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   · simp
   rw [succ_nsmul, add_lt_aleph_0_iff, ih, and_self_iff]
 #align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff
+-/
 
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+#print Cardinal.nsmul_lt_aleph0_iff_of_ne_zero /-
 /-- See also `cardinal.nsmul_lt_aleph_0_iff` for a hypothesis-free version. -/
 theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ :=
   nsmul_lt_aleph0_iff.trans <| or_iff_right h
 #align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero
+-/
 
 /- warning: cardinal.mul_lt_aleph_0 -> Cardinal.mul_lt_aleph0 is a dubious translation:
 lean 3 declaration is
@@ -2435,16 +2367,12 @@ theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
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+#print Cardinal.power_lt_aleph0 /-
 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← nat_cast_pow] <;> apply nat_lt_aleph_0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
+-/
 
 #print Cardinal.eq_one_iff_unique /-
 theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
@@ -2617,49 +2545,33 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat =
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
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+#print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
   rw [← nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
+-/
 
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+#print Cardinal.toNat_lt_iff_lt_of_lt_aleph0 /-
 theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat < d.toNat ↔ c < d := by
   rw [← nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0ₓ'. -/
+#print Cardinal.toNat_le_of_le_of_lt_aleph0 /-
 theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
     c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
 #align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0ₓ'. -/
+#print Cardinal.toNat_lt_of_lt_of_lt_aleph0 /-
 theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
     c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
+-/
 
 /- warning: cardinal.to_nat_cast -> Cardinal.toNat_cast is a dubious translation:
 lean 3 declaration is
@@ -2803,12 +2715,7 @@ theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
     exact aleph_0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)
 #align cardinal.to_nat_mul Cardinal.toNat_mul
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_hom Cardinal.toNatHomₓ'. -/
+#print Cardinal.toNatHom /-
 /-- `cardinal.to_nat` as a `monoid_with_zero_hom`. -/
 @[simps]
 def toNatHom : Cardinal →*₀ ℕ where
@@ -2817,12 +2724,13 @@ def toNatHom : Cardinal →*₀ ℕ where
   map_one' := one_toNat
   map_mul' := toNat_mul
 #align cardinal.to_nat_hom Cardinal.toNatHom
+-/
 
 /- warning: cardinal.to_nat_finset_prod -> Cardinal.toNat_finset_prod is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} (s : Finset.{u1} α) (f : α -> Cardinal.{u2}), Eq.{1} Nat (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} (Finset.prod.{succ u2, u1} Cardinal.{u2} α (CommSemiring.toCommMonoid.{succ u2} Cardinal.{u2} Cardinal.commSemiring.{u2}) s (fun (i : α) => f i))) (Finset.prod.{0, u1} Nat α Nat.commMonoid s (fun (i : α) => coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} (f i)))
 but is expected to have type
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+  forall {α : Type.{u2}} (s : Finset.{u2} α) (f : α -> Cardinal.{u1}), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.commMonoid.{u1} s (fun (i : α) => f i))) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (Finset.prod.{succ u1, u2} Cardinal.{u1} α Cardinal.commMonoid.{u1} s (fun (i : α) => f i))) (Finset.prod.{0, u2} Nat α Nat.commMonoid s (fun (i : α) => FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} (f i)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prodₓ'. -/
 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
@@ -2850,7 +2758,7 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
 lean 3 declaration is
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 but is expected to have type
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+  AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat Cardinal.toPartENatₓ'. -/
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to `⊤`. -/
@@ -2877,7 +2785,7 @@ def toPartENat : Cardinal →+ PartENat
 lean 3 declaration is
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
 but is expected to have type
-  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0ₓ'. -/
 theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPartENat = c.toNat :=
   if_pos h
@@ -2887,7 +2795,7 @@ theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPart
 lean 3 declaration is
   forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) (Top.top.{0} PartENat PartENat.hasTop))
 but is expected to have type
-  forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instTopPartENat))
+  forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instTopPartENat))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_leₓ'. -/
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPartENat = ⊤ :=
   if_neg h.not_lt
@@ -2897,7 +2805,7 @@ theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPa
 lean 3 declaration is
   forall (n : Nat), Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) n)
 but is expected to have type
-  forall (n : Nat), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) PartENat.instAddCommMonoidWithOnePartENat)) n)
+  forall (n : Nat), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) PartENat.instAddCommMonoidWithOnePartENat)) n)
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_cast Cardinal.toPartENat_castₓ'. -/
 @[simp]
 theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
@@ -2908,7 +2816,7 @@ theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
 lean 3 declaration is
   forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} PartENat PartENat.hasTop)
 but is expected to have type
-  forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instTopPartENat)
+  forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instTopPartENat)
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infiniteₓ'. -/
 @[simp]
 theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
@@ -2919,7 +2827,7 @@ theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
 lean 3 declaration is
   Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} PartENat PartENat.hasTop)
 but is expected to have type
-  Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) PartENat.instTopPartENat)
+  Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) PartENat.instTopPartENat)
 Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENatₓ'. -/
 @[simp]
 theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
@@ -2930,7 +2838,7 @@ theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
 lean 3 declaration is
   Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1})
 but is expected to have type
-  Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1})
+  Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjectiveₓ'. -/
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
@@ -2940,7 +2848,7 @@ theorem toPartENat_surjective : Surjective toPartENat := fun x =>
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (Fintype.card.{u1} α _inst_1))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} 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 Case conversion may be inaccurate. Consider using '#align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_cardₓ'. -/
 theorem mk_toPartENat_eq_coe_card [Fintype α] : (#α).toPartENat = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card
@@ -3034,29 +2942,21 @@ theorem mk_pLift_false : (#PLift False) = 0 :=
 #align cardinal.mk_plift_false Cardinal.mk_pLift_false
 -/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.mk_vector Cardinal.mk_vectorₓ'. -/
+#print Cardinal.mk_vector /-
 @[simp]
 theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
   (mk_congr (Equiv.vectorEquivFin α n)).trans <| by simp
 #align cardinal.mk_vector Cardinal.mk_vector
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_powₓ'. -/
+#print Cardinal.mk_list_eq_sum_pow /-
 theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α) ^ℕ n :=
   calc
     (#List α) = (#Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
     _ = sum fun n : ℕ => (#α) ^ℕ n := by simp
     
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
+-/
 
 #print Cardinal.mk_quot_le /-
 theorem mk_quot_le {α : Type u} {r : α → α → Prop} : (#Quot r) ≤ (#α) :=
@@ -3213,15 +3113,11 @@ theorem mk_bunionᵢ_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
   apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_le
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0ₓ'. -/
+#print Cardinal.finset_card_lt_aleph0 /-
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
+-/
 
 /- warning: cardinal.mk_set_eq_nat_iff_finset -> Cardinal.mk_set_eq_nat_iff_finset is a dubious translation:
 lean 3 declaration is
@@ -3575,25 +3471,16 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 -- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.le_powerlt Cardinal.le_powerltₓ'. -/
+#print Cardinal.le_powerlt /-
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
   apply @le_csupᵢ _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
+-/
 
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-Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_le Cardinal.powerlt_leₓ'. -/
+#print Cardinal.powerlt_le /-
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
   rw [powerlt, csupᵢ_le_iff']
@@ -3601,6 +3488,7 @@ theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) 
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.powerlt_le Cardinal.powerlt_le
+-/
 
 #print Cardinal.powerlt_le_powerlt_left /-
 theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a ^< c :=
@@ -3608,14 +3496,10 @@ theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a
 #align cardinal.powerlt_le_powerlt_left Cardinal.powerlt_le_powerlt_left
 -/
 
-/- warning: cardinal.powerlt_mono_left -> Cardinal.powerlt_mono_left is a dubious translation:
-lean 3 declaration is
-  forall (a : Cardinal.{u1}), Monotone.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (fun (c : Cardinal.{u1}) => Cardinal.powerlt.{u1} a c)
-but is expected to have type
-  forall (a : Cardinal.{u1}), Monotone.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (fun (c : Cardinal.{u1}) => Cardinal.powerlt.{u1} a c)
-Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_mono_left Cardinal.powerlt_mono_leftₓ'. -/
+#print Cardinal.powerlt_mono_left /-
 theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun b c => powerlt_le_powerlt_left
 #align cardinal.powerlt_mono_left Cardinal.powerlt_mono_left
+-/
 
 /- warning: cardinal.powerlt_succ -> Cardinal.powerlt_succ is a dubious translation:
 lean 3 declaration is

bump to nightly-2023-04-08-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/dc65937e7a0fff4677f0f5a7086f42da70a69575

cd9aed2d

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit e08a42b2dd544cf11eba72e5fc7bf199d4349925
+! leanprover-community/mathlib commit 9dba31df156d9d65b9d78db449542ca73d147c68
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1894,6 +1894,26 @@ theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a := by rw [← lift_natCast.{v} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
 
+@[simp]
+theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
+  simp only [← lift_nat_cast, lift_le]
+#align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
+
+@[simp]
+theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} :
+    (n : Cardinal) ≤ lift.{v} a ↔ (n : Cardinal) ≤ a := by simp only [← lift_nat_cast, lift_le]
+#align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
+
+@[simp]
+theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
+  simp only [← lift_nat_cast, lift_lt]
+#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
+
+@[simp]
+theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} :
+    (n : Cardinal) < lift.{v} a ↔ (n : Cardinal) < a := by simp only [← lift_nat_cast, lift_lt]
+#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
+
 /- warning: cardinal.lift_mk_fin -> Cardinal.lift_mk_fin is a dubious translation:
 lean 3 declaration is
   forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.lift.{u1, 0} (Cardinal.mk.{0} (Fin n))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)

e08a42b2

bump to nightly-2023-04-07-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/5ec62c8106221a3f9160e4e4fcc3eed79fe213e9

3f979973

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f
+! leanprover-community/mathlib commit e08a42b2dd544cf11eba72e5fc7bf199d4349925
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1839,6 +1839,16 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 -/
 
+@[simp]
+theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
+  rw [← lift_aleph_0, lift_lt]
+#align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
+
+@[simp]
+theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
+  rw [← lift_aleph_0, lift_lt]
+#align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
+
 /-! ### Properties about the cast from `ℕ` -/
 
 
@@ -2737,9 +2747,9 @@ theorem toNat_lift (c : Cardinal.{v}) : (lift.{u, v} c).toNat = c.toNat :=
   apply nat_cast_injective
   cases' lt_or_ge c ℵ₀ with hc hc
   · rw [cast_to_nat_of_lt_aleph_0, ← lift_nat_cast, cast_to_nat_of_lt_aleph_0 hc]
-    rwa [← lift_aleph_0, lift_lt]
+    rwa [lift_lt_aleph_0]
   · rw [cast_to_nat_of_aleph_0_le, ← lift_nat_cast, cast_to_nat_of_aleph_0_le hc, lift_zero]
-    rwa [← lift_aleph_0, lift_le]
+    rwa [aleph_0_le_lift]
 #align cardinal.to_nat_lift Cardinal.toNat_lift
 -/
 
@@ -2810,12 +2820,8 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
     (lift.{v, u} a + lift.{u, v} b).toNat = a.toNat + b.toNat :=
   by
   apply Cardinal.natCast_injective
-  replace ha : lift.{v, u} a < ℵ₀ := by
-    rw [← lift_aleph_0]
-    exact lift_lt.2 ha
-  replace hb : lift.{u, v} b < ℵ₀ := by
-    rw [← lift_aleph_0]
-    exact lift_lt.2 hb
+  replace ha : lift.{v, u} a < ℵ₀ := by rwa [lift_lt_aleph_0]
+  replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph_0]
   rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_to_nat_of_lt_aleph_0 ha,
     cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0 (add_lt_aleph_0 ha hb)]
 #align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0

7c2ce0c2

bump to nightly-2023-04-04-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/3cacc945118c8c637d89950af01da78307f59325

2ee17135

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit ea050b44c0f9aba9d16a948c7cc7d2e7c8493567
+! leanprover-community/mathlib commit 7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,7 +14,7 @@ import Mathbin.Data.Nat.PartEnat
 import Mathbin.Data.Set.Countable
 import Mathbin.Logic.Small.Basic
 import Mathbin.Order.ConditionallyCompleteLattice.Basic
-import Mathbin.Order.SuccPred.Basic
+import Mathbin.Order.SuccPred.Limit
 import Mathbin.SetTheory.Cardinal.SchroederBernstein
 import Mathbin.Tactic.Positivity
 
@@ -35,6 +35,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 * Multiplication `c₁ * c₂` is defined by `cardinal.mul_def : #α * #β = #(α × β)`.
 * The order `c₁ ≤ c₂` is defined by `cardinal.le_def α β : #α ≤ #β ↔ nonempty (α ↪ β)`.
 * Exponentiation `c₁ ^ c₂` is defined by `cardinal.power_def α β : #α ^ #β = #(β → α)`.
+* `cardinal.is_limit c` means that `c` is a (weak) limit cardinal: `c ≠ 0 ∧ ∀ x < c, succ x < c`.
 * `cardinal.aleph_0` or `ℵ₀` is the cardinality of `ℕ`. This definition is universe polymorphic:
   `cardinal.aleph_0.{u} : cardinal.{u}` (contrast with `ℕ : Type`, which lives in a specific
   universe). In some cases the universe level has to be given explicitly.
@@ -1105,6 +1106,40 @@ instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => infₛ { c' | c < c' }) fun a b =>
     ⟨lt_of_lt_of_le <| cinfₛ_mem <| exists_gt a, cinfₛ_le'⟩
 
+#print Cardinal.IsLimit /-
+/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
+  cardinal by this definition, but `0` isn't.
+
+  Use `is_succ_limit` if you want to include the `c = 0` case. -/
+def IsLimit (c : Cardinal) : Prop :=
+  c ≠ 0 ∧ IsSuccLimit c
+#align cardinal.is_limit Cardinal.IsLimit
+-/
+
+#print Cardinal.IsLimit.ne_zero /-
+protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
+  h.1
+#align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
+-/
+
+protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
+  h.2
+#align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
+
+/- warning: cardinal.is_limit.succ_lt -> Cardinal.IsLimit.succ_lt is a dubious translation:
+lean 3 declaration is
+  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} x) c)
+but is expected to have type
+  forall {x : Cardinal.{u1}} {c : Cardinal.{u1}}, (Cardinal.IsLimit.{u1} c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x c) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} x) c)
+Case conversion may be inaccurate. Consider using '#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_ltₓ'. -/
+theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
+  h.IsSuccLimit.succ_lt
+#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
+
+theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
+  isSuccLimit_bot
+#align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
+
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
   forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c c')))
@@ -2158,6 +2193,31 @@ Case conversion may be inaccurate. Consider using '#align set.finite.lt_aleph_0
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
+theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
+  isSuccLimit_of_succ_lt fun a ha =>
+    by
+    rcases lt_aleph_0.1 ha with ⟨n, rfl⟩
+    rw [← nat_succ]
+    apply nat_lt_aleph_0
+#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
+
+#print Cardinal.isLimit_aleph0 /-
+theorem isLimit_aleph0 : IsLimit ℵ₀ :=
+  ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
+#align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
+-/
+
+#print Cardinal.IsLimit.aleph0_le /-
+theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
+  by
+  by_contra' h'
+  rcases lt_aleph_0.1 h' with ⟨_ | n, rfl⟩
+  · exact h.ne_zero.irrefl
+  · rw [nat_succ] at h
+    exact not_is_succ_limit_succ _ h.is_succ_limit
+#align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
+-/
+
 /- warning: cardinal.lt_aleph_0_iff_subtype_finite -> Cardinal.lt_aleph0_iff_subtype_finite is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} {p : α -> Prop}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} (Subtype.{succ u1} α (fun (x : α) => p x))) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α (setOf.{u1} α (fun (x : α) => p x)))

ea050b44

bump to nightly-2023-04-03-02

mathlib commit https://github.com/leanprover-community/mathlib/commit/d95bef0d215ea58c0fd7bbc4b151bf3fe952c095

8477bcac

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 34ee86e6a59d911a8e4f89b68793ee7577ae79c7
+! leanprover-community/mathlib commit ea050b44c0f9aba9d16a948c7cc7d2e7c8493567
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -298,13 +298,17 @@ instance : LE Cardinal.{u} :=
     Quotient.liftOn₂ q₁ q₂ (fun α β => Nonempty <| α ↪ β) fun α β γ δ ⟨e₁⟩ ⟨e₂⟩ =>
       propext ⟨fun ⟨e⟩ => ⟨e.congr e₁ e₂⟩, fun ⟨e⟩ => ⟨e.congr e₁.symm e₂.symm⟩⟩⟩
 
-instance : PartialOrder Cardinal.{u} where
+instance : LinearOrder Cardinal.{u} where
   le := (· ≤ ·)
   le_refl := by rintro ⟨α⟩ <;> exact ⟨embedding.refl _⟩
   le_trans := by rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ ⟨e₁⟩ ⟨e₂⟩ <;> exact ⟨e₁.trans e₂⟩
   le_antisymm := by
     rintro ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩
     exact Quotient.sound (e₁.antisymm e₂)
+  le_total := by
+    rintro ⟨α⟩ ⟨β⟩
+    apply embedding.total
+  decidableLe := Classical.decRel _
 
 #print Cardinal.le_def /-
 theorem le_def (α β : Type u) : (#α) ≤ (#β) ↔ Nonempty (α ↪ β) :=
@@ -425,16 +429,20 @@ theorem lift_inj {a b : Cardinal} : lift a = lift b ↔ a = b :=
 #align cardinal.lift_inj Cardinal.lift_inj
 -/
 
-#print Cardinal.lift_lt /-
+/- warning: cardinal.lift_lt -> Cardinal.lift_lt is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (SemilatticeInf.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (Lattice.toSemilatticeInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (LinearOrder.toLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2}))))) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (SemilatticeInf.toPartialOrder.{succ u1} Cardinal.{u1} (Lattice.toSemilatticeInf.{succ u1} Cardinal.{u1} (LinearOrder.toLattice.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1}))))) a b)
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a b)
+Case conversion may be inaccurate. Consider using '#align cardinal.lift_lt Cardinal.lift_ltₓ'. -/
 @[simp]
 theorem lift_lt {a b : Cardinal} : lift a < lift b ↔ a < b :=
   liftOrderEmbedding.lt_iff_lt
 #align cardinal.lift_lt Cardinal.lift_lt
--/
 
 /- warning: cardinal.lift_strict_mono -> Cardinal.lift_strictMono is a dubious translation:
 lean 3 declaration is
-  StrictMono.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.lift.{u2, u1}
+  StrictMono.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (SemilatticeInf.toPartialOrder.{succ u1} Cardinal.{u1} (Lattice.toSemilatticeInf.{succ u1} Cardinal.{u1} (LinearOrder.toLattice.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1})))) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (SemilatticeInf.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (Lattice.toSemilatticeInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (LinearOrder.toLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2})))) Cardinal.lift.{u2, u1}
 but is expected to have type
   StrictMono.{succ u2, max (succ u2) (succ u1)} Cardinal.{u2} Cardinal.{max u2 u1} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (PartialOrder.toPreorder.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.partialOrder.{max u2 u1}) Cardinal.lift.{u1, u2}
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_strict_mono Cardinal.lift_strictMonoₓ'. -/
@@ -443,7 +451,7 @@ theorem lift_strictMono : StrictMono lift := fun a b => lift_lt.2
 
 /- warning: cardinal.lift_monotone -> Cardinal.lift_monotone is a dubious translation:
 lean 3 declaration is
-  Monotone.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.lift.{u2, u1}
+  Monotone.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (SemilatticeInf.toPartialOrder.{succ u1} Cardinal.{u1} (Lattice.toSemilatticeInf.{succ u1} Cardinal.{u1} (LinearOrder.toLattice.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1})))) (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (SemilatticeInf.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (Lattice.toSemilatticeInf.{succ (max u1 u2)} Cardinal.{max u1 u2} (LinearOrder.toLattice.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2})))) Cardinal.lift.{u2, u1}
 but is expected to have type
   Monotone.{succ u2, max (succ u2) (succ u1)} Cardinal.{u2} Cardinal.{max u2 u1} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (PartialOrder.toPreorder.{max (succ u2) (succ u1)} Cardinal.{max u2 u1} Cardinal.partialOrder.{max u2 u1}) Cardinal.lift.{u1, u2}
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_monotone Cardinal.lift_monotoneₓ'. -/
@@ -933,7 +941,7 @@ instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· +
 
 instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
   { Cardinal.commSemiring,
-    Cardinal.partialOrder with
+    Cardinal.linearOrder with
     bot := 0
     bot_le := Cardinal.zero_le
     add_le_add_left := fun a b => add_le_add_left
@@ -947,6 +955,15 @@ instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
     eq_zero_or_eq_zero_of_mul_eq_zero := fun a b =>
       inductionOn₂ a b fun α β => by simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }
 
+instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
+
+instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=
+  { Cardinal.commSemiring,
+    Cardinal.linearOrder with
+    mul_le_mul_left := @mul_le_mul_left' _ _ _ _
+    zero_le_one := zero_le _ }
+
 #print Cardinal.zero_power_le /-
 theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 :=
   by
@@ -978,7 +995,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 
 /- warning: cardinal.cantor -> Cardinal.cantor is a dubious translation:
 lean 3 declaration is
-  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
+  forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) a)
 but is expected to have type
   forall (a : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (instOfNat.{succ u1} Cardinal.{u1} 2 Cardinal.instNatCastCardinal.{u1} (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) a)
 Case conversion may be inaccurate. Consider using '#align cardinal.cantor Cardinal.cantorₓ'. -/
@@ -992,29 +1009,24 @@ theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
   exact cantor_injective f hf
 #align cardinal.cantor Cardinal.cantor
 
-instance : NoMaxOrder Cardinal.{u} :=
-  { Cardinal.partialOrder with exists_gt := fun a => ⟨_, cantor a⟩ }
-
-instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
-  { (inferInstance : CanonicallyOrderedAddMonoid Cardinal),
-    Cardinal.partialOrder with
-    le_total := by
-      rintro ⟨α⟩ ⟨β⟩
-      apply embedding.total
-    decidableLe := Classical.decRel _ }
+instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := by infer_instance
 
-#print Cardinal.one_lt_iff_nontrivial /-
+/- warning: cardinal.one_lt_iff_nontrivial -> Cardinal.one_lt_iff_nontrivial is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) (Cardinal.mk.{u1} α)) (Nontrivial.{u1} α)
+but is expected to have type
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) (Cardinal.mk.{u1} α)) (Nontrivial.{u1} α)
+Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivialₓ'. -/
 theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
--/
 
 /- warning: cardinal.power_le_max_power_one -> Cardinal.power_le_max_power_one is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} b c) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b) (LinearOrder.max.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toLinearOrder.{succ u1} Cardinal.{u1} Cardinal.canonicallyLinearOrderedAddMonoid.{u1}) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} b c) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b) (LinearOrder.max.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} b c) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b) (Max.max.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toMax.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u1}) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.power_le_max_power_one Cardinal.power_le_max_power_oneₓ'. -/
@@ -1031,15 +1043,24 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 -/
 
-#print Cardinal.power_pos /-
+/- warning: cardinal.power_pos -> Cardinal.power_pos is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} (b : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) a) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
+but is expected to have type
+  forall {a : Cardinal.{u1}} (b : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) a) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b))
+Case conversion may be inaccurate. Consider using '#align cardinal.power_pos Cardinal.power_posₓ'. -/
 theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
--/
 
 end OrderProperties
 
-#print Cardinal.lt_wf /-
+/- warning: cardinal.lt_wf -> Cardinal.lt_wf is a dubious translation:
+lean 3 declaration is
+  WellFounded.{succ (succ u1)} Cardinal.{u1} (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))
+but is expected to have type
+  WellFounded.{succ (succ u1)} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5969 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5971 : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5969 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.5971)
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_wf Cardinal.lt_wfₓ'. -/
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -1052,18 +1073,18 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [f, mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
--/
 
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
-instance : WellFoundedLT Cardinal.{u} :=
-  ⟨Cardinal.lt_wf⟩
-
-#print Cardinal.wo /-
+/- warning: cardinal.wo -> Cardinal.wo is a dubious translation:
+lean 3 declaration is
+  IsWellOrder.{succ u1} Cardinal.{u1} (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))
+but is expected to have type
+  IsWellOrder.{succ u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6233 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6235 : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6233 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.6235)
+Case conversion may be inaccurate. Consider using '#align cardinal.wo Cardinal.woₓ'. -/
 instance wo : @IsWellOrder Cardinal.{u} (· < ·) where
 #align cardinal.wo Cardinal.wo
--/
 
 instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
@@ -1086,7 +1107,7 @@ instance : SuccOrder Cardinal :=
 
 /- warning: cardinal.succ_def -> Cardinal.succ_def is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
+  forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasInf.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c c')))
 but is expected to have type
   forall (c : Cardinal.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c) (InfSet.infₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toInfSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) (setOf.{succ u1} Cardinal.{u1} (fun (c' : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c c')))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_def Cardinal.succ_defₓ'. -/
@@ -1096,7 +1117,7 @@ theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
 
 /- warning: cardinal.add_one_le_succ -> Cardinal.add_one_le_succ is a dubious translation:
 lean 3 declaration is
-  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} c)
+  forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
 but is expected to have type
   forall (c : Cardinal.{u1}), LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) c (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_one_le_succ Cardinal.add_one_le_succₓ'. -/
@@ -1116,7 +1137,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c :=
 
 /- warning: cardinal.succ_pos -> Cardinal.succ_pos is a dubious translation:
 lean 3 declaration is
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+  forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c)
 but is expected to have type
   forall (c : Cardinal.{u1}), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} c)
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_pos Cardinal.succ_posₓ'. -/
@@ -1126,7 +1147,7 @@ theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
 
 /- warning: cardinal.succ_ne_zero -> Cardinal.succ_ne_zero is a dubious translation:
 lean 3 declaration is
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+  forall (c : Cardinal.{u1}), Ne.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} c) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))
 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.succ_ne_zero Cardinal.succ_ne_zeroₓ'. -/
@@ -1268,7 +1289,12 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
         (hf b)
 #align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
 
-#print Cardinal.bddAbove_range /-
+/- warning: cardinal.bdd_above_range -> Cardinal.bddAbove_range is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), BddAbove.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2}))))) (Set.range.{succ (max u1 u2), succ u1} Cardinal.{max u1 u2} ι f)
+but is expected to have type
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{max u1 u2}), BddAbove.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) (Set.range.{max (succ u1) (succ u2), succ u1} Cardinal.{max u1 u2} ι f)
+Case conversion may be inaccurate. Consider using '#align cardinal.bdd_above_range Cardinal.bddAbove_rangeₓ'. -/
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
     inputs, is always bounded above. -/
 theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove (Set.range f) :=
@@ -1276,7 +1302,6 @@ theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove
     rintro a ⟨i, rfl⟩
     exact le_sum f i⟩
 #align cardinal.bdd_above_range Cardinal.bddAbove_range
--/
 
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) :=
   by
@@ -1288,7 +1313,12 @@ instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) :=
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=
   small_subset Iio_subset_Iic_self
 
-#print Cardinal.bddAbove_iff_small /-
+/- warning: cardinal.bdd_above_iff_small -> Cardinal.bddAbove_iff_small is a dubious translation:
+lean 3 declaration is
+  forall {s : Set.{succ u1} Cardinal.{u1}}, Iff (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) s) (Small.{u1, succ u1} (coeSort.{succ (succ u1), succ (succ (succ u1))} (Set.{succ u1} Cardinal.{u1}) Type.{succ u1} (Set.hasCoeToSort.{succ u1} Cardinal.{u1}) s))
+but is expected to have type
+  forall {s : Set.{succ u1} Cardinal.{u1}}, Iff (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) (Small.{u1, succ u1} (Set.Elem.{succ u1} Cardinal.{u1} s))
+Case conversion may be inaccurate. Consider using '#align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_smallₓ'. -/
 /-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to an usual ZFC set. -/
 theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :=
   ⟨fun ⟨a, ha⟩ => @small_subset _ (Iic a) s (fun x h => ha h) _,
@@ -1305,31 +1335,42 @@ theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :
     · simp_rw [Subtype.val_eq_coe, Equiv.symm_apply_apply]
       rfl⟩
 #align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_small
--/
 
-#print Cardinal.bddAbove_of_small /-
+/- warning: cardinal.bdd_above_of_small -> Cardinal.bddAbove_of_small is a dubious translation:
+lean 3 declaration is
+  forall (s : Set.{succ u1} Cardinal.{u1}) [h : Small.{u1, succ u1} (coeSort.{succ (succ u1), succ (succ (succ u1))} (Set.{succ u1} Cardinal.{u1}) Type.{succ u1} (Set.hasCoeToSort.{succ u1} Cardinal.{u1}) s)], BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) s
+but is expected to have type
+  forall (s : Set.{succ u1} Cardinal.{u1}) [h : Small.{u1, succ u1} (Set.Elem.{succ u1} Cardinal.{u1} s)], BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s
+Case conversion may be inaccurate. Consider using '#align cardinal.bdd_above_of_small Cardinal.bddAbove_of_smallₓ'. -/
 theorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s :=
   bddAbove_iff_small.2 h
 #align cardinal.bdd_above_of_small Cardinal.bddAbove_of_small
--/
 
-#print Cardinal.bddAbove_image /-
+/- warning: cardinal.bdd_above_image -> Cardinal.bddAbove_image is a dubious translation:
+lean 3 declaration is
+  forall (f : Cardinal.{u1} -> Cardinal.{max u1 u2}) {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) s) -> (BddAbove.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2}))))) (Set.image.{succ u1, succ (max u1 u2)} Cardinal.{u1} Cardinal.{max u1 u2} f s))
+but is expected to have type
+  forall (f : Cardinal.{u1} -> Cardinal.{max u1 u2}) {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) -> (BddAbove.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) (Set.image.{succ u1, max (succ u1) (succ u2)} Cardinal.{u1} Cardinal.{max u1 u2} f s))
+Case conversion may be inaccurate. Consider using '#align cardinal.bdd_above_image Cardinal.bddAbove_imageₓ'. -/
 theorem bddAbove_image (f : Cardinal.{u} → Cardinal.{max u v}) {s : Set Cardinal.{u}}
     (hs : BddAbove s) : BddAbove (f '' s) :=
   by
   rw [bdd_above_iff_small] at hs⊢
   exact small_lift _
 #align cardinal.bdd_above_image Cardinal.bddAbove_image
--/
 
-#print Cardinal.bddAbove_range_comp /-
+/- warning: cardinal.bdd_above_range_comp -> Cardinal.bddAbove_range_comp is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_compₓ'. -/
 theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbove (range f))
     (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) :=
   by
   rw [range_comp]
   exact bdd_above_image g hf
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
--/
 
 /- warning: cardinal.supr_le_sum -> Cardinal.supᵢ_le_sum is a dubious translation:
 lean 3 declaration is
@@ -1557,7 +1598,12 @@ theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
 #align cardinal.le_lift_iff Cardinal.le_lift_iff
 -/
 
-#print Cardinal.lt_lift_iff /-
+/- warning: cardinal.lt_lift_iff -> Cardinal.lt_lift_iff is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{max u1 u2}}, Iff (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2})))))) b (Cardinal.lift.{u2, u1} a)) (Exists.{succ (succ u1)} Cardinal.{u1} (fun (a' : Cardinal.{u1}) => And (Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} a') b) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a' a)))
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{max u1 u2}}, Iff (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) b (Cardinal.lift.{u2, u1} a)) (Exists.{succ (succ u1)} Cardinal.{u1} (fun (a' : Cardinal.{u1}) => And (Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} a') b) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a' a)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_lift_iff Cardinal.lt_lift_iffₓ'. -/
 theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
   ⟨fun h =>
@@ -1565,11 +1611,10 @@ theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
     fun ⟨a', e, h⟩ => e ▸ lift_lt.2 h⟩
 #align cardinal.lt_lift_iff Cardinal.lt_lift_iff
--/
 
 /- warning: cardinal.lift_succ -> Cardinal.lift_succ is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
   forall (a : Cardinal.{u1}), Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} a)) (Order.succ.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_succ Cardinal.lift_succₓ'. -/
@@ -1593,7 +1638,7 @@ theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
 
 /- warning: cardinal.lift_min -> Cardinal.lift_min is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (LinearOrder.min.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) a b)) (LinearOrder.min.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (LinearOrder.min.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} a b)) (LinearOrder.min.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b))
 but is expected to have type
   forall {a : Cardinal.{u2}} {b : Cardinal.{u2}}, Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (Min.min.{succ u2} Cardinal.{u2} (CanonicallyLinearOrderedAddMonoid.toMin.{succ u2} Cardinal.{u2} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u2}) a b)) (Min.min.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (CanonicallyLinearOrderedAddMonoid.toMin.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{max u1 u2}) (Cardinal.lift.{u1, u2} a) (Cardinal.lift.{u1, u2} b))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_min Cardinal.lift_minₓ'. -/
@@ -1604,7 +1649,7 @@ theorem lift_min {a b : Cardinal} : lift (min a b) = min (lift a) (lift b) :=
 
 /- warning: cardinal.lift_max -> Cardinal.lift_max is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (LinearOrder.max.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) a b)) (LinearOrder.max.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.conditionallyCompleteLinearOrderBot.{max u1 u2})) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (LinearOrder.max.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} a b)) (LinearOrder.max.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.linearOrder.{max u1 u2} (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u2, u1} b))
 but is expected to have type
   forall {a : Cardinal.{u2}} {b : Cardinal.{u2}}, Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (Max.max.{succ u2} Cardinal.{u2} (CanonicallyLinearOrderedAddMonoid.toMax.{succ u2} Cardinal.{u2} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u2}) a b)) (Max.max.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} (CanonicallyLinearOrderedAddMonoid.toMax.{max (succ u1) (succ u2)} Cardinal.{max u2 u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{max u1 u2}) (Cardinal.lift.{u1, u2} a) (Cardinal.lift.{u1, u2} b))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_max Cardinal.lift_maxₓ'. -/
@@ -1615,7 +1660,7 @@ theorem lift_max {a b : Cardinal} : lift (max a b) = max (lift a) (lift b) :=
 
 /- warning: cardinal.lift_Sup -> Cardinal.lift_supₛ is a dubious translation:
 lean 3 declaration is
-  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.supₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.supₛ.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
+  forall {s : Set.{succ u2} Cardinal.{u2}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) s) -> (Eq.{succ (succ (max u2 u1))} Cardinal.{max u2 u1} (Cardinal.lift.{u1, u2} (SupSet.supₛ.{succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) s)) (SupSet.supₛ.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u2 u1)} Cardinal.{max u2 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u2 u1}))) (Set.image.{succ u2, succ (max u2 u1)} Cardinal.{u2} Cardinal.{max u2 u1} Cardinal.lift.{u1, u2} s)))
 but is expected to have type
   forall {s : Set.{succ u1} Cardinal.{u1}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) s) -> (Eq.{max (succ (succ u2)) (succ (succ u1))} Cardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (SupSet.supₛ.{succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) s)) (SupSet.supₛ.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLattice.toSupSet.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u2) (succ u1)} Cardinal.{max u1 u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u2 u1}))) (Set.image.{succ u1, max (succ u2) (succ u1)} Cardinal.{u1} Cardinal.{max u1 u2} Cardinal.lift.{u2, u1} s)))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_Sup Cardinal.lift_supₛₓ'. -/
@@ -1634,7 +1679,7 @@ theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (supₛ s) =
 
 /- warning: cardinal.lift_supr -> Cardinal.lift_supᵢ is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (supᵢ.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{succ (succ (max u3 u1))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) (supᵢ.{succ (max u3 u1), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toHasSup.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ (max u3 u1)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.conditionallyCompleteLinearOrderBot.{max u3 u1}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (Eq.{max (succ (succ u1)) (succ (succ u3))} Cardinal.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) (supᵢ.{max (succ u1) (succ u3), succ u2} Cardinal.{max u3 u1} (ConditionallyCompleteLattice.toSupSet.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{max u1 u3}))) ι (fun (i : ι) => Cardinal.lift.{u1, u3} (f i))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr Cardinal.lift_supᵢₓ'. -/
@@ -1645,7 +1690,7 @@ theorem lift_supᵢ {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (rang
 
 /- warning: cardinal.lift_supr_le -> Cardinal.lift_supᵢ_le is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u3 u1}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t)
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}} {t : Cardinal.{max u1 u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t) -> (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t)
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le Cardinal.lift_supᵢ_leₓ'. -/
@@ -1660,7 +1705,7 @@ theorem lift_supᵢ_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (h
 
 /- warning: cardinal.lift_supr_le_iff -> Cardinal.lift_supᵢ_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
+  forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} (OrderedAddCommMonoid.toPartialOrder.{succ u3} Cardinal.{u3} (OrderedSemiring.toOrderedAddCommMonoid.{succ u3} Cardinal.{u3} (OrderedCommSemiring.toOrderedSemiring.{succ u3} Cardinal.{u3} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u3} Cardinal.{u3} Cardinal.canonicallyOrderedCommSemiring.{u3}))))) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u3 u1}}, Iff (LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toHasSup.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.conditionallyCompleteLinearOrderBot.{u3}))) ι f)) t) (forall (i : ι), LE.le.{succ (max u3 u1)} Cardinal.{max u3 u1} Cardinal.hasLe.{max u3 u1} (Cardinal.lift.{u1, u3} (f i)) t))
 but is expected to have type
   forall {ι : Type.{u2}} {f : ι -> Cardinal.{u3}}, (BddAbove.{succ u3} Cardinal.{u3} (PartialOrder.toPreorder.{succ u3} Cardinal.{u3} Cardinal.partialOrder.{u3}) (Set.range.{succ u3, succ u2} Cardinal.{u3} ι f)) -> (forall {t : Cardinal.{max u1 u3}}, Iff (LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (supᵢ.{succ u3, succ u2} Cardinal.{u3} (ConditionallyCompleteLattice.toSupSet.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u3} Cardinal.{u3} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u3} Cardinal.{u3} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u3}))) ι f)) t) (forall (i : ι), LE.le.{max (succ u1) (succ u3)} Cardinal.{max u3 u1} Cardinal.instLECardinal.{max u1 u3} (Cardinal.lift.{u1, u3} (f i)) t))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_iff Cardinal.lift_supᵢ_le_iffₓ'. -/
@@ -1676,7 +1721,7 @@ universe v' w'
 
 /- warning: cardinal.lift_supr_le_lift_supr -> Cardinal.lift_supᵢ_le_lift_supᵢ is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} (OrderedAddCommMonoid.toPartialOrder.{succ u4} Cardinal.{u4} (OrderedSemiring.toOrderedAddCommMonoid.{succ u4} Cardinal.{u4} (OrderedCommSemiring.toOrderedSemiring.{succ u4} Cardinal.{u4} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u4} Cardinal.{u4} Cardinal.canonicallyOrderedCommSemiring.{u4}))))) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{succ (max u2 u4)} Cardinal.{max u2 u4} Cardinal.hasLe.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toHasSup.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.conditionallyCompleteLinearOrderBot.{u4}))) ι' f'))))
 but is expected to have type
   forall {ι : Type.{u1}} {ι' : Type.{u3}} {f : ι -> Cardinal.{u2}} {f' : ι' -> Cardinal.{u4}}, (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u1} Cardinal.{u2} ι f)) -> (BddAbove.{succ u4} Cardinal.{u4} (PartialOrder.toPreorder.{succ u4} Cardinal.{u4} Cardinal.partialOrder.{u4}) (Set.range.{succ u4, succ u3} Cardinal.{u4} ι' f')) -> (forall {g : ι -> ι'}, (forall (i : ι), LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (f i)) (Cardinal.lift.{u2, u4} (f' (g i)))) -> (LE.le.{max (succ u2) (succ u4)} Cardinal.{max u2 u4} Cardinal.instLECardinal.{max u2 u4} (Cardinal.lift.{u4, u2} (supᵢ.{succ u2, succ u1} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι f)) (Cardinal.lift.{u2, u4} (supᵢ.{succ u4, succ u3} Cardinal.{u4} (ConditionallyCompleteLattice.toSupSet.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u4} Cardinal.{u4} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u4} Cardinal.{u4} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u4}))) ι' f'))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢₓ'. -/
@@ -1694,7 +1739,7 @@ theorem lift_supᵢ_le_lift_supᵢ {ι : Type v} {ι' : Type v'} {f : ι → Car
 
 /- warning: cardinal.lift_supr_le_lift_supr' -> Cardinal.lift_supᵢ_le_lift_supᵢ' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
+  forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2}))))) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasLe.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toHasSup.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toHasSup.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.conditionallyCompleteLinearOrderBot.{u2}))) ι' f'))))
 but is expected to have type
   forall {ι : Type.{u1}} {ι' : Type.{u2}} {f : ι -> Cardinal.{u1}} {f' : ι' -> Cardinal.{u2}}, (BddAbove.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (Set.range.{succ u1, succ u1} Cardinal.{u1} ι f)) -> (BddAbove.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2}) (Set.range.{succ u2, succ u2} Cardinal.{u2} ι' f')) -> (forall (g : ι -> ι'), (forall (i : ι), LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (f i)) (Cardinal.lift.{u1, u2} (f' (g i)))) -> (LE.le.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instLECardinal.{max u1 u2} (Cardinal.lift.{u2, u1} (supᵢ.{succ u1, succ u1} Cardinal.{u1} (ConditionallyCompleteLattice.toSupSet.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u1}))) ι f)) (Cardinal.lift.{u1, u2} (supᵢ.{succ u2, succ u2} Cardinal.{u2} (ConditionallyCompleteLattice.toSupSet.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u2} Cardinal.{u2} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u2} Cardinal.{u2} Cardinal.instConditionallyCompleteLinearOrderBotCardinal.{u2}))) ι' f'))))
 Case conversion may be inaccurate. Consider using '#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_supᵢ_le_lift_supᵢ'ₓ'. -/
@@ -1728,11 +1773,15 @@ theorem aleph0_ne_zero : ℵ₀ ≠ 0 :=
 #align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero
 -/
 
-#print Cardinal.aleph0_pos /-
+/- warning: cardinal.aleph_0_pos -> Cardinal.aleph0_pos is a dubious translation:
+lean 3 declaration is
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) Cardinal.aleph0.{u1}
+but is expected to have type
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_pos Cardinal.aleph0_posₓ'. -/
 theorem aleph0_pos : 0 < ℵ₀ :=
   pos_iff_ne_zero.2 aleph0_ne_zero
 #align cardinal.aleph_0_pos Cardinal.aleph0_pos
--/
 
 #print Cardinal.lift_aleph0 /-
 @[simp]
@@ -1884,7 +1933,7 @@ theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
 
 /- warning: cardinal.nat_cast_lt -> Cardinal.natCast_lt is a dubious translation:
 lean 3 declaration is
-  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
+  forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) m) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (LT.lt.{0} Nat Nat.hasLt m n)
 but is expected to have type
   forall {m : Nat} {n : Nat}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} m) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (LT.lt.{0} Nat instLTNat m n)
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_cast_lt Cardinal.natCast_ltₓ'. -/
@@ -1917,7 +1966,7 @@ theorem natCast_injective : Injective (coe : ℕ → Cardinal) :=
 
 /- warning: cardinal.nat_succ -> Cardinal.nat_succ is a dubious translation:
 lean 3 declaration is
-  forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
+  forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))
 but is expected to have type
   forall (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (Nat.succ n)) (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))
 Case conversion may be inaccurate. Consider using '#align cardinal.nat_succ Cardinal.nat_succₓ'. -/
@@ -1928,7 +1977,7 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ n :=
 
 /- warning: cardinal.succ_zero -> Cardinal.succ_zero is a dubious translation:
 lean 3 declaration is
-  Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))
+  Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))
 but is expected to have type
   Eq.{succ (succ u1)} Cardinal.{u1} (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.succ_zero Cardinal.succ_zeroₓ'. -/
@@ -1952,18 +2001,26 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
-#print Cardinal.cantor' /-
+/- warning: cardinal.cantor' -> Cardinal.cantor' is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall (a : Cardinal.{u1}) {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) b) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) b a))
+Case conversion may be inaccurate. Consider using '#align cardinal.cantor' Cardinal.cantor'ₓ'. -/
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
   by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
--/
 
-#print Cardinal.one_le_iff_pos /-
+/- warning: cardinal.one_le_iff_pos -> Cardinal.one_le_iff_pos is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) c) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1}))) c)
+but is expected to have type
+  forall {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (One.toOfNat1.{succ u1} Cardinal.{u1} Cardinal.instOneCardinal.{u1})) c) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1})) c)
+Case conversion may be inaccurate. Consider using '#align cardinal.one_le_iff_pos Cardinal.one_le_iff_posₓ'. -/
 theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by rw [← succ_zero, succ_le_iff]
 #align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos
--/
 
 #print Cardinal.one_le_iff_ne_zero /-
 theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le_iff_pos, pos_iff_ne_zero]
@@ -1972,7 +2029,7 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by rw [one_le
 
 /- warning: cardinal.nat_lt_aleph_0 -> Cardinal.nat_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
+  forall (n : Nat), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n) Cardinal.aleph0.{u1}
 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0ₓ'. -/
@@ -1983,11 +2040,15 @@ theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
       exact ⟨⟨coe, fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
-#print Cardinal.one_lt_aleph0 /-
+/- warning: cardinal.one_lt_aleph_0 -> Cardinal.one_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 1 (OfNat.mk.{succ u1} Cardinal.{u1} 1 (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1}))) Cardinal.aleph0.{u1}
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0ₓ'. -/
 @[simp]
 theorem one_lt_aleph0 : 1 < ℵ₀ := by simpa using nat_lt_aleph_0 1
 #align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0
--/
 
 #print Cardinal.one_le_aleph0 /-
 theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
@@ -1997,7 +2058,7 @@ theorem one_le_aleph0 : 1 ≤ ℵ₀ :=
 
 /- warning: cardinal.lt_aleph_0 -> Cardinal.lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {c : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)))
 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0 Cardinal.lt_aleph0ₓ'. -/
@@ -2028,7 +2089,7 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
 
 /- warning: cardinal.range_nat_cast -> Cardinal.range_natCast is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align cardinal.range_nat_cast Cardinal.range_natCastₓ'. -/
@@ -2047,40 +2108,66 @@ theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ F
   rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
-#print Cardinal.lt_aleph0_iff_finite /-
+/- warning: cardinal.lt_aleph_0_iff_finite -> Cardinal.lt_aleph0_iff_finite is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finiteₓ'. -/
 theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph_0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
--/
 
-#print Cardinal.lt_aleph0_iff_fintype /-
+/- warning: cardinal.lt_aleph_0_iff_fintype -> Cardinal.lt_aleph0_iff_fintype is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {α : Type.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}) (Nonempty.{succ u1} (Fintype.{u1} α))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintypeₓ'. -/
 theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
--/
 
-#print Cardinal.lt_aleph0_of_finite /-
+/- warning: cardinal.lt_aleph_0_of_finite -> Cardinal.lt_aleph0_of_finite is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : Finite.{succ u1} α], LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} α) Cardinal.aleph0.{u1}
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finiteₓ'. -/
 theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
--/
 
-#print Cardinal.lt_aleph0_iff_set_finite /-
+/- warning: cardinal.lt_aleph_0_iff_set_finite -> Cardinal.lt_aleph0_iff_set_finite is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {S : Set.{u1} α}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α S)
+but is expected to have type
+  forall {α : Type.{u1}} {S : Set.{u1} α}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α S)) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α S)
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finiteₓ'. -/
 @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
--/
 
+/- warning: set.finite.lt_aleph_0 -> Set.Finite.lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {S : Set.{u1} α}, (Set.Finite.{u1} α S) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) S)) Cardinal.aleph0.{u1})
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+  forall {α : Type.{u1}} {S : Set.{u1} α}, (Set.Finite.{u1} α S) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α S)) Cardinal.aleph0.{u1})
+Case conversion may be inaccurate. Consider using '#align set.finite.lt_aleph_0 Set.Finite.lt_aleph0ₓ'. -/
 alias lt_aleph_0_iff_set_finite ↔ _ _root_.set.finite.lt_aleph_0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
-#print Cardinal.lt_aleph0_iff_subtype_finite /-
+/- warning: cardinal.lt_aleph_0_iff_subtype_finite -> Cardinal.lt_aleph0_iff_subtype_finite is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {p : α -> Prop}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} (Subtype.{succ u1} α (fun (x : α) => p x))) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α (setOf.{u1} α (fun (x : α) => p x)))
+but is expected to have type
+  forall {α : Type.{u1}} {p : α -> Prop}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Subtype.{succ u1} α (fun (x : α) => p x))) Cardinal.aleph0.{u1}) (Set.Finite.{u1} α (setOf.{u1} α (fun (x : α) => p x)))
+Case conversion may be inaccurate. Consider using '#align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finiteₓ'. -/
 @[simp]
 theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
--/
 
 #print Cardinal.mk_le_aleph0_iff /-
 theorem mk_le_aleph0_iff : (#α) ≤ ℵ₀ ↔ Countable α := by
@@ -2115,7 +2202,7 @@ theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
 
 /- warning: cardinal.can_lift_cardinal_nat -> Cardinal.canLiftCardinalNat is a dubious translation:
 lean 3 declaration is
-  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
+  CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1})))) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) x Cardinal.aleph0.{u1})
 but is expected to have type
   CanLift.{succ (succ u1), 1} Cardinal.{u1} Nat (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1}) (fun (x : Cardinal.{u1}) => LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNatₓ'. -/
@@ -2127,7 +2214,7 @@ instance canLiftCardinalNat : CanLift Cardinal ℕ coe fun x => x < ℵ₀ :=
 
 /- warning: cardinal.add_lt_aleph_0 -> Cardinal.add_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0ₓ'. -/
@@ -2138,7 +2225,7 @@ theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a +
 
 /- warning: cardinal.add_lt_aleph_0_iff -> Cardinal.add_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iffₓ'. -/
@@ -2159,7 +2246,7 @@ theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a 
 
 /- warning: cardinal.nsmul_lt_aleph_0_iff -> Cardinal.nsmul_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+  forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}))
 but is expected to have type
   forall {n : Nat} {a : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HSMul.hSMul.{0, succ u1, succ u1} Nat Cardinal.{u1} Cardinal.{u1} (instHSMul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) n a) Cardinal.aleph0.{u1}) (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iffₓ'. -/
@@ -2176,7 +2263,7 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
 
 /- warning: cardinal.nsmul_lt_aleph_0_iff_of_ne_zero -> Cardinal.nsmul_lt_aleph0_iff_of_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall {n : Nat} {a : Cardinal.{u1}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
+  forall {n : Nat} {a : Cardinal.{u1}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (SMul.smul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) n a) Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}))
 but is expected to have type
   forall {n : Nat} {a : Cardinal.{u1}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HSMul.hSMul.{0, succ u1, succ u1} Nat Cardinal.{u1} Cardinal.{u1} (instHSMul.{0, succ u1} Nat Cardinal.{u1} (AddMonoid.SMul.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) n a) Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}))
 Case conversion may be inaccurate. Consider using '#align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zeroₓ'. -/
@@ -2187,7 +2274,7 @@ theorem nsmul_lt_aleph0_iff_of_ne_zero {n : ℕ} {a : Cardinal} (h : n ≠ 0) :
 
 /- warning: cardinal.mul_lt_aleph_0 -> Cardinal.mul_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1})
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0ₓ'. -/
@@ -2198,7 +2285,7 @@ theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a *
 
 /- warning: cardinal.mul_lt_aleph_0_iff -> Cardinal.mul_lt_aleph0_iff is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}))))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (Or (Eq.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (Or (Eq.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}))))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iffₓ'. -/
@@ -2250,7 +2337,7 @@ theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 
 
 /- warning: cardinal.mul_lt_aleph_0_iff_of_ne_zero -> Cardinal.mul_lt_aleph0_iff_of_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.hasMul.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1})))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Ne.{succ (succ u1)} Cardinal.{u1} b (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Iff (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HMul.hMul.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHMul.{succ u1} Cardinal.{u1} Cardinal.instMulCardinal.{u1}) a b) Cardinal.aleph0.{u1}) (And (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1})))
 Case conversion may be inaccurate. Consider using '#align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zeroₓ'. -/
@@ -2258,12 +2345,16 @@ theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph_0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
-#print Cardinal.power_lt_aleph0 /-
+/- warning: cardinal.power_lt_aleph_0 -> Cardinal.power_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b) Cardinal.aleph0.{u1})
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) b Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b) Cardinal.aleph0.{u1})
+Case conversion may be inaccurate. Consider using '#align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0ₓ'. -/
 theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← nat_cast_pow] <;> apply nat_lt_aleph_0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
--/
 
 #print Cardinal.eq_one_iff_unique /-
 theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
@@ -2401,7 +2492,7 @@ def toNat : ZeroHom Cardinal ℕ :=
 
 /- warning: cardinal.to_nat_apply_of_lt_aleph_0 -> Cardinal.toNat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {c : Cardinal.{u1}} (h : LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}), Eq.{1} Nat (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (Classical.choose.{1} Nat (fun (x : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) x)) (Iff.mp (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n))) (Cardinal.lt_aleph0.{u1} c) h))
 but is expected to have type
   forall {c : Cardinal.{u1}} (h : LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c) (Classical.choose.{1} Nat (fun (x : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} x)) (Iff.mp (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) (Exists.{1} Nat (fun (n : Nat) => Eq.{succ (succ u1)} Cardinal.{u1} c (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n))) (Cardinal.lt_aleph0.{u1} c) h))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0ₓ'. -/
@@ -2418,7 +2509,7 @@ theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toNat = 0
 
 /- warning: cardinal.cast_to_nat_of_lt_aleph_0 -> Cardinal.cast_toNat_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (Eq.{succ (succ u1)} Cardinal.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)) c)
 but is expected to have type
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)) c)
 Case conversion may be inaccurate. Consider using '#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0ₓ'. -/
@@ -2436,33 +2527,49 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑c.toNat =
   rw [to_nat_apply_of_aleph_0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
-#print Cardinal.toNat_le_iff_le_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_le_iff_le_of_lt_aleph_0 -> Cardinal.toNat_le_iff_le_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) d Cardinal.aleph0.{u1}) -> (Iff (LE.le.{0} Nat Nat.hasLe (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d)) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} c d))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0ₓ'. -/
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat ≤ d.toNat ↔ c ≤ d := by
   rw [← nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
--/
 
-#print Cardinal.toNat_lt_iff_lt_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 -> Cardinal.toNat_lt_iff_lt_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) d Cardinal.aleph0.{u1}) -> (Iff (LT.lt.{0} Nat Nat.hasLt (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d)) (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c d))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0ₓ'. -/
 theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     c.toNat < d.toNat ↔ c < d := by
   rw [← nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd]
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
--/
 
-#print Cardinal.toNat_le_of_le_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_le_of_le_of_lt_aleph_0 -> Cardinal.toNat_le_of_le_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) d Cardinal.aleph0.{u1}) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} c d) -> (LE.le.{0} Nat Nat.hasLe (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0ₓ'. -/
 theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
     c.toNat ≤ d.toNat :=
   (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
 #align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
--/
 
-#print Cardinal.toNat_lt_of_lt_of_lt_aleph0 /-
+/- warning: cardinal.to_nat_lt_of_lt_of_lt_aleph_0 -> Cardinal.toNat_lt_of_lt_of_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {c : Cardinal.{u1}} {d : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) d Cardinal.aleph0.{u1}) -> (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c d) -> (LT.lt.{0} Nat Nat.hasLt (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} d))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0ₓ'. -/
 theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
     c.toNat < d.toNat :=
   (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
--/
 
 /- warning: cardinal.to_nat_cast -> Cardinal.toNat_cast is a dubious translation:
 lean 3 declaration is
@@ -2634,7 +2741,7 @@ theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
 
 /- warning: cardinal.to_nat_add_of_lt_aleph_0 -> Cardinal.toNat_add_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
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+  forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2})))))) b Cardinal.aleph0.{u2}) -> (Eq.{1} Nat (coeFn.{succ (succ (max u1 u2)), succ (succ (max u1 u2))} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) (fun (_x : ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) => Cardinal.{max u1 u2} -> Nat) (ZeroHom.hasCoeToFun.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.hasZero.{max u1 u2} Nat.hasZero) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{succ (max u1 u2), succ (max u1 u2), succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.{max u1 u2} Cardinal.{max u1 u2} (instHAdd.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.hasAdd.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} a) (coeFn.{succ (succ u2), succ (succ u2)} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) (fun (_x : ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) => Cardinal.{u2} -> Nat) (ZeroHom.hasCoeToFun.{succ u2, 0} Cardinal.{u2} Nat Cardinal.hasZero.{u2} Nat.hasZero) Cardinal.toNat.{u2} b)))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u2}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) a Cardinal.aleph0.{u1}) -> (LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) b Cardinal.aleph0.{u2}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{max u1 u2}) => Nat) (HAdd.hAdd.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u2 u1} Cardinal.{max u1 u2} (instHAdd.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instAddCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (FunLike.coe.{succ (succ (max u1 u2)), succ (succ (max u1 u2)), 1} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{max u1 u2} (fun (_x : Cardinal.{max u1 u2}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{max u1 u2}) => Nat) _x) (ZeroHomClass.toFunLike.{succ (max u1 u2), succ (max u1 u2), 0} (ZeroHom.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat Cardinal.instZeroCardinal.{max u1 u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{max u1 u2} (HAdd.hAdd.{max (succ u1) (succ u2), max (succ u1) (succ u2), max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.{max u2 u1} Cardinal.{max u1 u2} (instHAdd.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.instAddCardinal.{max u1 u2}) (Cardinal.lift.{u2, u1} a) (Cardinal.lift.{u1, u2} b))) (HAdd.hAdd.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u2}) => Nat) b) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) (instHAdd.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) a) instAddNat) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} a) (FunLike.coe.{succ (succ u2), succ (succ u2), 1} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u2} (fun (_x : Cardinal.{u2}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u2}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u2, succ u2, 0} (ZeroHom.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u2, 0} Cardinal.{u2} Nat Cardinal.instZeroCardinal.{u2} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u2} b)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0ₓ'. -/
@@ -2682,7 +2789,7 @@ def toPartENat : Cardinal →+ PartENat
 
 /- warning: cardinal.to_part_enat_apply_of_lt_aleph_0 -> Cardinal.toPartENat_apply_of_lt_aleph0 is a dubious translation:
 lean 3 declaration is
-  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
 but is expected to have type
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0ₓ'. -/
@@ -2766,7 +2873,7 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 
 /- warning: cardinal.sum_lt_prod -> Cardinal.sum_lt_prod is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} Cardinal.partialOrder.{u2})) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
+  forall {ι : Type.{u1}} (f : ι -> Cardinal.{u2}) (g : ι -> Cardinal.{u2}), (forall (i : ι), LT.lt.{succ u2} Cardinal.{u2} (Preorder.toLT.{succ u2} Cardinal.{u2} (PartialOrder.toPreorder.{succ u2} Cardinal.{u2} (OrderedAddCommMonoid.toPartialOrder.{succ u2} Cardinal.{u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ u2} Cardinal.{u2} (OrderedCommSemiring.toOrderedSemiring.{succ u2} Cardinal.{u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u2} Cardinal.{u2} Cardinal.canonicallyOrderedCommSemiring.{u2})))))) (f i) (g i)) -> (LT.lt.{succ (max u1 u2)} Cardinal.{max u1 u2} (Preorder.toLT.{succ (max u1 u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedAddCommMonoid.toPartialOrder.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toOrderedAddCommMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.canonicallyOrderedCommSemiring.{max u1 u2})))))) (Cardinal.sum.{u1, u2} ι f) (Cardinal.prod.{u1, u2} ι g))
 but is expected to have type
   forall {ι : Type.{u2}} (f : ι -> Cardinal.{u1}) (g : ι -> Cardinal.{u1}), (forall (i : ι), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (f i) (g i)) -> (LT.lt.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (Preorder.toLT.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} (PartialOrder.toPreorder.{max (succ u1) (succ u2)} Cardinal.{max u1 u2} Cardinal.partialOrder.{max u1 u2})) (Cardinal.sum.{u2, u1} ι f) (Cardinal.prod.{u2, u1} ι g))
 Case conversion may be inaccurate. Consider using '#align cardinal.sum_lt_prod Cardinal.sum_lt_prodₓ'. -/
@@ -3020,11 +3127,15 @@ theorem mk_bunionᵢ_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
   apply mk_Union_le
 #align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_le
 
-#print Cardinal.finset_card_lt_aleph0 /-
+/- warning: cardinal.finset_card_lt_aleph_0 -> Cardinal.finset_card_lt_aleph0 is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} (s : Finset.{u1} α), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) (Cardinal.mk.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} α) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (Finset.Set.hasCoeT.{u1} α))) s))) Cardinal.aleph0.{u1}
+but is expected to have type
+  forall {α : Type.{u1}} (s : Finset.{u1} α), LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Cardinal.mk.{u1} (Set.Elem.{u1} α (Finset.toSet.{u1} α s))) Cardinal.aleph0.{u1}
+Case conversion may be inaccurate. Consider using '#align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0ₓ'. -/
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
--/
 
 /- warning: cardinal.mk_set_eq_nat_iff_finset -> Cardinal.mk_set_eq_nat_iff_finset is a dubious translation:
 lean 3 declaration is
@@ -3339,7 +3450,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
 
 /- warning: cardinal.exists_not_mem_of_length_lt -> Cardinal.exists_not_mem_of_length_lt is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
+  forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.Mem.{u1, u1} α (List.{u1} α) (List.hasMem.{u1} α) z l)))
 but is expected to have type
   forall {α : Type.{u1}} (l : List.{u1} α), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} (List.length.{u1} α l)) (Cardinal.mk.{u1} α)) -> (Exists.{succ u1} α (fun (z : α) => Not (Membership.mem.{u1, u1} α (List.{u1} α) (List.instMembershipList.{u1} α) z l)))
 Case conversion may be inaccurate. Consider using '#align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_ltₓ'. -/
@@ -3378,16 +3489,25 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 -- mathport name: «expr ^< »
 infixl:80 " ^< " => powerlt
 
-#print Cardinal.le_powerlt /-
+/- warning: cardinal.le_powerlt -> Cardinal.le_powerlt is a dubious translation:
+lean 3 declaration is
+  forall {b : Cardinal.{u1}} {c : Cardinal.{u1}} (a : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) c b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a c) (Cardinal.powerlt.{u1} a b))
+but is expected to have type
+  forall {b : Cardinal.{u1}} {c : Cardinal.{u1}} (a : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a c) (Cardinal.powerlt.{u1} a b))
+Case conversion may be inaccurate. Consider using '#align cardinal.le_powerlt Cardinal.le_powerltₓ'. -/
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
   by
   apply @le_csupᵢ _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
--/
 
-#print Cardinal.powerlt_le /-
+/- warning: cardinal.powerlt_le -> Cardinal.powerlt_le is a dubious translation:
+lean 3 declaration is
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (Cardinal.powerlt.{u1} a b) c) (forall (x : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))) x b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a x) c))
+but is expected to have type
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Iff (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (Cardinal.powerlt.{u1} a b) c) (forall (x : Cardinal.{u1}), (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) x b) -> (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a x) c))
+Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_le Cardinal.powerlt_leₓ'. -/
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
   by
   rw [powerlt, csupᵢ_le_iff']
@@ -3395,7 +3515,6 @@ theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) 
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.powerlt_le Cardinal.powerlt_le
--/
 
 #print Cardinal.powerlt_le_powerlt_left /-
 theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a ^< c :=
@@ -3403,14 +3522,18 @@ theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a
 #align cardinal.powerlt_le_powerlt_left Cardinal.powerlt_le_powerlt_left
 -/
 
-#print Cardinal.powerlt_mono_left /-
+/- warning: cardinal.powerlt_mono_left -> Cardinal.powerlt_mono_left is a dubious translation:
+lean 3 declaration is
+  forall (a : Cardinal.{u1}), Monotone.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) (fun (c : Cardinal.{u1}) => Cardinal.powerlt.{u1} a c)
+but is expected to have type
+  forall (a : Cardinal.{u1}), Monotone.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) (fun (c : Cardinal.{u1}) => Cardinal.powerlt.{u1} a c)
+Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_mono_left Cardinal.powerlt_mono_leftₓ'. -/
 theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun b c => powerlt_le_powerlt_left
 #align cardinal.powerlt_mono_left Cardinal.powerlt_mono_left
--/
 
 /- warning: cardinal.powerlt_succ -> Cardinal.powerlt_succ is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.succOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (OfNat.mk.{succ u1} Cardinal.{u1} 0 (Zero.zero.{succ u1} Cardinal.{u1} Cardinal.hasZero.{u1})))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} (OrderedAddCommMonoid.toPartialOrder.{succ u1} Cardinal.{u1} (OrderedSemiring.toOrderedAddCommMonoid.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))) Cardinal.succOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a b))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}}, (Ne.{succ (succ u1)} Cardinal.{u1} a (OfNat.ofNat.{succ u1} Cardinal.{u1} 0 (Zero.toOfNat0.{succ u1} Cardinal.{u1} Cardinal.instZeroCardinal.{u1}))) -> (Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Order.succ.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1}) Cardinal.instSuccOrderCardinalToPreorderPartialOrder.{u1} b)) (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a b))
 Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_succ Cardinal.powerlt_succₓ'. -/
@@ -3421,7 +3544,7 @@ theorem powerlt_succ {a b : Cardinal} (h : a ≠ 0) : a ^< succ b = (a^b) :=
 
 /- warning: cardinal.powerlt_min -> Cardinal.powerlt_min is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (LinearOrder.min.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) b c)) (LinearOrder.min.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (LinearOrder.min.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} b c)) (LinearOrder.min.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Min.min.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toMin.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u1}) b c)) (Min.min.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toMin.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u1}) (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
 Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_min Cardinal.powerlt_minₓ'. -/
@@ -3431,7 +3554,7 @@ theorem powerlt_min {a b c : Cardinal} : a ^< min b c = min (a ^< b) (a ^< c) :=
 
 /- warning: cardinal.powerlt_max -> Cardinal.powerlt_max is a dubious translation:
 lean 3 declaration is
-  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (LinearOrder.max.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) b c)) (LinearOrder.max.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrder.toLinearOrder.{succ u1} Cardinal.{u1} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u1} Cardinal.{u1} Cardinal.conditionallyCompleteLinearOrderBot.{u1})) (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
+  forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (LinearOrder.max.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} b c)) (LinearOrder.max.{succ u1} Cardinal.{u1} Cardinal.linearOrder.{u1} (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
 but is expected to have type
   forall {a : Cardinal.{u1}} {b : Cardinal.{u1}} {c : Cardinal.{u1}}, Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.powerlt.{u1} a (Max.max.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toMax.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u1}) b c)) (Max.max.{succ u1} Cardinal.{u1} (CanonicallyLinearOrderedAddMonoid.toMax.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyLinearOrderedAddMonoidCardinal.{u1}) (Cardinal.powerlt.{u1} a b) (Cardinal.powerlt.{u1} a c))
 Case conversion may be inaccurate. Consider using '#align cardinal.powerlt_max Cardinal.powerlt_maxₓ'. -/

bump to nightly-2023-03-11-22

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

a8ee822f

Diff

@@ -2684,7 +2684,7 @@ def toPartENat : Cardinal →+ PartENat
 lean 3 declaration is
   forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (coeFn.{succ (succ u1), succ (succ u1)} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) (fun (_x : ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) => Cardinal.{u1} -> Nat) (ZeroHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} Nat Cardinal.hasZero.{u1} Nat.hasZero) Cardinal.toNat.{u1} c)))
 but is expected to have type
-  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
+  forall {c : Cardinal.{u1}}, (LT.lt.{succ u1} Cardinal.{u1} (Preorder.toLT.{succ u1} Cardinal.{u1} (PartialOrder.toPreorder.{succ u1} Cardinal.{u1} Cardinal.partialOrder.{u1})) c Cardinal.aleph0.{u1}) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instAddCommMonoidWithOnePartENat)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.124 : Cardinal.{u1}) => Nat) _x) (ZeroHomClass.toFunLike.{succ u1, succ u1, 0} (ZeroHom.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (ZeroHom.zeroHomClass.{succ u1, 0} Cardinal.{u1} Nat Cardinal.instZeroCardinal.{u1} (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero))) Cardinal.toNat.{u1} c)))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0ₓ'. -/
 theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPartENat = c.toNat :=
   if_pos h
@@ -2694,7 +2694,7 @@ theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : c.toPart
 lean 3 declaration is
   forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} c) (Top.top.{0} PartENat PartENat.hasTop))
 but is expected to have type
-  forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) c) PartENat.instTopPartENat))
+  forall {c : Cardinal.{u1}}, (LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} Cardinal.aleph0.{u1} c) -> (Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} c) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) c) PartENat.instTopPartENat))
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_leₓ'. -/
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPartENat = ⊤ :=
   if_neg h.not_lt
@@ -2704,7 +2704,7 @@ theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : c.toPa
 lean 3 declaration is
   forall (n : Nat), Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) n)
 but is expected to have type
-  forall (n : Nat), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) PartENat.instAddCommMonoidWithOnePartENat)) n)
+  forall (n : Nat), Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) PartENat.instAddCommMonoidWithOnePartENat)) n)
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_cast Cardinal.toPartENat_castₓ'. -/
 @[simp]
 theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
@@ -2715,7 +2715,7 @@ theorem toPartENat_cast (n : ℕ) : Cardinal.toPartENat n = n := by
 lean 3 declaration is
   forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} PartENat PartENat.hasTop)
 but is expected to have type
-  forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instTopPartENat)
+  forall {α : Type.{u1}} [h : Infinite.{succ u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instTopPartENat)
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infiniteₓ'. -/
 @[simp]
 theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
@@ -2726,7 +2726,7 @@ theorem mk_toPartENat_of_infinite [h : Infinite α] : (#α).toPartENat = ⊤ :=
 lean 3 declaration is
   Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} PartENat PartENat.hasTop)
 but is expected to have type
-  Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) PartENat.instTopPartENat)
+  Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} Cardinal.aleph0.{u1}) (Top.top.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) Cardinal.aleph0.{u1}) PartENat.instTopPartENat)
 Case conversion may be inaccurate. Consider using '#align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENatₓ'. -/
 @[simp]
 theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
@@ -2737,7 +2737,7 @@ theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
 lean 3 declaration is
   Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1})
 but is expected to have type
-  Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1})
+  Function.Surjective.{succ (succ u1), 1} Cardinal.{u1} PartENat (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1})
 Case conversion may be inaccurate. Consider using '#align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjectiveₓ'. -/
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
@@ -2747,7 +2747,7 @@ theorem toPartENat_surjective : Surjective toPartENat := fun x =>
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{1} PartENat (coeFn.{succ (succ u1), succ (succ u1)} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) (fun (_x : AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) => Cardinal.{u1} -> PartENat) (AddMonoidHom.hasCoeToFun.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne)))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat PartENat (HasLiftT.mk.{1, 1} Nat PartENat (CoeTCₓ.coe.{1, 1} Nat PartENat (Nat.castCoe.{0} PartENat (AddMonoidWithOne.toNatCast.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.addCommMonoidWithOne))))) (Fintype.card.{u1} α _inst_1))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instAddCommMonoidWithOnePartENat)) (Fintype.card.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (FunLike.coe.{succ (succ u1), succ (succ u1), 1} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} (fun (_x : Cardinal.{u1}) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) _x) (AddHomClass.toFunLike.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddZeroClass.toAdd.{succ u1} Cardinal.{u1} (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))))) (AddZeroClass.toAdd.{0} PartENat (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) (AddMonoidHomClass.toAddHomClass.{succ u1, succ u1, 0} (AddMonoidHom.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))) Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat))) (AddMonoidHom.addMonoidHomClass.{succ u1, 0} Cardinal.{u1} PartENat (AddMonoid.toAddZeroClass.{succ u1} Cardinal.{u1} (AddMonoidWithOne.toAddMonoid.{succ u1} Cardinal.{u1} (AddCommMonoidWithOne.toAddMonoidWithOne.{succ u1} Cardinal.{u1} (NonAssocSemiring.toAddCommMonoidWithOne.{succ u1} Cardinal.{u1} (Semiring.toNonAssocSemiring.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))))) (AddMonoid.toAddZeroClass.{0} PartENat (AddMonoidWithOne.toAddMonoid.{0} PartENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} PartENat PartENat.instAddCommMonoidWithOnePartENat)))))) Cardinal.toPartENat.{u1} (Cardinal.mk.{u1} α)) (Nat.cast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (AddMonoidWithOne.toNatCast.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Cardinal.{u1}) => PartENat) (Cardinal.mk.{u1} α)) PartENat.instAddCommMonoidWithOnePartENat)) (Fintype.card.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_cardₓ'. -/
 theorem mk_toPartENat_eq_coe_card [Fintype α] : (#α).toPartENat = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card

bump to nightly-2023-03-11-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

cd44fb92

Diff

@@ -2217,7 +2217,8 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
       refine' (mul_le_mul' le_rfl hb).trans_lt h
     · rw [← one_mul b]
       refine' (mul_le_mul' ha le_rfl).trans_lt h
-  rintro (rfl | rfl | ⟨ha, hb⟩) <;> simp only [*, mul_lt_aleph_0, aleph_0_pos, zero_mul, mul_zero]
+  rintro (rfl | rfl | ⟨ha, hb⟩) <;>
+    simp only [*, mul_lt_aleph_0, aleph_0_pos, MulZeroClass.zero_mul, MulZeroClass.mul_zero]
 #align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff
 
 /- warning: cardinal.aleph_0_le_mul_iff -> Cardinal.aleph0_le_mul_iff is a dubious translation:
@@ -2591,17 +2592,17 @@ Case conversion may be inaccurate. Consider using '#align cardinal.to_nat_mul Ca
 theorem toNat_mul (x y : Cardinal) : (x * y).toNat = x.toNat * y.toNat :=
   by
   rcases eq_or_ne x 0 with (rfl | hx1)
-  · rw [zero_mul, zero_to_nat, zero_mul]
+  · rw [MulZeroClass.zero_mul, zero_to_nat, MulZeroClass.zero_mul]
   rcases eq_or_ne y 0 with (rfl | hy1)
-  · rw [mul_zero, zero_to_nat, mul_zero]
+  · rw [MulZeroClass.mul_zero, zero_to_nat, MulZeroClass.mul_zero]
   cases' lt_or_le x ℵ₀ with hx2 hx2
   · cases' lt_or_le y ℵ₀ with hy2 hy2
     · lift x to ℕ using hx2
       lift y to ℕ using hy2
       rw [← Nat.cast_mul, to_nat_cast, to_nat_cast, to_nat_cast]
-    · rw [to_nat_apply_of_aleph_0_le hy2, mul_zero, to_nat_apply_of_aleph_0_le]
+    · rw [to_nat_apply_of_aleph_0_le hy2, MulZeroClass.mul_zero, to_nat_apply_of_aleph_0_le]
       exact aleph_0_le_mul_iff'.2 (Or.inl ⟨hx1, hy2⟩)
-  · rw [to_nat_apply_of_aleph_0_le hx2, zero_mul, to_nat_apply_of_aleph_0_le]
+  · rw [to_nat_apply_of_aleph_0_le hx2, MulZeroClass.zero_mul, to_nat_apply_of_aleph_0_le]
     exact aleph_0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)
 #align cardinal.to_nat_mul Cardinal.toNat_mul

bump to nightly-2023-03-09-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

75cac28a

Diff

@@ -408,7 +408,7 @@ theorem lift_le {a b : Cardinal} : lift a ≤ lift b ↔ a ≤ b :=
 /-- `cardinal.lift` as an `order_embedding`. -/
 @[simps (config := { fullyApplied := false })]
 def liftOrderEmbedding : Cardinal.{v} ↪o Cardinal.{max v u} :=
-  OrderEmbedding.ofMapLeIff lift fun _ _ => lift_le
+  OrderEmbedding.ofMapLEIff lift fun _ _ => lift_le
 #align cardinal.lift_order_embedding Cardinal.liftOrderEmbedding
 -/

bump to nightly-2023-03-01-20

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

20cadee0

Diff

@@ -3347,7 +3347,7 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
   contrapose! h
   calc
     (#α) = (#(Set.univ : Set α)) := mk_univ.symm
-    _ ≤ (#l.to_finset) := mk_le_mk_of_subset fun x _ => list.mem_to_finset.mpr (h x)
+    _ ≤ (#l.to_finset) := (mk_le_mk_of_subset fun x _ => list.mem_to_finset.mpr (h x))
     _ = l.to_finset.card := Cardinal.mk_coe_finset
     _ ≤ l.length := cardinal.nat_cast_le.mpr (List.toFinset_card_le l)

bump to nightly-2023-02-23-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/3ade05ac9447ae31a22d2ea5423435e054131240

6ff36814

Diff

@@ -795,7 +795,7 @@ theorem power_mul {a b c : Cardinal} : (a^b * c) = ((a^b)^c) :=
 lean 3 declaration is
   forall (a : Cardinal.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.hasPow.{u1}) a ((fun (a : Type) (b : Type.{succ u1}) [self : HasLiftT.{1, succ (succ u1)} a b] => self.0) Nat Cardinal.{u1} (HasLiftT.mk.{1, succ (succ u1)} Nat Cardinal.{u1} (CoeTCₓ.coe.{1, succ (succ u1)} Nat Cardinal.{u1} (Nat.castCoe.{succ u1} Cardinal.{u1} Cardinal.hasNatCast.{u1}))) n)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (CommSemiring.toSemiring.{succ u1} Cardinal.{u1} Cardinal.commSemiring.{u1}))))) a n)
 but is expected to have type
-  forall (a : Cardinal.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (Pow.pow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (CommSemiring.toSemiring.{succ u1} Cardinal.{u1} Cardinal.commSemiring.{u1})))) a n)
+  forall (a : Cardinal.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (HPow.hPow.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHPow.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.instPowCardinal.{u1}) a (Nat.cast.{succ u1} Cardinal.{u1} Cardinal.instNatCastCardinal.{u1} n)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (CommSemiring.toSemiring.{succ u1} Cardinal.{u1} Cardinal.commSemiring.{u1}))))) a n)
 Case conversion may be inaccurate. Consider using '#align cardinal.pow_cast_right Cardinal.pow_cast_rightₓ'. -/
 @[simp]
 theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : (a^(↑n : Cardinal.{u})) = a ^ℕ n :=
@@ -915,7 +915,7 @@ private theorem add_le_add' : ∀ {a b c d : Cardinal}, a ≤ b → c ≤ d →
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1})) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4877 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4879 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4877 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4879) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4892 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4894 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4892 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4894)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4896 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4898 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4896 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4898) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4911 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4913 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4911 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4913)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_covariant_class Cardinal.add_covariantClassₓ'. -/
 instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤ ·) :=
   ⟨fun a b c => add_le_add' le_rfl⟩
@@ -925,7 +925,7 @@ instance add_covariantClass : CovariantClass Cardinal Cardinal (· + ·) (· ≤
 lean 3 declaration is
   CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1}))) (LE.le.{succ u1} Cardinal.{u1} Cardinal.hasLe.{u1})
 but is expected to have type
-  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4944 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4946 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4944 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4946)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4959 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4961 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4959 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4961)
+  CovariantClass.{succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} (Function.swap.{succ (succ u1), succ (succ u1), succ (succ u1)} Cardinal.{u1} Cardinal.{u1} (fun (ᾰ : Cardinal.{u1}) (ᾰ : Cardinal.{u1}) => Cardinal.{u1}) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4963 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4965 : Cardinal.{u1}) => HAdd.hAdd.{succ u1, succ u1, succ u1} Cardinal.{u1} Cardinal.{u1} Cardinal.{u1} (instHAdd.{succ u1} Cardinal.{u1} Cardinal.instAddCardinal.{u1}) x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4963 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4965)) (fun (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4978 : Cardinal.{u1}) (x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4980 : Cardinal.{u1}) => LE.le.{succ u1} Cardinal.{u1} Cardinal.instLECardinal.{u1} x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4978 x._@.Mathlib.SetTheory.Cardinal.Basic._hyg.4980)
 Case conversion may be inaccurate. Consider using '#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClassₓ'. -/
 instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· + ·)) (· ≤ ·) :=
   ⟨fun a b c h => add_le_add' h le_rfl⟩
@@ -1822,7 +1822,7 @@ theorem mk_coe_finset {α : Type u} {s : Finset α} : (#s) = ↑(Finset.card s)
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finset.{u1} α)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (OfNat.mk.{succ u1} Cardinal.{u1} 2 (bit0.{succ u1} Cardinal.{u1} Cardinal.hasAdd.{u1} (One.one.{succ u1} Cardinal.{u1} Cardinal.hasOne.{u1})))) (Fintype.card.{u1} α _inst_1))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finset.{u1} α)) (Pow.pow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (instOfNat.{succ u1} Cardinal.{u1} 2 Cardinal.instNatCastCardinal.{u1} (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (Fintype.card.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : Fintype.{u1} α], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finset.{u1} α)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))) (OfNat.ofNat.{succ u1} Cardinal.{u1} 2 (instOfNat.{succ u1} Cardinal.{u1} 2 Cardinal.instNatCastCardinal.{u1} (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) (Fintype.card.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintypeₓ'. -/
 theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
@@ -1831,7 +1831,7 @@ theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card 
 lean 3 declaration is
   forall (α : Type.{u1}) (β : Type.{u2}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u2} β], Eq.{succ (succ (max u1 u2))} Cardinal.{max u1 u2} (Cardinal.mk.{max u1 u2} (Finsupp.{u1, u2} α β _inst_2)) (HPow.hPow.{succ (max u2 u1), 0, succ (max u2 u1)} Cardinal.{max u2 u1} Nat Cardinal.{max u2 u1} (instHPow.{succ (max u2 u1), 0} Cardinal.{max u2 u1} Nat (Monoid.Pow.{succ (max u2 u1)} Cardinal.{max u2 u1} (MonoidWithZero.toMonoid.{succ (max u2 u1)} Cardinal.{max u2 u1} (Semiring.toMonoidWithZero.{succ (max u2 u1)} Cardinal.{max u2 u1} (OrderedSemiring.toSemiring.{succ (max u2 u1)} Cardinal.{max u2 u1} (OrderedCommSemiring.toOrderedSemiring.{succ (max u2 u1)} Cardinal.{max u2 u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u2 u1)} Cardinal.{max u2 u1} Cardinal.canonicallyOrderedCommSemiring.{max u2 u1}))))))) (Cardinal.lift.{u1, u2} (Cardinal.mk.{u2} β)) (Fintype.card.{u1} α _inst_1))
 but is expected to have type
-  forall (α : Type.{u1}) (β : Type.{u2}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u2} β], Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.mk.{max u2 u1} (Finsupp.{u1, u2} α β _inst_2)) (Pow.pow.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat (Monoid.Pow.{succ (max u1 u2)} Cardinal.{max u1 u2} (MonoidWithZero.toMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (Semiring.toMonoidWithZero.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{max u1 u2})))))) (Cardinal.lift.{u1, u2} (Cardinal.mk.{u2} β)) (Fintype.card.{u1} α _inst_1))
+  forall (α : Type.{u1}) (β : Type.{u2}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u2} β], Eq.{max (succ (succ u1)) (succ (succ u2))} Cardinal.{max u2 u1} (Cardinal.mk.{max u2 u1} (Finsupp.{u1, u2} α β _inst_2)) (HPow.hPow.{succ (max u1 u2), 0, succ (max u1 u2)} Cardinal.{max u1 u2} Nat Cardinal.{max u1 u2} (instHPow.{succ (max u1 u2), 0} Cardinal.{max u1 u2} Nat (Monoid.Pow.{succ (max u1 u2)} Cardinal.{max u1 u2} (MonoidWithZero.toMonoid.{succ (max u1 u2)} Cardinal.{max u1 u2} (Semiring.toMonoidWithZero.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedSemiring.toSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (OrderedCommSemiring.toOrderedSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ (max u1 u2)} Cardinal.{max u1 u2} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{max u1 u2}))))))) (Cardinal.lift.{u1, u2} (Cardinal.mk.{u2} β)) (Fintype.card.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintypeₓ'. -/
 @[simp]
 theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zero β] :
@@ -1843,7 +1843,7 @@ theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zer
 lean 3 declaration is
   forall (α : Type.{u1}) (β : Type.{u1}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u1} β], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finsupp.{u1, u1} α β _inst_2)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))) (Cardinal.mk.{u1} β) (Fintype.card.{u1} α _inst_1))
 but is expected to have type
-  forall (α : Type.{u1}) (β : Type.{u1}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u1} β], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finsupp.{u1, u1} α β _inst_2)) (Pow.pow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))) (Cardinal.mk.{u1} β) (Fintype.card.{u1} α _inst_1))
+  forall (α : Type.{u1}) (β : Type.{u1}) [_inst_1 : Fintype.{u1} α] [_inst_2 : Zero.{u1} β], Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Finsupp.{u1, u1} α β _inst_2)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))) (Cardinal.mk.{u1} β) (Fintype.card.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintypeₓ'. -/
 theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
     (#α →₀ β) = (#β) ^ℕ Fintype.card α := by simp
@@ -2844,7 +2844,7 @@ theorem mk_pLift_false : (#PLift False) = 0 :=
 lean 3 declaration is
   forall (α : Type.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Vector.{u1} α n)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))) (Cardinal.mk.{u1} α) n)
 but is expected to have type
-  forall (α : Type.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Vector.{u1} α n)) (Pow.pow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))) (Cardinal.mk.{u1} α) n)
+  forall (α : Type.{u1}) (n : Nat), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (Vector.{u1} α n)) (HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))) (Cardinal.mk.{u1} α) n)
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_vector Cardinal.mk_vectorₓ'. -/
 @[simp]
 theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
@@ -2855,7 +2855,7 @@ theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
 lean 3 declaration is
   forall (α : Type.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (List.{u1} α)) (Cardinal.sum.{0, u1} Nat (fun (n : Nat) => HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.canonicallyOrderedCommSemiring.{u1}))))))) (Cardinal.mk.{u1} α) n))
 but is expected to have type
-  forall (α : Type.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (List.{u1} α)) (Cardinal.sum.{0, u1} Nat (fun (n : Nat) => Pow.pow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1})))))) (Cardinal.mk.{u1} α) n))
+  forall (α : Type.{u1}), Eq.{succ (succ u1)} Cardinal.{u1} (Cardinal.mk.{u1} (List.{u1} α)) (Cardinal.sum.{0, u1} Nat (fun (n : Nat) => HPow.hPow.{succ u1, 0, succ u1} Cardinal.{u1} Nat Cardinal.{u1} (instHPow.{succ u1, 0} Cardinal.{u1} Nat (Monoid.Pow.{succ u1} Cardinal.{u1} (MonoidWithZero.toMonoid.{succ u1} Cardinal.{u1} (Semiring.toMonoidWithZero.{succ u1} Cardinal.{u1} (OrderedSemiring.toSemiring.{succ u1} Cardinal.{u1} (OrderedCommSemiring.toOrderedSemiring.{succ u1} Cardinal.{u1} (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{succ u1} Cardinal.{u1} Cardinal.instCanonicallyOrderedCommSemiringCardinal.{u1}))))))) (Cardinal.mk.{u1} α) n))
 Case conversion may be inaccurate. Consider using '#align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_powₓ'. -/
 theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α) ^ℕ n :=
   calc

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

chore: split Algebra.Module.Basic (#12501)

Similar to #12486, which did this for Algebra.Algebra.Basic.

Splits Algebra.Module.Defs off Algebra.Module.Basic. Most imports only need the Defs file, which has significantly smaller imports. The remaining Algebra.Module.Basic is now a grab-bag of unrelated results, and should probably be split further or rehomed.

This is mostly motivated by the wasted effort during minimization upon encountering Algebra.Module.Basic.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

7d5102df

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 -/
-import Mathlib.Algebra.Module.Basic
+import Mathlib.Algebra.Module.Defs
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.Finsupp.Defs
 import Mathlib.Data.Set.Countable

chore(SetTheory/Cardinal/Basic): add sections structuring the file better (#11603)

This file is overly long: add some sections to structure the material better.

And tweak one line break to be more logical.

68a86285

Diff

@@ -1308,6 +1308,7 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
 #align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
 
 /-! ### Properties about the cast from `ℕ` -/
+section castFromN
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
@@ -1666,8 +1667,8 @@ theorem add_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a +
 #align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0
 
 theorem add_lt_aleph0_iff {a b : Cardinal} : a + b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ :=
-  ⟨fun h => ⟨(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h⟩, fun ⟨h1, h2⟩ =>
-    add_lt_aleph0 h1 h2⟩
+  ⟨fun h => ⟨(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h⟩,
+   fun ⟨h1, h2⟩ => add_lt_aleph0 h1 h2⟩
 #align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iff
 
 theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b := by
@@ -1829,8 +1830,6 @@ theorem ofNat_add_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n)
 theorem aleph0_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ + no_index (OfNat.ofNat n) = ℵ₀ :=
   aleph0_add_nat n
 
-variable {c : Cardinal}
-
 theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m := by
   lift c to ℕ using h.trans_lt (nat_lt_aleph0 _)
   exact ⟨c, mod_cast h, rfl⟩
@@ -1844,6 +1843,10 @@ theorem mk_pNat : #ℕ+ = ℵ₀ :=
   mk_denumerable ℕ+
 #align cardinal.mk_pnat Cardinal.mk_pNat
 
+end castFromN
+
+variable {c : Cardinal}
+
 /-- **König's theorem** -/
 theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
   lt_of_not_ge fun ⟨F⟩ => by
@@ -1864,6 +1867,9 @@ theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f
     exact hc i a (congr_fun h _)
 #align cardinal.sum_lt_prod Cardinal.sum_lt_prod
 
+/-! Cardinalities of sets: cardinality of empty, finite sets, unions, subsets etc. -/
+section sets
+
 -- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_empty : #Empty = 0 :=
@@ -2151,7 +2157,6 @@ theorem mk_union_le_aleph0 {α} {P Q : Set α} :
     ← countable_union]
 #align cardinal.mk_union_le_aleph_0 Cardinal.mk_union_le_aleph0
 
-
 theorem mk_subtype_of_equiv {α β : Type u} (p : β → Prop) (e : α ≃ β) :
     #{ a : α // p (e a) } = #{ b : β // p b } :=
   mk_congr (Equiv.subtypeEquivOfSubtype e)
@@ -2270,6 +2275,10 @@ theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z
   simpa [not_or] using this
 #align cardinal.three_le Cardinal.three_le
 
+end sets
+
+section powerlt
+
 /-- The function `a ^< b`, defined as the supremum of `a ^ c` for `c < b`. -/
 def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
   ⨆ c : Iio b, a ^ (c : Cardinal)
@@ -2323,6 +2332,8 @@ theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 := by
   exact Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt
 #align cardinal.powerlt_zero Cardinal.powerlt_zero
 
+end powerlt
+
 /-- The cardinality of a nontrivial module over a ring is at least the cardinality of the ring if
 there are no zero divisors (for instance if the ring is a field) -/
 theorem mk_le_of_module (R : Type u) (E : Type v)

chore: remove mathport name: <expression> lines (#11928)

Quoting [@digama0](https://github.com/digama0):

These were actually never meant to go in the file, they are basically debugging information and only useful on significantly broken mathport files. You can safely remove all of them.

11431c65

Diff

@@ -1267,7 +1267,6 @@ def aleph0 : Cardinal.{u} :=
   lift #ℕ
 #align cardinal.aleph_0 Cardinal.aleph0
 
--- mathport name: cardinal.aleph_0
 @[inherit_doc]
 scoped notation "ℵ₀" => Cardinal.aleph0

chore: tidy various files (#11624)

483848e8

Diff

@@ -60,7 +60,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
   `Cardinal.{u} : Type (u + 1)` is the quotient of types in `Type u`.
   The operation `Cardinal.lift` lifts cardinal numbers to a higher level.
 * Cardinal arithmetic specifically for infinite cardinals (like `κ * κ = κ`) is in the file
-  `SetTheory/Cardinal/Ordinal.lean`.
+  `Mathlib/SetTheory/Cardinal/Ordinal.lean`.
 * There is an instance `Pow Cardinal`, but this will only fire if Lean already knows that both
   the base and the exponent live in the same universe. As a workaround, you can add
   ```

feat: add Set.countable_infinite_iff_nonempty_denumerable (#11605)

From sphere-eversion; I'm just submitting this.

2c870c1a

Diff

@@ -1771,6 +1771,10 @@ theorem mk_denumerable (α : Type u) [Denumerable α] : #α = ℵ₀ :=
   denumerable_iff.1 ⟨‹_›⟩
 #align cardinal.mk_denumerable Cardinal.mk_denumerable
 
+theorem _root_.Set.countable_infinite_iff_nonempty_denumerable {α : Type*} {s : Set α} :
+    s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable s) := by
+  rw [nonempty_denumerable_iff, ← Set.infinite_coe_iff, countable_coe_iff]
+
 @[simp]
 theorem aleph0_add_aleph0 : ℵ₀ + ℵ₀ = ℵ₀ :=
   mk_denumerable _

feat: Cardinal.mk_insert_le (#11749)

e6b2fea1

Diff

@@ -2104,6 +2104,11 @@ theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
   simpa
 #align cardinal.mk_insert Cardinal.mk_insert
 
+theorem mk_insert_le {α : Type u} {s : Set α} {a : α} : #(insert a s : Set α) ≤ #s + 1 := by
+  by_cases h : a ∈ s
+  · simp only [insert_eq_of_mem h, self_le_add_right]
+  · rw [mk_insert h]
+
 theorem mk_sum_compl {α} (s : Set α) : #s + #(sᶜ : Set α) = #α :=
   mk_congr (Equiv.Set.sumCompl s)
 #align cardinal.mk_sum_compl Cardinal.mk_sum_compl

change the order of operation in zsmulRec and nsmulRec (#11451)

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

pow_succ and pow_succ' have switched their meanings.
Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

9a3feeae

Diff

@@ -533,7 +533,7 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   nsmul := nsmulRec
   npow n c := c ^ (n : Cardinal)
   npow_zero := @power_zero
-  npow_succ n c := show c ^ (↑(n + 1) : Cardinal) = c * c ^ (↑n : Cardinal)
+  npow_succ n c := show c ^ (↑(n + 1) : Cardinal) = c ^ (↑n : Cardinal) * c
     by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']
   natCast := (fun n => lift.{u} #(Fin n) : ℕ → Cardinal.{u})
   natCast_zero := rfl
@@ -1450,7 +1450,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
 -- @[simp, norm_cast]
 @[norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^ (↑n : Cardinal) := by
-  induction n <;> simp [pow_succ', power_add, *, Pow.pow]
+  induction n <;> simp [pow_succ, power_add, *, Pow.pow]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 
 -- porting note (#10618): simp can prove this

fix: improve simp lemmas around PartENat.ofENat (#11553)

5d242aa4

Diff

@@ -37,7 +37,7 @@ theorem partENatOfENat_toENat (c : Cardinal) : (toENat c : PartENat) = toPartENa
 
 @[simp]
 theorem toPartENat_natCast (n : ℕ) : toPartENat n = n := by
-  simp only [← partENatOfENat_toENat, toENat_nat]; rfl
+  simp only [← partENatOfENat_toENat, toENat_nat, PartENat.ofENat_coe]
 #align cardinal.to_part_enat_cast Cardinal.toPartENat_natCast
 
 theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c := by
@@ -48,7 +48,7 @@ theorem toPartENat_eq_top {c : Cardinal} :
     toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
   rw [← partENatOfENat_toENat, ← PartENat.withTopEquiv_symm_top, ← toENat_eq_top,
     ← PartENat.withTopEquiv.symm.injective.eq_iff]
-  rfl
+  simp
 #align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top
 
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toPartENat c = ⊤ :=
@@ -80,17 +80,15 @@ theorem toPartENat_strictMonoOn : StrictMonoOn toPartENat (Set.Iic ℵ₀) :=
 lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
     toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
   lift c to ℕ∞ using h
-  simp only [← partENatOfENat_toENat, toENat_ofENat, enat_gc _,
-   ← PartENat.withTopOrderIso.symm.le_iff_le]
-  rfl
+  simp_rw [← partENatOfENat_toENat, toENat_ofENat, enat_gc _,
+   ← PartENat.withTopOrderIso.symm.le_iff_le, PartENat.ofENat_le, map_le_map_iff]
 #align to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_of_le_aleph0
 
 lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
     toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
   lift c' to ℕ using hc'
-  simp only [← partENatOfENat_toENat, toENat_nat, ← toENat_le_nat,
-   ← PartENat.withTopOrderIso.symm.le_iff_le]
-  rfl
+  simp_rw [← partENatOfENat_toENat, toENat_nat, ← toENat_le_nat,
+   ← PartENat.withTopOrderIso.symm.le_iff_le, PartENat.ofENat_le, map_le_map_iff]
 #align to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_of_lt_aleph0
 
 lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :

feat: four small lemmas about successors of finite cardinals (#11544)

Co-authored-by: @fpvandoorn

8c5b9da8

Diff

@@ -1488,6 +1488,17 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n := by
   rw [← Nat.cast_succ]
   exact natCast_lt.2 (Nat.lt_succ_self _)
 
+lemma succ_natCast (n : ℕ) : Order.succ (n : Cardinal) = n + 1 := by
+  rw [← Cardinal.nat_succ]
+  norm_cast
+
+lemma natCast_add_one_le_iff {n : ℕ} {c : Cardinal} : n + 1 ≤ c ↔ n < c := by
+  rw [← Order.succ_le_iff, Cardinal.succ_natCast]
+
+lemma two_le_iff_one_lt {c : Cardinal} : 2 ≤ c ↔ 1 < c := by
+  convert natCast_add_one_le_iff
+  norm_cast
+
 @[simp]
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
@@ -1551,6 +1562,10 @@ theorem lt_aleph0 {c : Cardinal} : c < ℵ₀ ↔ ∃ n : ℕ, c = n :=
     exact ⟨Infinite.natEmbedding S⟩, fun ⟨n, e⟩ => e.symm ▸ nat_lt_aleph0 _⟩
 #align cardinal.lt_aleph_0 Cardinal.lt_aleph0
 
+lemma succ_eq_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : Order.succ c = c + 1 := by
+  obtain ⟨n, hn⟩ := Cardinal.lt_aleph0.mp h
+  rw [hn, succ_natCast]
+
 theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
   ⟨fun h n => (nat_lt_aleph0 _).le.trans h, fun h =>
     le_of_not_lt fun hn => by

refactor: do not allow nsmul and zsmul to default automatically (#6262)

This PR removes the default values for nsmul and zsmul, forcing the user to populate them manually. The previous behavior can be obtained by writing nsmul := nsmulRec and zsmul := zsmulRec, which is now in the docstring for these fields.

The motivation here is to make it more obvious when module diamonds are being introduced, or at least where they might be hiding; you can now simply search for nsmulRec in the source code.

Arguably we should do the same thing for intCast, natCast, pow, and zpow too, but diamonds are less common in those fields, so I'll leave them to a subsequent PR.

Co-authored-by: Matthew Ballard <matt@mrb.email>

e42f78a9

Diff

@@ -530,6 +530,7 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   mul_comm := mul_comm'
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
+  nsmul := nsmulRec
   npow n c := c ^ (n : Cardinal)
   npow_zero := @power_zero
   npow_succ n c := show c ^ (↑(n + 1) : Cardinal) = c * c ^ (↑n : Cardinal)

chore: classify deprecated porting notes (#11230)

Classifies by adding issue number #11229 to porting notes claiming:

deprecated

1c5c3fac

Diff

@@ -538,7 +538,7 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   natCast_zero := rfl
   natCast_succ := Cardinal.cast_succ
 
-/-! Porting note: Deprecated section. Remove. -/
+/-! Porting note (#11229): Deprecated section. Remove. -/
 section deprecated
 set_option linter.deprecated false
 
@@ -618,7 +618,7 @@ theorem lift_mul (a b : Cardinal.{u}) : lift.{v} (a * b) = lift.{v} a * lift.{v}
     mk_congr <| Equiv.ulift.trans (Equiv.prodCongr Equiv.ulift Equiv.ulift).symm
 #align cardinal.lift_mul Cardinal.lift_mul
 
-/-! Porting note: Deprecated section. Remove. -/
+/-! Porting note (#11229): Deprecated section. Remove. -/
 section deprecated
 set_option linter.deprecated false

chore: Remove ball and bex from lemma names (#10816)

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

c697e040

Diff

@@ -63,7 +63,7 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c)
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
 theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by
-  simp only [← range_natCast, StrictMonoOn, forall_range_iff, toNat_natCast, Nat.cast_lt]
+  simp only [← range_natCast, StrictMonoOn, forall_mem_range, toNat_natCast, Nat.cast_lt]
   exact fun _ _ ↦ id
 
 theorem toNat_monotoneOn : MonotoneOn toNat (Iio ℵ₀) := toNat_strictMonoOn.monotoneOn

chore: scope open Classical (#11199)

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

c747be0a

Diff

@@ -81,7 +81,8 @@ Cantor's theorem, König's theorem, Konig's theorem
 -/
 
 
-open Function Set Order BigOperators Classical
+open scoped Classical
+open Function Set Order BigOperators
 
 noncomputable section
 
@@ -766,13 +767,13 @@ end OrderProperties
 
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
-    byContradiction fun h => by
+    by_contradiction fun h => by
       let ι := { c : Cardinal // ¬Acc (· < ·) c }
       let f : ι → Cardinal := Subtype.val
       haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩
       obtain ⟨⟨c : Cardinal, hc : ¬Acc (· < ·) c⟩, ⟨h_1 : ∀ j, (f ⟨c, hc⟩).out ↪ (f j).out⟩⟩ :=
         Embedding.min_injective fun i => (f i).out
-      refine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_)
+      refine hc (Acc.intro _ fun j h' => by_contradiction fun hj => h'.2 ?_)
       have : #_ ≤ #_ := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -705,7 +705,7 @@ instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=
 instance : CommMonoidWithZero Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
 
--- porting note: new
+-- Porting note: new
 -- Computable instance to prevent a non-computable one being found via the one above
 instance : CommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
@@ -1444,7 +1444,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
   @mk_coe_finset _ s ▸ mk_set_le _
 #align cardinal.card_le_of_finset Cardinal.card_le_of_finset
 
--- porting note: was `simp`. LHS is not normal form.
+-- Porting note: was `simp`. LHS is not normal form.
 -- @[simp, norm_cast]
 @[norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^ (↑n : Cardinal) := by

feat(SetTheory/Cardinal): sInf_eq_zero_iff, iInf_eq_zero_iff (#11132)

Add the two lemmas

lemma sInf_eq_zero_iff {s : Set Cardinal} : sInf s = 0 ↔ s = ∅ ∨ ∃ a ∈ s, a = 0 := by

and

lemma iInf_eq_zero_iff {ι : Sort*} {f : ι → Cardinal} :
    (⨅ i, f i) = 0 ↔ IsEmpty ι ∨ ∃ i, f i = 0 := by

Feel free to golf.

From AperiodicMonotilesLean.

8102736d

Diff

@@ -795,6 +795,19 @@ theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg Set.not_nonempty_empty
 #align cardinal.Inf_empty Cardinal.sInf_empty
 
+lemma sInf_eq_zero_iff {s : Set Cardinal} : sInf s = 0 ↔ s = ∅ ∨ ∃ a ∈ s, a = 0 := by
+  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
+  · rcases s.eq_empty_or_nonempty with rfl | hne
+    · exact Or.inl rfl
+    · exact Or.inr ⟨sInf s, csInf_mem hne, h⟩
+  · rcases h with rfl | ⟨a, ha, rfl⟩
+    · exact Cardinal.sInf_empty
+    · exact eq_bot_iff.2 (csInf_le' ha)
+
+lemma iInf_eq_zero_iff {ι : Sort*} {f : ι → Cardinal} :
+    (⨅ i, f i) = 0 ↔ IsEmpty ι ∨ ∃ i, f i = 0 := by
+  simp [iInf, sInf_eq_zero_iff]
+
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' })

chore: more backporting of simp changes from #10995 (#11001)

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

d9589c14

Diff

@@ -2091,7 +2091,7 @@ theorem mk_le_iff_forall_finset_subset_card_le {α : Type u} {n : ℕ} {t : Set
     Finset.card_image_of_injOn (injOn_of_injective Subtype.coe_injective _)
   rw [← this]
   apply H
-  simp only [Finset.coe_image, image_subset_iff, Subtype.coe_preimage_self, subset_univ]
+  simp only [u, Finset.coe_image, image_subset_iff, Subtype.coe_preimage_self, subset_univ]
 
 theorem mk_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) :
     #{ x // p x } ≤ #{ x // q x } :=

style: fix typos in porting notes (#10931)

8db59278

Diff

@@ -1034,7 +1034,7 @@ lemma exists_eq_of_iSup_eq_of_not_isLimit
   rw [← le_zero_iff] at h ⊢
   exact (le_ciSup hf _).trans h
 
--- Portin note: simpNF is not happy with universe levels.
+-- Porting note: simpNF is not happy with universe levels.
 @[simp, nolint simpNF]
 theorem lift_mk_shrink (α : Type u) [Small.{v} α] :
     Cardinal.lift.{max u w} #(Shrink.{v} α) = Cardinal.lift.{max v w} #α :=

chore: add lemmas for nat literals corresponding to lemmas for nat casts (#8006)

I loogled for every occurrence of "cast", Nat and "natCast" and where the casted nat was n, and made sure there were corresponding @[simp] lemmas for 0, 1, and OfNat.ofNat n. This is necessary in general for simp confluence. Example:

import Mathlib

variable {α : Type*} [LinearOrderedRing α] (m n : ℕ) [m.AtLeastTwo] [n.AtLeastTwo]

example : ((OfNat.ofNat m : ℕ) : α) ≤ ((OfNat.ofNat n : ℕ) : α) ↔ (OfNat.ofNat m : ℕ) ≤ (OfNat.ofNat n : ℕ) := by
  simp only [Nat.cast_le] -- this `@[simp]` lemma can apply

example : ((OfNat.ofNat m : ℕ) : α) ≤ ((OfNat.ofNat n : ℕ) : α) ↔ (OfNat.ofNat m : α) ≤ (OfNat.ofNat n : α) := by
  simp only [Nat.cast_ofNat] -- and so can this one

example : (OfNat.ofNat m : α) ≤ (OfNat.ofNat n : α) ↔ (OfNat.ofNat m : ℕ) ≤ (OfNat.ofNat n : ℕ) := by
  simp -- fails! `simp` doesn't have a lemma to bridge their results. confluence issue.

As far as I know, the only file this PR leaves with ofNat gaps is PartENat.lean. #8002 is addressing that file in parallel.

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

bbe9baad

Diff

@@ -601,6 +601,10 @@ theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : a ^ (↑n : Cardinal.{u})
 theorem lift_one : lift 1 = 1 := mk_eq_one _
 #align cardinal.lift_one Cardinal.lift_one
 
+@[simp]
+theorem lift_eq_one {a : Cardinal.{v}} : lift.{u} a = 1 ↔ a = 1 :=
+  lift_injective.eq_iff' lift_one
+
 @[simp]
 theorem lift_add (a b : Cardinal.{u}) : lift.{v} (a + b) = lift.{v} a + lift.{v} b :=
   inductionOn₂ a b fun _ _ =>
@@ -1300,37 +1304,109 @@ theorem mk_fin (n : ℕ) : #(Fin n) = n := by simp
 theorem lift_natCast (n : ℕ) : lift.{u} (n : Cardinal.{v}) = n := by induction n <;> simp [*]
 #align cardinal.lift_nat_cast Cardinal.lift_natCast
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem lift_ofNat (n : ℕ) [n.AtLeastTwo] :
+    lift.{u} (no_index (OfNat.ofNat n : Cardinal.{v})) = OfNat.ofNat n :=
+  lift_natCast n
+
 @[simp]
 theorem lift_eq_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a = n ↔ a = n :=
   lift_injective.eq_iff' (lift_natCast n)
 #align cardinal.lift_eq_nat_iff Cardinal.lift_eq_nat_iff
 
+@[simp]
+theorem lift_eq_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    lift.{v} a = (no_index (OfNat.ofNat n)) ↔ a = OfNat.ofNat n :=
+  lift_eq_nat_iff
+
 @[simp]
 theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
     (n : Cardinal) = lift.{v} a ↔ (n : Cardinal) = a := by
   rw [← lift_natCast.{v,u} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
 
+@[simp]
+theorem zero_eq_lift_iff {a : Cardinal.{u}} :
+    (0 : Cardinal) = lift.{v} a ↔ 0 = a := by
+  simpa using nat_eq_lift_iff (n := 0)
+
+@[simp]
+theorem one_eq_lift_iff {a : Cardinal.{u}} :
+    (1 : Cardinal) = lift.{v} a ↔ 1 = a := by
+  simpa using nat_eq_lift_iff (n := 1)
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem ofNat_eq_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    (no_index (OfNat.ofNat n : Cardinal)) = lift.{v} a ↔ (OfNat.ofNat n : Cardinal) = a :=
+  nat_eq_lift_iff
+
 @[simp]
 theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
   rw [← lift_natCast.{v,u}, lift_le]
 #align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
 
+@[simp]
+theorem lift_le_one_iff {a : Cardinal.{u}} :
+    lift.{v} a ≤ 1 ↔ a ≤ 1 := by
+  simpa using lift_le_nat_iff (n := 1)
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem lift_le_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    lift.{v} a ≤ (no_index (OfNat.ofNat n)) ↔ a ≤ OfNat.ofNat n :=
+  lift_le_nat_iff
+
 @[simp]
 theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} : n ≤ lift.{v} a ↔ n ≤ a := by
   rw [← lift_natCast.{v,u}, lift_le]
 #align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
 
+@[simp]
+theorem one_le_lift_iff {a : Cardinal.{u}} :
+    (1 : Cardinal) ≤ lift.{v} a ↔ 1 ≤ a := by
+  simpa using nat_le_lift_iff (n := 1)
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem ofNat_le_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    (no_index (OfNat.ofNat n : Cardinal)) ≤ lift.{v} a ↔ (OfNat.ofNat n : Cardinal) ≤ a :=
+  nat_le_lift_iff
+
 @[simp]
 theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
   rw [← lift_natCast.{v,u}, lift_lt]
 #align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem lift_lt_ofNat_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    lift.{v} a < (no_index (OfNat.ofNat n)) ↔ a < OfNat.ofNat n :=
+  lift_lt_nat_iff
+
 @[simp]
 theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} : n < lift.{v} a ↔ n < a := by
   rw [← lift_natCast.{v,u}, lift_lt]
 #align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem zero_lt_lift_iff {a : Cardinal.{u}} :
+    (0 : Cardinal) < lift.{v} a ↔ 0 < a := by
+  simpa using nat_lt_lift_iff (n := 0)
+
+@[simp]
+theorem one_lt_lift_iff {a : Cardinal.{u}} :
+    (1 : Cardinal) < lift.{v} a ↔ 1 < a := by
+  simpa using nat_lt_lift_iff (n := 1)
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem ofNat_lt_lift_iff {a : Cardinal.{u}} {n : ℕ} [n.AtLeastTwo] :
+    (no_index (OfNat.ofNat n : Cardinal)) < lift.{v} a ↔ (OfNat.ofNat n : Cardinal) < a :=
+  nat_lt_lift_iff
+
 theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n := rfl
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
 
@@ -1429,6 +1505,10 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by
   rw [one_le_iff_pos, pos_iff_ne_zero]
 #align cardinal.one_le_iff_ne_zero Cardinal.one_le_iff_ne_zero
 
+@[simp]
+theorem lt_one_iff_zero {c : Cardinal} : c < 1 ↔ c = 0 := by
+  simpa using lt_succ_bot_iff (a := c)
+
 theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
   succ_le_iff.1
     (by

feat: small_iUnion and small_sUnion (#10921)

Also moves the other results about Small on sets to their own file.

89415b39

Diff

@@ -7,7 +7,7 @@ import Mathlib.Algebra.Module.Basic
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.Finsupp.Defs
 import Mathlib.Data.Set.Countable
-import Mathlib.Logic.Small.Basic
+import Mathlib.Logic.Small.Set
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 import Mathlib.Order.SuccPred.CompleteLinearOrder
 import Mathlib.SetTheory.Cardinal.SchroederBernstein

style: reduce spacing variation in "porting note" comments (#10886)

In this pull request, I have systematically eliminated the leading whitespace preceding the colon (:) within all unlabelled or unclassified porting notes. This adjustment facilitates a more efficient review process for the remaining notes by ensuring no entries are overlooked due to formatting inconsistencies.

86b84c52

Diff

@@ -194,7 +194,7 @@ theorem mk_uLift (α) : #(ULift.{v, u} α) = lift.{v} #α :=
   rfl
 #align cardinal.mk_ulift Cardinal.mk_uLift
 
--- Porting note : simpNF is not happy with universe levels, but this is needed as simp lemma
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
 -- further down in this file
 /-- `lift.{max u v, u}` equals `lift.{v, u}`. -/
 @[simp, nolint simpNF]
@@ -202,7 +202,7 @@ theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
   funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
 #align cardinal.lift_umax Cardinal.lift_umax
 
--- Porting note : simpNF is not happy with universe levels, but this is needed as simp lemma
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
 -- further down in this file
 /-- `lift.{max v u, u}` equals `lift.{v, u}`. -/
 @[simp, nolint simpNF]
@@ -210,7 +210,7 @@ theorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=
   lift_umax
 #align cardinal.lift_umax' Cardinal.lift_umax'
 
--- Porting note : simpNF is not happy with universe levels, but this is needed as simp lemma
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
 -- further down in this file
 /-- A cardinal lifted to a lower or equal universe equals itself. -/
 @[simp, nolint simpNF]

chore: remove terminal, terminal refines (#10762)

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

ea36aa7d

Diff

@@ -1597,9 +1597,9 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
     rw [← Ne, ← one_le_iff_ne_zero] at ha hb
     constructor
     · rw [← mul_one a]
-      refine' (mul_le_mul' le_rfl hb).trans_lt h
+      exact (mul_le_mul' le_rfl hb).trans_lt h
     · rw [← one_mul b]
-      refine' (mul_le_mul' ha le_rfl).trans_lt h
+      exact (mul_le_mul' ha le_rfl).trans_lt h
   rintro (rfl | rfl | ⟨ha, hb⟩) <;> simp only [*, mul_lt_aleph0, aleph0_pos, zero_mul, mul_zero]
 #align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff

chore: classify simp can do this porting notes (#10619)

Classify by adding issue number (#10618) to porting notes claiming anything semantically equivalent to simp can prove this or simp can simplify this.

de765f60

Diff

@@ -135,6 +135,8 @@ theorem aleph0_toNat : toNat ℵ₀ = 0 :=
 theorem mk_toNat_eq_card [Fintype α] : toNat #α = Fintype.card α := by simp
 #align cardinal.mk_to_nat_eq_card Cardinal.mk_toNat_eq_card
 
+-- porting note (#10618): simp can prove this
+-- @[simp]
 theorem zero_toNat : toNat 0 = 0 := map_zero _
 #align cardinal.zero_to_nat Cardinal.zero_toNat

chore: classify simp can do this porting notes (#10619)

Classify by adding issue number (#10618) to porting notes claiming anything semantically equivalent to simp can prove this or simp can simplify this.

de765f60

Diff

@@ -225,7 +225,7 @@ theorem lift_id (a : Cardinal) : lift.{u, u} a = a :=
 #align cardinal.lift_id Cardinal.lift_id
 
 /-- A cardinal lifted to the zero universe equals itself. -/
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=
   lift_id'.{0, u} a
@@ -558,12 +558,12 @@ theorem one_power {a : Cardinal} : (1 : Cardinal) ^ a = 1 :=
   inductionOn a fun _ => mk_eq_one _
 #align cardinal.one_power Cardinal.one_power
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_bool : #Bool = 2 := by simp
 #align cardinal.mk_bool Cardinal.mk_bool
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_Prop : #Prop = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
@@ -1291,7 +1291,7 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
 
 /-! ### Properties about the cast from `ℕ` -/
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_fin (n : ℕ) : #(Fin n) = n := by simp
 #align cardinal.mk_fin Cardinal.mk_fin
@@ -1362,7 +1362,7 @@ theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^
   induction n <;> simp [pow_succ', power_add, *, Pow.pow]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp, norm_cast]
 @[norm_cast]
 theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
@@ -1370,7 +1370,7 @@ theorem natCast_le {m n : ℕ} : (m : Cardinal) ≤ n ↔ m ≤ n := by
     Fintype.card_fin, Fintype.card_fin]
 #align cardinal.nat_cast_le Cardinal.natCast_le
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp, norm_cast]
 @[norm_cast]
 theorem natCast_lt {m n : ℕ} : (m : Cardinal) < n ↔ m < n := by
@@ -1508,7 +1508,7 @@ theorem lt_aleph0_of_finite (α : Type u) [Finite α] : #α < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem lt_aleph0_iff_set_finite {S : Set α} : #S < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
@@ -1531,7 +1531,7 @@ theorem mk_le_aleph0 [Countable α] : #α ≤ ℵ₀ :=
   mk_le_aleph0_iff.mpr ‹_›
 #align cardinal.mk_le_aleph_0 Cardinal.mk_le_aleph0
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable := mk_le_aleph0_iff
 #align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable
@@ -1655,7 +1655,7 @@ theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ #α = ℵ
     exact ⟨Denumerable.mk' <| f.trans Equiv.ulift⟩⟩
 #align cardinal.denumerable_iff Cardinal.denumerable_iff
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_denumerable (α : Type u) [Denumerable α] : #α = ℵ₀ :=
   denumerable_iff.1 ⟨‹_›⟩
@@ -1751,19 +1751,19 @@ theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f
     exact hc i a (congr_fun h _)
 #align cardinal.sum_lt_prod Cardinal.sum_lt_prod
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_empty : #Empty = 0 :=
   mk_eq_zero _
 #align cardinal.mk_empty Cardinal.mk_empty
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_pempty : #PEmpty = 0 :=
   mk_eq_zero _
 #align cardinal.mk_pempty Cardinal.mk_pempty
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_punit : #PUnit = 1 :=
   mk_eq_one PUnit
@@ -1773,19 +1773,19 @@ theorem mk_unit : #Unit = 1 :=
   mk_punit
 #align cardinal.mk_unit Cardinal.mk_unit
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_singleton {α : Type u} (x : α) : #({x} : Set α) = 1 :=
   mk_eq_one _
 #align cardinal.mk_singleton Cardinal.mk_singleton
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_plift_true : #(PLift True) = 1 :=
   mk_eq_one _
 #align cardinal.mk_plift_true Cardinal.mk_plift_true
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_plift_false : #(PLift False) = 0 :=
   mk_eq_zero _
@@ -1815,7 +1815,7 @@ theorem mk_subtype_le_of_subset {α : Type u} {p q : α → Prop} (h : ∀ ⦃x
   ⟨Embedding.subtypeMap (Embedding.refl α) h⟩
 #align cardinal.mk_subtype_le_of_subset Cardinal.mk_subtype_le_of_subset
 
--- Porting note : simp can prove this
+-- porting note (#10618): simp can prove this
 -- @[simp]
 theorem mk_emptyCollection (α : Type u) : #(∅ : Set α) = 0 :=
   mk_eq_zero _

chore: classify was simp porting notes (#10746)

Classifies by adding issue number (#10745) to porting notes claiming was simp.

71e045ac

Diff

@@ -1355,7 +1355,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
   @mk_coe_finset _ s ▸ mk_set_le _
 #align cardinal.card_le_of_finset Cardinal.card_le_of_finset
 
--- Porting note: was `simp`. LHS is not normal form.
+-- porting note: was `simp`. LHS is not normal form.
 -- @[simp, norm_cast]
 @[norm_cast]
 theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^ (↑n : Cardinal) := by

chore(SetTheory/Cardinal/Basic): Cardinal.exists_finset_le_card (#10651)

10ab06c9

Diff

@@ -1401,13 +1401,19 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n := by
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
 
+theorem exists_finset_le_card (α : Type*) (n : ℕ) (h : n ≤ #α) :
+    ∃ s : Finset α, n ≤ s.card := by
+  obtain hα|hα := finite_or_infinite α
+  · let hα := Fintype.ofFinite α
+    use Finset.univ
+    simpa only [mk_fintype, Nat.cast_le] using h
+  · obtain ⟨s, hs⟩ := Infinite.exists_subset_card_eq α n
+    exact ⟨s, hs.ge⟩
+
 theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : #α ≤ n := by
-  refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
-  rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn
-  cases' hn with f
-  refine' (H <| Finset.univ.map f).not_lt _
-  rw [Finset.card_map, ← Fintype.card, Fintype.card_ulift, Fintype.card_fin]
-  exact n.lt_succ_self
+  contrapose! H
+  apply exists_finset_le_card α (n+1)
+  simpa only [nat_succ, succ_le_iff] using H
 #align cardinal.card_le_of Cardinal.card_le_of
 
 theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a := by

chore: remove stream-of-consciousness uses of have, replace and suffices (#10640)

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

a13e8040

Diff

@@ -2140,8 +2140,8 @@ theorem exists_not_mem_of_length_lt {α : Type*} (l : List α) (h : ↑l.length
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
 theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
-  have : ↑(3 : ℕ) ≤ #α; simpa using h
-  have : ↑(2 : ℕ) < #α; rwa [← succ_le_iff, ← Cardinal.nat_succ]
+  have : ↑(3 : ℕ) ≤ #α := by simpa using h
+  have : ↑(2 : ℕ) < #α := by rwa [← succ_le_iff, ← Cardinal.nat_succ]
   have := exists_not_mem_of_length_lt [x, y] this
   simpa [not_or] using this
 #align cardinal.three_le Cardinal.three_le

refactor(Cardinal): redefine toNat and toPartENat (#10472)

Redefine these operations in terms of toENat.

6af8f3b3

Diff

@@ -3,7 +3,7 @@ Copyright (c) 2021 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
 -/
-import Mathlib.SetTheory.Cardinal.Basic
+import Mathlib.SetTheory.Cardinal.ENat
 
 #align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
 
@@ -16,101 +16,115 @@ We also prove basic lemmas about this definition.
 -/
 
 universe u v
-open Function
+open Function Set
 open scoped BigOperators
 
 namespace Cardinal
 
-variable {α : Type u} {c : Cardinal}
+variable {α : Type u} {c d : Cardinal.{u}}
 
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to 0. -/
-noncomputable def toNat : ZeroHom Cardinal ℕ where
-  toFun c := if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0
-  map_zero' := by
-    have h : 0 < ℵ₀ := nat_lt_aleph0 0
-    dsimp only
-    rw [dif_pos h, ← Cardinal.natCast_inj, ← Classical.choose_spec (lt_aleph0.1 h),
-      Nat.cast_zero]
+noncomputable def toNat : Cardinal →*₀ ℕ :=
+  ENat.toNat.comp toENat
 #align cardinal.to_nat Cardinal.toNat
+#align cardinal.to_nat_hom Cardinal.toNat
+
+@[simp] lemma toNat_toENat (a : Cardinal) : ENat.toNat (toENat a) = toNat a := rfl
+
+@[simp]
+theorem toNat_ofENat (n : ℕ∞) : toNat n = ENat.toNat n :=
+  congr_arg ENat.toNat <| toENat_ofENat n
+
+@[simp, norm_cast] theorem toNat_natCast (n : ℕ) : toNat n = n := toNat_ofENat n
 
 @[simp]
 lemma toNat_eq_zero : toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c := by
-  simp only [toNat, ZeroHom.coe_mk, dite_eq_right_iff, or_iff_not_imp_right, not_le]
-  refine' forall_congr' fun h => _
-  rw [← @Nat.cast_eq_zero Cardinal, ← Classical.choose_spec (p := fun n : ℕ ↦ c = n)]
+  rw [← toNat_toENat, ENat.toNat_eq_zero, toENat_eq_zero, toENat_eq_top]
 
 lemma toNat_ne_zero : toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀ := by simp [not_or]
 @[simp] lemma toNat_pos : 0 < toNat c ↔ c ≠ 0 ∧ c < ℵ₀ := pos_iff_ne_zero.trans toNat_ne_zero
 
+theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by
+  lift c to ℕ using h
+  rw [toNat_natCast]
+#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0
+
 theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :
     toNat c = Classical.choose (lt_aleph0.1 h) :=
-  dif_pos h
+  Nat.cast_injective <| by rw [cast_toNat_of_lt_aleph0 h, ← Classical.choose_spec (lt_aleph0.1 h)]
 #align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0
 
-theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toNat c = 0 :=
-  dif_neg h.not_lt
+theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toNat c = 0 := by simp [h]
 #align cardinal.to_nat_apply_of_aleph_0_le Cardinal.toNat_apply_of_aleph0_le
 
-theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by
-  rw [toNat_apply_of_lt_aleph0 h, ← Classical.choose_spec (lt_aleph0.1 h)]
-#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0
-
 theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c) = (0 : Cardinal) := by
   rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
-/-- Two finite cardinals are equal iff they are equal their to_nat are equal -/
-theorem toNat_eq_iff_eq_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c = toNat d ↔ c = d := by
-  rw [← natCast_inj, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
+theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by
+  simp only [← range_natCast, StrictMonoOn, forall_range_iff, toNat_natCast, Nat.cast_lt]
+  exact fun _ _ ↦ id
+
+theorem toNat_monotoneOn : MonotoneOn toNat (Iio ℵ₀) := toNat_strictMonoOn.monotoneOn
+
+theorem toNat_injOn : InjOn toNat (Iio ℵ₀) := toNat_strictMonoOn.injOn
+
+/-- Two finite cardinals are equal
+iff they are equal their `Cardinal.toNat` projections are equal. -/
+theorem toNat_eq_iff_eq_of_lt_aleph0 (hc : c < ℵ₀) (hd : d < ℵ₀) :
+    toNat c = toNat d ↔ c = d :=
+  toNat_injOn.eq_iff hc hd
 #align cardinal.to_nat_eq_iff_eq_of_lt_aleph_0 Cardinal.toNat_eq_iff_eq_of_lt_aleph0
 
-theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c ≤ toNat d ↔ c ≤ d := by
-  rw [← natCast_le, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
+theorem toNat_le_iff_le_of_lt_aleph0 (hc : c < ℵ₀) (hd : d < ℵ₀) :
+    toNat c ≤ toNat d ↔ c ≤ d :=
+  toNat_strictMonoOn.le_iff_le hc hd
 #align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
 
-theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c < toNat d ↔ c < d := by
-  rw [← natCast_lt, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
+theorem toNat_lt_iff_lt_of_lt_aleph0 (hc : c < ℵ₀) (hd : d < ℵ₀) :
+    toNat c < toNat d ↔ c < d :=
+  toNat_strictMonoOn.lt_iff_lt hc hd
 #align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
 
-theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
+@[gcongr]
+theorem toNat_le_toNat (hcd : c ≤ d) (hd : d < ℵ₀) : toNat c ≤ toNat d :=
+  toNat_monotoneOn (hcd.trans_lt hd) hd hcd
+#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_toNat
+
+@[deprecated toNat_le_toNat]
+theorem toNat_le_of_le_of_lt_aleph0 (hd : d < ℵ₀) (hcd : c ≤ d) :
     toNat c ≤ toNat d :=
-  (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
-#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
+  toNat_le_toNat hcd hd
 
-theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
-    toNat c < toNat d :=
-  (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
-#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
+theorem toNat_lt_toNat (hcd : c < d) (hd : d < ℵ₀) : toNat c < toNat d :=
+  toNat_strictMonoOn (hcd.trans hd) hd hcd
+#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_toNat
 
-@[simp]
-theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n := by
-  rw [toNat_apply_of_lt_aleph0 (nat_lt_aleph0 n), ← natCast_inj]
-  exact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm
-#align cardinal.to_nat_cast Cardinal.toNat_cast
+@[deprecated toNat_lt_toNat]
+theorem toNat_lt_of_lt_of_lt_aleph0 (hd : d < ℵ₀) (hcd : c < d) : toNat c < toNat d :=
+  toNat_lt_toNat hcd hd
+
+@[deprecated] alias toNat_cast := toNat_natCast
+#align cardinal.to_nat_cast Cardinal.toNat_natCast
 
 -- See note [no_index around OfNat.ofNat]
 @[simp]
 theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] :
     Cardinal.toNat (no_index (OfNat.ofNat n)) = OfNat.ofNat n :=
-  toNat_cast n
+  toNat_natCast n
 
 /-- `toNat` has a right-inverse: coercion. -/
 theorem toNat_rightInverse : Function.RightInverse ((↑) : ℕ → Cardinal) toNat :=
-  toNat_cast
+  toNat_natCast
 #align cardinal.to_nat_right_inverse Cardinal.toNat_rightInverse
 
 theorem toNat_surjective : Surjective toNat :=
   toNat_rightInverse.surjective
 #align cardinal.to_nat_surjective Cardinal.toNat_surjective
 
-
 @[simp]
-theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 :=
-  dif_neg (infinite_iff.1 h).not_lt
+theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 := by simp
 #align cardinal.mk_to_nat_of_infinite Cardinal.mk_toNat_of_infinite
 
 @[simp]
@@ -121,45 +135,33 @@ theorem aleph0_toNat : toNat ℵ₀ = 0 :=
 theorem mk_toNat_eq_card [Fintype α] : toNat #α = Fintype.card α := by simp
 #align cardinal.mk_to_nat_eq_card Cardinal.mk_toNat_eq_card
 
--- Porting note : simp can prove this
--- @[simp]
-theorem zero_toNat : toNat 0 = 0 := by rw [← toNat_cast 0, Nat.cast_zero]
+theorem zero_toNat : toNat 0 = 0 := map_zero _
 #align cardinal.zero_to_nat Cardinal.zero_toNat
 
-@[simp]
-theorem one_toNat : toNat 1 = 1 := by rw [← toNat_cast 1, Nat.cast_one]
+theorem one_toNat : toNat 1 = 1 := map_one _
 #align cardinal.one_to_nat Cardinal.one_toNat
 
-theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c = n :=
-  ⟨fun h =>
-    (cast_toNat_of_lt_aleph0
-            (lt_of_not_ge (hn ∘ h.symm.trans ∘ toNat_apply_of_aleph0_le))).symm.trans
-      (congr_arg _ h),
-    fun h => (congr_arg toNat h).trans (toNat_cast n)⟩
+theorem toNat_eq_iff {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c = n := by
+  rw [← toNat_toENat, ENat.toNat_eq_iff hn, toENat_eq_nat]
 #align cardinal.to_nat_eq_iff Cardinal.toNat_eq_iff
 
 /-- A version of `toNat_eq_iff` for literals -/
-theorem toNat_eq_ofNat {c : Cardinal} {n : ℕ} [Nat.AtLeastTwo n] :
+theorem toNat_eq_ofNat {n : ℕ} [Nat.AtLeastTwo n] :
     toNat c = OfNat.ofNat n ↔ c = OfNat.ofNat n :=
-  toNat_eq_iff <| Nat.cast_ne_zero.mpr <| OfNat.ofNat_ne_zero n
+  toNat_eq_iff <| OfNat.ofNat_ne_zero n
 
 @[simp]
-theorem toNat_eq_one {c : Cardinal} : toNat c = 1 ↔ c = 1 := by
+theorem toNat_eq_one : toNat c = 1 ↔ c = 1 := by
   rw [toNat_eq_iff one_ne_zero, Nat.cast_one]
 #align cardinal.to_nat_eq_one Cardinal.toNat_eq_one
 
-theorem toNat_eq_one_iff_unique {α : Type*} : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
+theorem toNat_eq_one_iff_unique : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
   toNat_eq_one.trans eq_one_iff_unique
 #align cardinal.to_nat_eq_one_iff_unique Cardinal.toNat_eq_one_iff_unique
 
 @[simp]
 theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by
-  apply natCast_injective
-  cases' lt_or_ge c ℵ₀ with hc hc
-  · rw [cast_toNat_of_lt_aleph0, ← lift_natCast.{u,v}, cast_toNat_of_lt_aleph0 hc]
-    rwa [lift_lt_aleph0]
-  · rw [cast_toNat_of_aleph0_le, ← lift_natCast.{u,v}, cast_toNat_of_aleph0_le hc, lift_zero]
-    rwa [aleph0_le_lift]
+  simp only [← toNat_toENat, toENat_lift]
 #align cardinal.to_nat_lift Cardinal.toNat_lift
 
 theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat #α = toNat #β := by
@@ -167,45 +169,28 @@ theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat #α = toNat #β := by
   rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]
 #align cardinal.to_nat_congr Cardinal.toNat_congr
 
-@[simp]
-theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y := by
-  rcases eq_or_ne x 0 with (rfl | hx1)
-  · rw [zero_mul, zero_toNat, zero_mul]
-  rcases eq_or_ne y 0 with (rfl | hy1)
-  · rw [mul_zero, zero_toNat, mul_zero]
-  cases' lt_or_le x ℵ₀ with hx2 hx2
-  · cases' lt_or_le y ℵ₀ with hy2 hy2
-    · lift x to ℕ using hx2
-      lift y to ℕ using hy2
-      rw [← Nat.cast_mul, toNat_cast, toNat_cast, toNat_cast]
-    · rw [toNat_apply_of_aleph0_le hy2, mul_zero, toNat_apply_of_aleph0_le]
-      exact aleph0_le_mul_iff'.2 (Or.inl ⟨hx1, hy2⟩)
-  · rw [toNat_apply_of_aleph0_le hx2, zero_mul, toNat_apply_of_aleph0_le]
-    exact aleph0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)
+theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y := map_mul toNat x y
 #align cardinal.to_nat_mul Cardinal.toNat_mul
 
-/-- `Cardinal.toNat` as a `MonoidWithZeroHom`. -/
-@[simps]
-noncomputable def toNatHom : Cardinal →*₀ ℕ where
-  toFun := toNat
-  map_zero' := zero_toNat
-  map_one' := one_toNat
-  map_mul' := toNat_mul
-#align cardinal.to_nat_hom Cardinal.toNatHom
-
+@[deprecated map_prod]
 theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
     toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
-  map_prod toNatHom _ _
+  map_prod toNat _ _
 #align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prod
 
 @[simp]
-theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
-    toNat (lift.{v, u} a + lift.{u, v} b) = toNat a + toNat b := by
-  apply Cardinal.natCast_injective
-  replace ha : lift.{v, u} a < ℵ₀ := by rwa [lift_lt_aleph0]
-  replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph0]
-  rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_toNat_of_lt_aleph0 ha,
-    cast_toNat_of_lt_aleph0 hb, cast_toNat_of_lt_aleph0 (add_lt_aleph0 ha hb)]
-#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0
+theorem toNat_add (hc : c < ℵ₀) (hd : d < ℵ₀) : toNat (c + d) = toNat c + toNat d := by
+  lift c to ℕ using hc
+  lift d to ℕ using hd
+  norm_cast
+
+@[simp]
+theorem toNat_lift_add_lift {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
+    toNat (lift.{v} a + lift.{u} b) = toNat a + toNat b := by
+  simp [*]
+
+@[deprecated]
+alias toNat_add_of_lt_aleph0 := toNat_lift_add_lift
+#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_lift_add_lift
 
 end Cardinal

refactor(Cardinal): redefine toNat and toPartENat (#10472)

Redefine these operations in terms of toENat.

6af8f3b3

Diff

@@ -25,40 +25,38 @@ namespace Cardinal
 
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to `⊤`. -/
-noncomputable def toPartENat : Cardinal →+ PartENat where
-  toFun c := if c < ℵ₀ then toNat c else ⊤
-  map_zero' := by simp [if_pos (zero_lt_one.trans one_lt_aleph0)]
-  map_add' x y := by
-    by_cases hx : x < ℵ₀
-    · obtain ⟨x0, rfl⟩ := lt_aleph0.1 hx
-      by_cases hy : y < ℵ₀
-      · obtain ⟨y0, rfl⟩ := lt_aleph0.1 hy
-        simp only [add_lt_aleph0 hx hy, hx, hy, toNat_cast, if_true]
-        rw [← Nat.cast_add, toNat_cast, Nat.cast_add]
-      · simp_rw [if_neg hy, PartENat.add_top]
-        contrapose! hy
-        simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
-          and_true, not_false_eq_true] at hy
-        exact le_add_self.trans_lt hy
-    · simp_rw [if_neg hx, PartENat.top_add]
-      contrapose! hx
-      simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
-        and_true, not_false_eq_true] at hx
-      exact le_self_add.trans_lt hx
+noncomputable def toPartENat : Cardinal →+o PartENat :=
+  .comp
+    { (PartENat.withTopAddEquiv.symm : ℕ∞ →+ PartENat),
+      (PartENat.withTopOrderIso.symm : ℕ∞ →o PartENat) with }
+    toENat
 #align cardinal.to_part_enat Cardinal.toPartENat
 
-theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c :=
-  if_pos h
+@[simp]
+theorem partENatOfENat_toENat (c : Cardinal) : (toENat c : PartENat) = toPartENat c := rfl
+
+@[simp]
+theorem toPartENat_natCast (n : ℕ) : toPartENat n = n := by
+  simp only [← partENatOfENat_toENat, toENat_nat]; rfl
+#align cardinal.to_part_enat_cast Cardinal.toPartENat_natCast
+
+theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c := by
+  lift c to ℕ using h; simp
 #align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0
 
+theorem toPartENat_eq_top {c : Cardinal} :
+    toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
+  rw [← partENatOfENat_toENat, ← PartENat.withTopEquiv_symm_top, ← toENat_eq_top,
+    ← PartENat.withTopEquiv.symm.injective.eq_iff]
+  rfl
+#align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top
+
 theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toPartENat c = ⊤ :=
-  if_neg h.not_lt
+  congr_arg PartENat.ofENat (toENat_eq_top.2 h)
 #align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_le
 
-@[simp]
-theorem toPartENat_cast (n : ℕ) : toPartENat n = n := by
-  rw [toPartENat_apply_of_lt_aleph0 (nat_lt_aleph0 n), toNat_cast]
-#align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
+@[deprecated]
+alias toPartENat_cast := toPartENat_natCast
 
 @[simp]
 theorem mk_toPartENat_of_infinite [h : Infinite α] : toPartENat #α = ⊤ :=
@@ -71,82 +69,42 @@ theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
 #align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
 
 theorem toPartENat_surjective : Surjective toPartENat := fun x =>
-  PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
+  PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_natCast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 
-theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} :
-    toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]
-  | inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc
-#align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0
+@[deprecated] alias toPartENat_eq_top_iff_le_aleph0 := toPartENat_eq_top
+
+theorem toPartENat_strictMonoOn : StrictMonoOn toPartENat (Set.Iic ℵ₀) :=
+  PartENat.withTopOrderIso.symm.strictMono.comp_strictMonoOn toENat_strictMonoOn
 
 lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
     toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    cases lt_or_ge c' ℵ₀ with
-    | inl hc' =>
-      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-      exact toNat_le_iff_le_of_lt_aleph0 hc hc'
-    | inr hc' =>
-      simp only [toPartENat_apply_of_aleph0_le hc',
-      le_top, true_iff]
-      exact le_trans h hc'
-  | inr hc =>
-    rw [toPartENat_apply_of_aleph0_le hc]
-    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0,
-    le_antisymm h hc]
+  lift c to ℕ∞ using h
+  simp only [← partENatOfENat_toENat, toENat_ofENat, enat_gc _,
+   ← PartENat.withTopOrderIso.symm.le_iff_le]
+  rfl
 #align to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_of_le_aleph0
 
 lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
     toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-    exact toNat_le_iff_le_of_lt_aleph0 hc hc'
-  | inr hc =>
-    rw [toPartENat_apply_of_aleph0_le hc]
-    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]
-    rw [← not_iff_not, not_le, not_le]
-    simp only [hc', lt_of_lt_of_le hc' hc]
+  lift c' to ℕ using hc'
+  simp only [← partENatOfENat_toENat, toENat_nat, ← toENat_le_nat,
+   ← PartENat.withTopOrderIso.symm.le_iff_le]
+  rfl
 #align to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_of_lt_aleph0
 
 lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :
-    toPartENat c = toPartENat c' ↔ c = c' := by
-  rw [le_antisymm_iff, le_antisymm_iff, toPartENat_le_iff_of_le_aleph0 hc,
-    toPartENat_le_iff_of_le_aleph0 hc']
+    toPartENat c = toPartENat c' ↔ c = c' :=
+  toPartENat_strictMonoOn.injOn.eq_iff hc hc'
 #align to_part_enat_eq_iff_eq_of_le_aleph_0 Cardinal.toPartENat_eq_iff_of_le_aleph0
 
 theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') :
-    toPartENat c ≤ toPartENat c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    cases lt_or_ge c' ℵ₀ with
-    | inl hc' =>
-      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-      exact toNat_le_of_le_of_lt_aleph0 hc' h
-    | inr hc' =>
-        rw [toPartENat_apply_of_aleph0_le hc']
-        exact le_top
-  | inr hc =>
-      rw [toPartENat_apply_of_aleph0_le hc,
-      toPartENat_apply_of_aleph0_le (le_trans hc h)]
+    toPartENat c ≤ toPartENat c' :=
+  OrderHomClass.mono _ h
 #align cardinal.to_part_enat_mono Cardinal.toPartENat_mono
 
 theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c := by
-  cases' lt_or_ge c ℵ₀ with hc hc
-  · rw [toPartENat_apply_of_lt_aleph0 hc, Cardinal.toPartENat_apply_of_lt_aleph0 _]
-    simp only [toNat_lift]
-    rw [lift_lt_aleph0]
-    exact hc
-  · rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]
-    rw [aleph0_le_lift]
-    exact hc
+  simp only [← partENatOfENat_toENat, toENat_lift]
 #align cardinal.to_part_enat_lift Cardinal.toPartENat_lift
 
 theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat #α = toPartENat #β := by

chore(Cardinal/Basic): split (#10466)

Move toNat and toPartENat to new files.

No changes in the code moved to the new files. One lemma remains in Basic but used toNat in the proof, so I changed the proof.

I'm going to redefine them in terms of toENat, so I need to move them out of Basic first.

c2320208

chore(Cardinal/Basic): split (#10466)

Move toNat and toPartENat to new files.

No changes in the code moved to the new files. One lemma remains in Basic but used toNat in the proof, so I changed the proof.

I'm going to redefine them in terms of toENat, so I need to move them out of Basic first.

c2320208

chore(Cardinal/Basic): split (#10466)

Move toNat and toPartENat to new files.

No changes in the code moved to the new files. One lemma remains in Basic but used toNat in the proof, so I changed the proof.

I'm going to redefine them in terms of toENat, so I need to move them out of Basic first.

c2320208

Diff

@@ -6,7 +6,6 @@ Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 import Mathlib.Algebra.Module.Basic
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.Finsupp.Defs
-import Mathlib.Data.Nat.PartENat
 import Mathlib.Data.Set.Countable
 import Mathlib.Logic.Small.Basic
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
@@ -1713,328 +1712,11 @@ theorem aleph0_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ + no_index (OfNat
 
 variable {c : Cardinal}
 
-/-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
-  to 0. -/
-def toNat : ZeroHom Cardinal ℕ where
-  toFun c := if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0
-  map_zero' := by
-    have h : 0 < ℵ₀ := nat_lt_aleph0 0
-    dsimp only
-    rw [dif_pos h, ← Cardinal.natCast_inj, ← Classical.choose_spec (lt_aleph0.1 h),
-      Nat.cast_zero]
-#align cardinal.to_nat Cardinal.toNat
-
-@[simp]
-lemma toNat_eq_zero : toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c := by
-  simp only [toNat, ZeroHom.coe_mk, dite_eq_right_iff, or_iff_not_imp_right, not_le]
-  refine' forall_congr' fun h => _
-  rw [← @Nat.cast_eq_zero Cardinal, ← Classical.choose_spec (p := fun n : ℕ ↦ c = n)]
-
-lemma toNat_ne_zero : toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀ := by simp [not_or]
-@[simp] lemma toNat_pos : 0 < toNat c ↔ c ≠ 0 ∧ c < ℵ₀ := pos_iff_ne_zero.trans toNat_ne_zero
-
-theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :
-    toNat c = Classical.choose (lt_aleph0.1 h) :=
-  dif_pos h
-#align cardinal.to_nat_apply_of_lt_aleph_0 Cardinal.toNat_apply_of_lt_aleph0
-
-theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toNat c = 0 :=
-  dif_neg h.not_lt
-#align cardinal.to_nat_apply_of_aleph_0_le Cardinal.toNat_apply_of_aleph0_le
-
-theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by
-  rw [toNat_apply_of_lt_aleph0 h, ← Classical.choose_spec (lt_aleph0.1 h)]
-#align cardinal.cast_to_nat_of_lt_aleph_0 Cardinal.cast_toNat_of_lt_aleph0
-
-theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c) = (0 : Cardinal) := by
-  rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]
-#align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
-
-/-- Two finite cardinals are equal iff they are equal their to_nat are equal -/
-theorem toNat_eq_iff_eq_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c = toNat d ↔ c = d := by
-  rw [← natCast_inj, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
-#align cardinal.to_nat_eq_iff_eq_of_lt_aleph_0 Cardinal.toNat_eq_iff_eq_of_lt_aleph0
-
-theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c ≤ toNat d ↔ c ≤ d := by
-  rw [← natCast_le, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
-#align cardinal.to_nat_le_iff_le_of_lt_aleph_0 Cardinal.toNat_le_iff_le_of_lt_aleph0
-
-theorem toNat_lt_iff_lt_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
-    toNat c < toNat d ↔ c < d := by
-  rw [← natCast_lt, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
-#align cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 Cardinal.toNat_lt_iff_lt_of_lt_aleph0
-
-theorem toNat_le_of_le_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
-    toNat c ≤ toNat d :=
-  (toNat_le_iff_le_of_lt_aleph0 (hcd.trans_lt hd) hd).mpr hcd
-#align cardinal.to_nat_le_of_le_of_lt_aleph_0 Cardinal.toNat_le_of_le_of_lt_aleph0
-
-theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c < d) :
-    toNat c < toNat d :=
-  (toNat_lt_iff_lt_of_lt_aleph0 (hcd.trans hd) hd).mpr hcd
-#align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
-
-@[simp]
-theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n := by
-  rw [toNat_apply_of_lt_aleph0 (nat_lt_aleph0 n), ← natCast_inj]
-  exact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm
-#align cardinal.to_nat_cast Cardinal.toNat_cast
-
--- See note [no_index around OfNat.ofNat]
-@[simp]
-theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] :
-    Cardinal.toNat (no_index (OfNat.ofNat n)) = OfNat.ofNat n :=
-  toNat_cast n
-
-/-- `toNat` has a right-inverse: coercion. -/
-theorem toNat_rightInverse : Function.RightInverse ((↑) : ℕ → Cardinal) toNat :=
-  toNat_cast
-#align cardinal.to_nat_right_inverse Cardinal.toNat_rightInverse
-
-theorem toNat_surjective : Surjective toNat :=
-  toNat_rightInverse.surjective
-#align cardinal.to_nat_surjective Cardinal.toNat_surjective
-
-theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m :=
-  let he := cast_toNat_of_lt_aleph0 (h.trans_lt <| nat_lt_aleph0 n)
-  ⟨toNat c, natCast_le.1 (he.trans_le h), he.symm⟩
+theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m := by
+  lift c to ℕ using h.trans_lt (nat_lt_aleph0 _)
+  exact ⟨c, mod_cast h, rfl⟩
 #align cardinal.exists_nat_eq_of_le_nat Cardinal.exists_nat_eq_of_le_nat
 
-@[simp]
-theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 :=
-  dif_neg (infinite_iff.1 h).not_lt
-#align cardinal.mk_to_nat_of_infinite Cardinal.mk_toNat_of_infinite
-
-@[simp]
-theorem aleph0_toNat : toNat ℵ₀ = 0 :=
-  toNat_apply_of_aleph0_le le_rfl
-#align cardinal.aleph_0_to_nat Cardinal.aleph0_toNat
-
-theorem mk_toNat_eq_card [Fintype α] : toNat #α = Fintype.card α := by simp
-#align cardinal.mk_to_nat_eq_card Cardinal.mk_toNat_eq_card
-
--- Porting note : simp can prove this
--- @[simp]
-theorem zero_toNat : toNat 0 = 0 := by rw [← toNat_cast 0, Nat.cast_zero]
-#align cardinal.zero_to_nat Cardinal.zero_toNat
-
-@[simp]
-theorem one_toNat : toNat 1 = 1 := by rw [← toNat_cast 1, Nat.cast_one]
-#align cardinal.one_to_nat Cardinal.one_toNat
-
-theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c = n :=
-  ⟨fun h =>
-    (cast_toNat_of_lt_aleph0
-            (lt_of_not_ge (hn ∘ h.symm.trans ∘ toNat_apply_of_aleph0_le))).symm.trans
-      (congr_arg _ h),
-    fun h => (congr_arg toNat h).trans (toNat_cast n)⟩
-#align cardinal.to_nat_eq_iff Cardinal.toNat_eq_iff
-
-/-- A version of `toNat_eq_iff` for literals -/
-theorem toNat_eq_ofNat {c : Cardinal} {n : ℕ} [Nat.AtLeastTwo n] :
-    toNat c = OfNat.ofNat n ↔ c = OfNat.ofNat n :=
-  toNat_eq_iff <| Nat.cast_ne_zero.mpr <| OfNat.ofNat_ne_zero n
-
-@[simp]
-theorem toNat_eq_one {c : Cardinal} : toNat c = 1 ↔ c = 1 := by
-  rw [toNat_eq_iff one_ne_zero, Nat.cast_one]
-#align cardinal.to_nat_eq_one Cardinal.toNat_eq_one
-
-theorem toNat_eq_one_iff_unique {α : Type*} : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
-  toNat_eq_one.trans eq_one_iff_unique
-#align cardinal.to_nat_eq_one_iff_unique Cardinal.toNat_eq_one_iff_unique
-
-@[simp]
-theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by
-  apply natCast_injective
-  cases' lt_or_ge c ℵ₀ with hc hc
-  · rw [cast_toNat_of_lt_aleph0, ← lift_natCast.{u,v}, cast_toNat_of_lt_aleph0 hc]
-    rwa [lift_lt_aleph0]
-  · rw [cast_toNat_of_aleph0_le, ← lift_natCast.{u,v}, cast_toNat_of_aleph0_le hc, lift_zero]
-    rwa [aleph0_le_lift]
-#align cardinal.to_nat_lift Cardinal.toNat_lift
-
-theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat #α = toNat #β := by
-  -- Porting note: Inserted universe hint below
-  rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]
-#align cardinal.to_nat_congr Cardinal.toNat_congr
-
-@[simp]
-theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y := by
-  rcases eq_or_ne x 0 with (rfl | hx1)
-  · rw [zero_mul, zero_toNat, zero_mul]
-  rcases eq_or_ne y 0 with (rfl | hy1)
-  · rw [mul_zero, zero_toNat, mul_zero]
-  cases' lt_or_le x ℵ₀ with hx2 hx2
-  · cases' lt_or_le y ℵ₀ with hy2 hy2
-    · lift x to ℕ using hx2
-      lift y to ℕ using hy2
-      rw [← Nat.cast_mul, toNat_cast, toNat_cast, toNat_cast]
-    · rw [toNat_apply_of_aleph0_le hy2, mul_zero, toNat_apply_of_aleph0_le]
-      exact aleph0_le_mul_iff'.2 (Or.inl ⟨hx1, hy2⟩)
-  · rw [toNat_apply_of_aleph0_le hx2, zero_mul, toNat_apply_of_aleph0_le]
-    exact aleph0_le_mul_iff'.2 (Or.inr ⟨hx2, hy1⟩)
-#align cardinal.to_nat_mul Cardinal.toNat_mul
-
-/-- `Cardinal.toNat` as a `MonoidWithZeroHom`. -/
-@[simps]
-def toNatHom : Cardinal →*₀ ℕ where
-  toFun := toNat
-  map_zero' := zero_toNat
-  map_one' := one_toNat
-  map_mul' := toNat_mul
-#align cardinal.to_nat_hom Cardinal.toNatHom
-
-theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
-    toNat (∏ i in s, f i) = ∏ i in s, toNat (f i) :=
-  map_prod toNatHom _ _
-#align cardinal.to_nat_finset_prod Cardinal.toNat_finset_prod
-
-@[simp]
-theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
-    toNat (lift.{v, u} a + lift.{u, v} b) = toNat a + toNat b := by
-  apply Cardinal.natCast_injective
-  replace ha : lift.{v, u} a < ℵ₀ := by rwa [lift_lt_aleph0]
-  replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph0]
-  rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_toNat_of_lt_aleph0 ha,
-    cast_toNat_of_lt_aleph0 hb, cast_toNat_of_lt_aleph0 (add_lt_aleph0 ha hb)]
-#align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0
-
-/-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
-  to `⊤`. -/
-def toPartENat : Cardinal →+ PartENat where
-  toFun c := if c < ℵ₀ then toNat c else ⊤
-  map_zero' := by simp [if_pos (zero_lt_one.trans one_lt_aleph0)]
-  map_add' x y := by
-    by_cases hx : x < ℵ₀
-    · obtain ⟨x0, rfl⟩ := lt_aleph0.1 hx
-      by_cases hy : y < ℵ₀
-      · obtain ⟨y0, rfl⟩ := lt_aleph0.1 hy
-        simp only [add_lt_aleph0 hx hy, hx, hy, toNat_cast, if_true]
-        rw [← Nat.cast_add, toNat_cast, Nat.cast_add]
-      · simp_rw [if_neg hy, PartENat.add_top]
-        contrapose! hy
-        simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
-          and_true, not_false_eq_true] at hy
-        exact le_add_self.trans_lt hy
-    · simp_rw [if_neg hx, PartENat.top_add]
-      contrapose! hx
-      simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
-        and_true, not_false_eq_true] at hx
-      exact le_self_add.trans_lt hx
-#align cardinal.to_part_enat Cardinal.toPartENat
-
-theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c :=
-  if_pos h
-#align cardinal.to_part_enat_apply_of_lt_aleph_0 Cardinal.toPartENat_apply_of_lt_aleph0
-
-theorem toPartENat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toPartENat c = ⊤ :=
-  if_neg h.not_lt
-#align cardinal.to_part_enat_apply_of_aleph_0_le Cardinal.toPartENat_apply_of_aleph0_le
-
-@[simp]
-theorem toPartENat_cast (n : ℕ) : toPartENat n = n := by
-  rw [toPartENat_apply_of_lt_aleph0 (nat_lt_aleph0 n), toNat_cast]
-#align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
-
-@[simp]
-theorem mk_toPartENat_of_infinite [h : Infinite α] : toPartENat #α = ⊤ :=
-  toPartENat_apply_of_aleph0_le (infinite_iff.1 h)
-#align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infinite
-
-@[simp]
-theorem aleph0_toPartENat : toPartENat ℵ₀ = ⊤ :=
-  toPartENat_apply_of_aleph0_le le_rfl
-#align cardinal.aleph_0_to_part_enat Cardinal.aleph0_toPartENat
-
-theorem toPartENat_surjective : Surjective toPartENat := fun x =>
-  PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
-#align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
-
-theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} :
-    toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]
-  | inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc
-#align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0
-
-lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
-    toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    cases lt_or_ge c' ℵ₀ with
-    | inl hc' =>
-      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-      exact toNat_le_iff_le_of_lt_aleph0 hc hc'
-    | inr hc' =>
-      simp only [toPartENat_apply_of_aleph0_le hc',
-      le_top, true_iff]
-      exact le_trans h hc'
-  | inr hc =>
-    rw [toPartENat_apply_of_aleph0_le hc]
-    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0,
-    le_antisymm h hc]
-#align to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_of_le_aleph0
-
-lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
-    toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-    exact toNat_le_iff_le_of_lt_aleph0 hc hc'
-  | inr hc =>
-    rw [toPartENat_apply_of_aleph0_le hc]
-    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]
-    rw [← not_iff_not, not_le, not_le]
-    simp only [hc', lt_of_lt_of_le hc' hc]
-#align to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_of_lt_aleph0
-
-lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :
-    toPartENat c = toPartENat c' ↔ c = c' := by
-  rw [le_antisymm_iff, le_antisymm_iff, toPartENat_le_iff_of_le_aleph0 hc,
-    toPartENat_le_iff_of_le_aleph0 hc']
-#align to_part_enat_eq_iff_eq_of_le_aleph_0 Cardinal.toPartENat_eq_iff_of_le_aleph0
-
-theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') :
-    toPartENat c ≤ toPartENat c' := by
-  cases lt_or_ge c ℵ₀ with
-  | inl hc =>
-    rw [toPartENat_apply_of_lt_aleph0 hc]
-    cases lt_or_ge c' ℵ₀ with
-    | inl hc' =>
-      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
-      exact toNat_le_of_le_of_lt_aleph0 hc' h
-    | inr hc' =>
-        rw [toPartENat_apply_of_aleph0_le hc']
-        exact le_top
-  | inr hc =>
-      rw [toPartENat_apply_of_aleph0_le hc,
-      toPartENat_apply_of_aleph0_le (le_trans hc h)]
-#align cardinal.to_part_enat_mono Cardinal.toPartENat_mono
-
-theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c := by
-  cases' lt_or_ge c ℵ₀ with hc hc
-  · rw [toPartENat_apply_of_lt_aleph0 hc, Cardinal.toPartENat_apply_of_lt_aleph0 _]
-    simp only [toNat_lift]
-    rw [lift_lt_aleph0]
-    exact hc
-  · rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]
-    rw [aleph0_le_lift]
-    exact hc
-#align cardinal.to_part_enat_lift Cardinal.toPartENat_lift
-
-theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat #α = toPartENat #β := by
-  rw [← toPartENat_lift, lift_mk_eq.{_, _,v}.mpr ⟨e⟩, toPartENat_lift]
-#align cardinal.to_part_enat_congr Cardinal.toPartENat_congr
-
-theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat #α = Fintype.card α := by simp
-#align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card
-
 theorem mk_int : #ℤ = ℵ₀ :=
   mk_denumerable ℤ
 #align cardinal.mk_int Cardinal.mk_int

fix(Cardinal): fix theorem names (#10465)

Remove unnecessary of_lt_aleph_0 from 2 theorem names. Also fix typos in the comments of 2 other files.

Co-authored-by: damiano <adomani@gmail.com>

a6f14574

Diff

@@ -61,7 +61,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
   `Cardinal.{u} : Type (u + 1)` is the quotient of types in `Type u`.
   The operation `Cardinal.lift` lifts cardinal numbers to a higher level.
 * Cardinal arithmetic specifically for infinite cardinals (like `κ * κ = κ`) is in the file
-  `SetTheory/CardinalOrdinal.lean`.
+  `SetTheory/Cardinal/Ordinal.lean`.
 * There is an instance `Pow Cardinal`, but this will only fire if Lean already knows that both
   the base and the exponent live in the same universe. As a workaround, you can add
   ```

chore: tidy various files (#10311)

890dba74

Diff

@@ -1014,11 +1014,12 @@ protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f
 lemma exists_eq_of_iSup_eq_of_not_isSuccLimit
     {ι : Type u} (f : ι → Cardinal.{v}) (ω : Cardinal.{v})
     (hω : ¬ Order.IsSuccLimit ω)
-    (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω :=
-  (Classical.em <| BddAbove <| range f).elim
-    (fun hf ↦ IsLUB.exists_of_not_isSuccLimit (h ▸ isLUB_csSup' hf) hω) fun hf ↦ by
-      rw [iSup, csSup_of_not_bddAbove hf, csSup_empty] at h
-      exact (hω <| h ▸ Order.isSuccLimit_bot).elim
+    (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω := by
+  subst h
+  refine (isLUB_csSup' ?_).exists_of_not_isSuccLimit hω
+  contrapose! hω with hf
+  rw [iSup, csSup_of_not_bddAbove hf, csSup_empty]
+  exact Order.isSuccLimit_bot
 
 lemma exists_eq_of_iSup_eq_of_not_isLimit
     {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{v}) (hf : BddAbove (range f))

refactor(Set/Countable): redefine Set.Countable (#9831)

Redefine Set.Countable s as _root_.Countable s. Fix compile, golf some of the broken proofs.

c4152bb6

Diff

@@ -1527,8 +1527,7 @@ theorem mk_le_aleph0 [Countable α] : #α ≤ ℵ₀ :=
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable := by
-  rw [mk_le_aleph0_iff, countable_coe_iff]
+theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable := mk_le_aleph0_iff
 #align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable
 
 alias ⟨_, _root_.Set.Countable.le_aleph0⟩ := le_aleph0_iff_set_countable

chore: remove spurious imports of positivity (#9924)

Some of these are already transitively imported, others aren't used at all (but not handled by noshake in #9772).

Mostly I wanted to avoid needing all of algebra imported (but unused!) in FilteredColimitCommutesFiniteLimit; there are now some assert_not_exists to preserve this.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

26c5ad69

Diff

@@ -12,7 +12,6 @@ import Mathlib.Logic.Small.Basic
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 import Mathlib.Order.SuccPred.CompleteLinearOrder
 import Mathlib.SetTheory.Cardinal.SchroederBernstein
-import Mathlib.Tactic.Positivity
 import Mathlib.Tactic.PPWithUniv
 
 #align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"

chore(*): rename FunLike to DFunLike (#9785)

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

82656743

Diff

@@ -926,7 +926,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
                   rw [Equiv.image_eq_preimage]
                   /- Porting note: Need to insert the following `have` b/c bad fun coercion
                    behaviour for Equivs -/
-                  have : FunLike.coe (Equiv.symm (Equiv.ulift (α := α))) = ULift.up (α := α) := rfl
+                  have : DFunLike.coe (Equiv.symm (Equiv.ulift (α := α))) = ULift.up (α := α) := rfl
                   rw [this]
                   simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf]
                   exact Equiv.refl _)

feat(LinearAlgebra, Cardinal): new cardinal lemmas to generalize some results about Module.rank in #9151 (#9253)

Proves that Sup (ciSup) commutes with cardinal addition (ciSup_add_ciSup) and multiplication. Generalize results in Cardinal/Basic introduced in #8842 to achieve this.
Use ciSup_add_ciSup to prove that the rank of a module is always at least the rank of a submodule plus the rank of the quotient by the submodule. Deduce that the rank of a product module is at least the sum of the ranks of the two factors.
Show that quotienting by a torsion submodule preserves the rank.
Golf rank_zero_iff_forall_zero using a recently added lemma.

Co-authored-by: Andrew Yang <the.erd.one@gmail.com>

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

4ad935a4

Diff

@@ -1013,20 +1013,23 @@ protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f
 #align cardinal.supr_of_empty Cardinal.iSup_of_empty
 
 lemma exists_eq_of_iSup_eq_of_not_isSuccLimit
-    {ι : Type u} (f : ι → Cardinal.{max u v}) (ω : Cardinal.{max u v})
+    {ι : Type u} (f : ι → Cardinal.{v}) (ω : Cardinal.{v})
     (hω : ¬ Order.IsSuccLimit ω)
     (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω :=
-  IsLUB.exists_of_not_isSuccLimit (h ▸ isLUB_csSup' (bddAbove_range.{u, v} f)) hω
+  (Classical.em <| BddAbove <| range f).elim
+    (fun hf ↦ IsLUB.exists_of_not_isSuccLimit (h ▸ isLUB_csSup' hf) hω) fun hf ↦ by
+      rw [iSup, csSup_of_not_bddAbove hf, csSup_empty] at h
+      exact (hω <| h ▸ Order.isSuccLimit_bot).elim
 
 lemma exists_eq_of_iSup_eq_of_not_isLimit
-    {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{max u v}) (ω : Cardinal.{max u v})
-    (hω : ¬ ω.IsLimit)
+    {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{v}) (hf : BddAbove (range f))
+    (ω : Cardinal.{v}) (hω : ¬ ω.IsLimit)
     (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω := by
   refine (not_and_or.mp hω).elim (fun e ↦ ⟨hι.some, ?_⟩)
     (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f ω · h)
   cases not_not.mp e
   rw [← le_zero_iff] at h ⊢
-  exact (le_ciSup (Cardinal.bddAbove_range.{u, v} f) _).trans h
+  exact (le_ciSup hf _).trans h
 
 -- Portin note: simpNF is not happy with universe levels.
 @[simp, nolint simpNF]
@@ -1475,9 +1478,9 @@ theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by
   rcases lt_aleph0.1 h' with ⟨n, rfl⟩
   exact not_isLimit_natCast n h
 
-lemma exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{max u v})
-    (n : ℕ) (h : ⨆ i, f i = n) : ∃ i, f i = n :=
-  exists_eq_of_iSup_eq_of_not_isLimit.{u, v} f _ (not_isLimit_natCast n) h
+lemma exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{v})
+    (hf : BddAbove (range f)) (n : ℕ) (h : ⨆ i, f i = n) : ∃ i, f i = n :=
+  exists_eq_of_iSup_eq_of_not_isLimit.{u, v} f hf _ (not_isLimit_natCast n) h
 
 @[simp]
 theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ :=

refactor: Use Pairwise wherever possible (#9236)

Performed with a regex search for ∀ (.) (.), \1 ≠ \2 →, and a few variants to catch implicit binders and explicit types.

I have deliberately avoided trying to make the analogous Set.Pairwise transformation (or any Pairwise (foo on bar) transformations) in this PR, to keep the diff small.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

86393c89

Diff

@@ -2215,14 +2215,14 @@ theorem mk_iUnion_le_sum_mk_lift {α : Type u} {ι : Type v} {f : ι → Set α}
     _ = sum fun i => #(f i) := mk_sigma _
 
 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
-    (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : #(⋃ i, f i) = sum fun i => #(f i) :=
+    (h : Pairwise fun i j => Disjoint (f i) (f j)) : #(⋃ i, f i) = sum fun i => #(f i) :=
   calc
     #(⋃ i, f i) = #(Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #(f i) := mk_sigma _
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
 theorem mk_iUnion_eq_sum_mk_lift {α : Type u} {ι : Type v} {f : ι → Set α}
-    (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) :
+    (h : Pairwise fun i j => Disjoint (f i) (f j)) :
     lift.{v} #(⋃ i, f i) = sum fun i => #(f i) :=
   calc
     lift.{v} #(⋃ i, f i) = #(Σi, f i) :=

feat(SetTheory/Cardinal/Basic): add missing lift versions of iUnion lemmas (#9187)

Also move some lift lemmas to be next to their non-lift counterparts.

For now this does not generalize to ι : Sort v as this would make a mess with PLift.

a9c72c70

Diff

@@ -2167,6 +2167,11 @@ theorem mk_range_eq (f : α → β) (h : Injective f) : #(range f) = #α :=
   mk_congr (Equiv.ofInjective f h).symm
 #align cardinal.mk_range_eq Cardinal.mk_range_eq
 
+theorem mk_range_eq_lift {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
+    lift.{max u w} #(range f) = lift.{max v w} #α :=
+  lift_mk_eq.{v,u,w}.mpr ⟨(Equiv.ofInjective f hf).symm⟩
+#align cardinal.mk_range_eq_lift Cardinal.mk_range_eq_lift
+
 theorem mk_range_eq_of_injective {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
     lift.{u} #(range f) = lift.{v} #α :=
   lift_mk_eq'.mpr ⟨(Equiv.ofInjective f hf).symm⟩
@@ -2177,21 +2182,38 @@ lemma lift_mk_le_lift_mk_of_injective {α : Type u} {β : Type v} {f : α → β
   rw [← Cardinal.mk_range_eq_of_injective hf]
   exact Cardinal.lift_le.2 (Cardinal.mk_set_le _)
 
-theorem mk_range_eq_lift {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
-    lift.{max u w} #(range f) = lift.{max v w} #α :=
-  lift_mk_eq.{v,u,w}.mpr ⟨(Equiv.ofInjective f hf).symm⟩
-#align cardinal.mk_range_eq_lift Cardinal.mk_range_eq_lift
+theorem mk_image_eq_of_injOn {α β : Type u} (f : α → β) (s : Set α) (h : InjOn f s) :
+    #(f '' s) = #s :=
+  mk_congr (Equiv.Set.imageOfInjOn f s h).symm
+#align cardinal.mk_image_eq_of_inj_on Cardinal.mk_image_eq_of_injOn
+
+theorem mk_image_eq_of_injOn_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α)
+    (h : InjOn f s) : lift.{u} #(f '' s) = lift.{v} #s :=
+  lift_mk_eq.{v, u, 0}.mpr ⟨(Equiv.Set.imageOfInjOn f s h).symm⟩
+#align cardinal.mk_image_eq_of_inj_on_lift Cardinal.mk_image_eq_of_injOn_lift
 
 theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injective f) : #(f '' s) = #s :=
-  mk_congr (Equiv.Set.image f s hf).symm
+  mk_image_eq_of_injOn _ _ <| hf.injOn _
 #align cardinal.mk_image_eq Cardinal.mk_image_eq
 
+theorem mk_image_eq_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α) (h : Injective f) :
+    lift.{u} #(f '' s) = lift.{v} #s :=
+  mk_image_eq_of_injOn_lift _ _ <| h.injOn _
+#align cardinal.mk_image_eq_lift Cardinal.mk_image_eq_lift
+
 theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : #(⋃ i, f i) ≤ sum fun i => #(f i) :=
   calc
     #(⋃ i, f i) ≤ #(Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
     _ = sum fun i => #(f i) := mk_sigma _
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 
+theorem mk_iUnion_le_sum_mk_lift {α : Type u} {ι : Type v} {f : ι → Set α} :
+    lift.{v} #(⋃ i, f i) ≤ sum fun i => #(f i) :=
+  calc
+    lift.{v} #(⋃ i, f i) ≤ #(Σi, f i) :=
+      mk_le_of_surjective <| ULift.up_surjective.comp (Set.sigmaToiUnion_surjective f)
+    _ = sum fun i => #(f i) := mk_sigma _
+
 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : #(⋃ i, f i) = sum fun i => #(f i) :=
   calc
@@ -2199,10 +2221,23 @@ theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     _ = sum fun i => #(f i) := mk_sigma _
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
+theorem mk_iUnion_eq_sum_mk_lift {α : Type u} {ι : Type v} {f : ι → Set α}
+    (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) :
+    lift.{v} #(⋃ i, f i) = sum fun i => #(f i) :=
+  calc
+    lift.{v} #(⋃ i, f i) = #(Σi, f i) :=
+      mk_congr <| .trans Equiv.ulift (Set.unionEqSigmaOfDisjoint h)
+    _ = sum fun i => #(f i) := mk_sigma _
+
 theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : #(⋃ i, f i) ≤ #ι * ⨆ i, #(f i) :=
   mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
 #align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
+theorem mk_iUnion_le_lift {α : Type u} {ι : Type v} (f : ι → Set α) :
+    lift.{v} #(⋃ i, f i) ≤ lift.{u} #ι * ⨆ i, lift.{v} #(f i) := by
+  refine mk_iUnion_le_sum_mk_lift.trans <| Eq.trans_le ?_ (sum_le_iSup_lift _)
+  rw [← lift_sum, lift_id'.{_,u}]
+
 theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : #(⋃₀ A) ≤ #A * ⨆ s : A, #s := by
   rw [sUnion_eq_iUnion]
   apply mk_iUnion_le
@@ -2214,6 +2249,11 @@ theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
   apply mk_iUnion_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
+theorem mk_biUnion_le_lift {α : Type u} {ι : Type v} (A : ι → Set α) (s : Set ι) :
+    lift.{v} #(⋃ x ∈ s, A x) ≤ lift.{u} #s * ⨆ x : s, lift.{v} #(A x.1) := by
+  rw [biUnion_eq_iUnion]
+  apply mk_iUnion_le_lift
+
 theorem finset_card_lt_aleph0 (s : Finset α) : #(↑s : Set α) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
@@ -2303,20 +2343,6 @@ theorem mk_union_le_aleph0 {α} {P Q : Set α} :
     ← countable_union]
 #align cardinal.mk_union_le_aleph_0 Cardinal.mk_union_le_aleph0
 
-theorem mk_image_eq_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α) (h : Injective f) :
-    lift.{u} #(f '' s) = lift.{v} #s :=
-  lift_mk_eq.{v, u, 0}.mpr ⟨(Equiv.Set.image f s h).symm⟩
-#align cardinal.mk_image_eq_lift Cardinal.mk_image_eq_lift
-
-theorem mk_image_eq_of_injOn_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α)
-    (h : InjOn f s) : lift.{u} #(f '' s) = lift.{v} #s :=
-  lift_mk_eq.{v, u, 0}.mpr ⟨(Equiv.Set.imageOfInjOn f s h).symm⟩
-#align cardinal.mk_image_eq_of_inj_on_lift Cardinal.mk_image_eq_of_injOn_lift
-
-theorem mk_image_eq_of_injOn {α β : Type u} (f : α → β) (s : Set α) (h : InjOn f s) :
-    #(f '' s) = #s :=
-  mk_congr (Equiv.Set.imageOfInjOn f s h).symm
-#align cardinal.mk_image_eq_of_inj_on Cardinal.mk_image_eq_of_injOn
 
 theorem mk_subtype_of_equiv {α β : Type u} (p : β → Prop) (e : α ≃ β) :
     #{ a : α // p (e a) } = #{ b : β // p b } :=
@@ -2353,6 +2379,11 @@ theorem mk_preimage_of_injective_of_subset_range_lift {β : Type v} (f : α →
   le_antisymm (mk_preimage_of_injective_lift f s h) (mk_preimage_of_subset_range_lift f s h2)
 #align cardinal.mk_preimage_of_injective_of_subset_range_lift Cardinal.mk_preimage_of_injective_of_subset_range_lift
 
+theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h : Injective f)
+    (h2 : s ⊆ range f) : #(f ⁻¹' s) = #s := by
+  convert mk_preimage_of_injective_of_subset_range_lift.{u, u} f s h h2 using 1 <;> rw [lift_id]
+#align cardinal.mk_preimage_of_injective_of_subset_range Cardinal.mk_preimage_of_injective_of_subset_range
+
 theorem mk_preimage_of_injective (f : α → β) (s : Set β) (h : Injective f) :
     #(f ⁻¹' s) ≤ #s := by
   rw [← lift_id #(↑(f ⁻¹' s)), ← lift_id #(↑s)]
@@ -2365,11 +2396,6 @@ theorem mk_preimage_of_subset_range (f : α → β) (s : Set β) (h : s ⊆ rang
   exact mk_preimage_of_subset_range_lift f s h
 #align cardinal.mk_preimage_of_subset_range Cardinal.mk_preimage_of_subset_range
 
-theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h : Injective f)
-    (h2 : s ⊆ range f) : #(f ⁻¹' s) = #s := by
-  convert mk_preimage_of_injective_of_subset_range_lift.{u, u} f s h h2 using 1 <;> rw [lift_id]
-#align cardinal.mk_preimage_of_injective_of_subset_range Cardinal.mk_preimage_of_injective_of_subset_range
-
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
     {t : Set β} (h : t ⊆ f '' s) : lift.{u} #t ≤ lift.{v} #({ x ∈ s | f x ∈ t } : Set α) := by
   rw [image_eq_range] at h

feat: Relation between IsSuccLimit and iSup. (#8842)

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

fdcf8b66

Diff

@@ -10,7 +10,7 @@ import Mathlib.Data.Nat.PartENat
 import Mathlib.Data.Set.Countable
 import Mathlib.Logic.Small.Basic
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
-import Mathlib.Order.SuccPred.Limit
+import Mathlib.Order.SuccPred.CompleteLinearOrder
 import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.Tactic.Positivity
 import Mathlib.Tactic.PPWithUniv
@@ -1012,6 +1012,22 @@ protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f
   ciSup_of_empty f
 #align cardinal.supr_of_empty Cardinal.iSup_of_empty
 
+lemma exists_eq_of_iSup_eq_of_not_isSuccLimit
+    {ι : Type u} (f : ι → Cardinal.{max u v}) (ω : Cardinal.{max u v})
+    (hω : ¬ Order.IsSuccLimit ω)
+    (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω :=
+  IsLUB.exists_of_not_isSuccLimit (h ▸ isLUB_csSup' (bddAbove_range.{u, v} f)) hω
+
+lemma exists_eq_of_iSup_eq_of_not_isLimit
+    {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{max u v}) (ω : Cardinal.{max u v})
+    (hω : ¬ ω.IsLimit)
+    (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω := by
+  refine (not_and_or.mp hω).elim (fun e ↦ ⟨hι.some, ?_⟩)
+    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f ω · h)
+  cases not_not.mp e
+  rw [← le_zero_iff] at h ⊢
+  exact (le_ciSup (Cardinal.bddAbove_range.{u, v} f) _).trans h
+
 -- Portin note: simpNF is not happy with universe levels.
 @[simp, nolint simpNF]
 theorem lift_mk_shrink (α : Type u) [Small.{v} α] :
@@ -1450,13 +1466,18 @@ theorem isLimit_aleph0 : IsLimit ℵ₀ :=
   ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
 #align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
 
+lemma not_isLimit_natCast : (n : ℕ) → ¬ IsLimit (n : Cardinal.{u})
+  | 0, e => e.1 rfl
+  | Nat.succ n, e => Order.not_isSuccLimit_succ _ (nat_succ n ▸ e.2)
+
 theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by
   by_contra! h'
-  rcases lt_aleph0.1 h' with ⟨_ | n, rfl⟩
-  · exact h.ne_zero.irrefl
-  · rw [nat_succ] at h
-    exact not_isSuccLimit_succ _ h.isSuccLimit
-#align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
+  rcases lt_aleph0.1 h' with ⟨n, rfl⟩
+  exact not_isLimit_natCast n h
+
+lemma exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{max u v})
+    (n : ℕ) (h : ⨆ i, f i = n) : ∃ i, f i = n :=
+  exists_eq_of_iSup_eq_of_not_isLimit.{u, v} f _ (not_isLimit_natCast n) h
 
 @[simp]
 theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ :=

feat: demote the instance Fintype.ofIsEmpty to a def (#8816)

Rationale: this instance creates (empty) data out of nothing, which may conflict with other data. If you have in the context [Fintype i] and case on whether i is empty or not, then this gave two non-defeq instances of [Fintype i] around.

8eef7faf

Diff

@@ -379,6 +379,7 @@ instance : Zero Cardinal.{u} :=
 instance : Inhabited Cardinal.{u} :=
   ⟨0⟩
 
+@[simp]
 theorem mk_eq_zero (α : Type u) [IsEmpty α] : #α = 0 :=
   (Equiv.equivOfIsEmpty α (ULift (Fin 0))).cardinal_eq
 #align cardinal.mk_eq_zero Cardinal.mk_eq_zero

chore: Sink Algebra.Support down the import tree (#8919)

Function.support is a very basic definition. Nevertheless, it is a pretty heavy import because it imports most objects a support lemma can be written about.

This PR reverses the dependencies between those objects and Function.support, so that the latter can become a much more lightweight import.

Only two import could not easily be reversed, namely the ones to Data.Set.Finite and Order.ConditionallyCompleteLattice.Basic, so I created two new files instead.

I credit:

Yury for https://github.com/leanprover-community/mathlib/pull/6791
Oliver for https://github.com/leanprover-community/mathlib/pull/7439

82ddb54f

Diff

@@ -3,6 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 -/
+import Mathlib.Algebra.Module.Basic
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.Finsupp.Defs
 import Mathlib.Data.Nat.PartENat

fix: add missing no_index around OfNat.ofNat (#8317)

Co-authored-by: timotree3 <timorcb@gmail.com>

070be719

Diff

@@ -1757,8 +1757,10 @@ theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n := by
   exact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm
 #align cardinal.to_nat_cast Cardinal.toNat_cast
 
+-- See note [no_index around OfNat.ofNat]
 @[simp]
-theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] : Cardinal.toNat (OfNat.ofNat n) = OfNat.ofNat n :=
+theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] :
+    Cardinal.toNat (no_index (OfNat.ofNat n)) = OfNat.ofNat n :=
   toNat_cast n
 
 /-- `toNat` has a right-inverse: coercion. -/

chore: rename by_contra' to by_contra! (#8797)

To fit with the "please try harder" convention of ! tactics.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

738b1a97

Diff

@@ -1449,7 +1449,7 @@ theorem isLimit_aleph0 : IsLimit ℵ₀ :=
 #align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
 
 theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by
-  by_contra' h'
+  by_contra! h'
   rcases lt_aleph0.1 h' with ⟨_ | n, rfl⟩
   · exact h.ne_zero.irrefl
   · rw [nat_succ] at h

chore: space after ← (#8178)

Co-authored-by: Moritz Firsching <firsching@google.com>

5fb9beab

Diff

@@ -629,11 +629,11 @@ theorem lift_bit1 (a : Cardinal) : lift.{v} (bit1 a) = bit1 (lift.{v} a) := by s
 end deprecated
 
 -- Porting note: Proof used to be simp, needed to remind simp that 1 + 1 = 2
-theorem lift_two : lift.{u, v} 2 = 2 := by simp [←one_add_one_eq_two]
+theorem lift_two : lift.{u, v} 2 = 2 := by simp [← one_add_one_eq_two]
 #align cardinal.lift_two Cardinal.lift_two
 
 @[simp]
-theorem mk_set {α : Type u} : #(Set α) = 2 ^ #α := by simp [←one_add_one_eq_two, Set, mk_arrow]
+theorem mk_set {α : Type u} : #(Set α) = 2 ^ #α := by simp [← one_add_one_eq_two, Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
 
 /-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/
@@ -643,7 +643,7 @@ theorem mk_powerset {α : Type u} (s : Set α) : #(↥(𝒫 s)) = 2 ^ #(↥s) :=
 #align cardinal.mk_powerset Cardinal.mk_powerset
 
 theorem lift_two_power (a : Cardinal) : lift.{v} (2 ^ a) = 2 ^ lift.{v} a := by
-  simp [←one_add_one_eq_two]
+  simp [← one_add_one_eq_two]
 #align cardinal.lift_two_power Cardinal.lift_two_power
 
 section OrderProperties
@@ -1103,7 +1103,7 @@ theorem lift_sInf (s : Set Cardinal) : lift.{u,v} (sInf s) = sInf (lift.{u,v} ''
 theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u,v} (iInf f) = ⨅ i, lift.{u,v} (f i) := by
   unfold iInf
   convert lift_sInf (range f)
-  simp_rw [←comp_apply (f := lift), range_comp]
+  simp_rw [← comp_apply (f := lift), range_comp]
 #align cardinal.lift_infi Cardinal.lift_iInf
 
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
@@ -1703,7 +1703,7 @@ def toNat : ZeroHom Cardinal ℕ where
 lemma toNat_eq_zero : toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c := by
   simp only [toNat, ZeroHom.coe_mk, dite_eq_right_iff, or_iff_not_imp_right, not_le]
   refine' forall_congr' fun h => _
-  rw [←@Nat.cast_eq_zero Cardinal, ← Classical.choose_spec (p := fun n : ℕ ↦ c = n)]
+  rw [← @Nat.cast_eq_zero Cardinal, ← Classical.choose_spec (p := fun n : ℕ ↦ c = n)]
 
 lemma toNat_ne_zero : toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀ := by simp [not_or]
 @[simp] lemma toNat_pos : 0 < toNat c ↔ c ≠ 0 ∧ c < ℵ₀ := pos_iff_ne_zero.trans toNat_ne_zero

fix: remove remaining ^ fixes (#8463)

b0e957dd

Diff

@@ -484,15 +484,6 @@ private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
 instance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=
   ⟨map₂ (fun α β => β → α) fun _ _ _ _ e₁ e₂ => e₂.arrowCongr e₁⟩
 
--- Porting note: This "workaround" does not work and break everything.
--- I changed it now from `^` to `^'` to prevent a clash
--- with `HPow`, but somebody should figure out
--- if this is still relevant in Lean4.
--- mathport name: cardinal.pow
-local infixr:80 " ^' " => @HPow.hPow Cardinal Cardinal Cardinal _
--- -- mathport name: cardinal.pow.nat
-local infixr:80 " ^ℕ " => @HPow.hPow Cardinal ℕ Cardinal instHPow
-
 theorem power_def (α β : Type u) : #α ^ #β = #(β → α) :=
   rfl
 #align cardinal.power_def Cardinal.power_def
@@ -602,7 +593,7 @@ theorem power_mul {a b c : Cardinal} : a ^ (b * c) = (a ^ b) ^ c := by
 #align cardinal.power_mul Cardinal.power_mul
 
 @[simp]
-theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : a ^ (↑n : Cardinal.{u}) = a ^ℕ n :=
+theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : a ^ (↑n : Cardinal.{u}) = a ^ n :=
   rfl
 #align cardinal.pow_cast_right Cardinal.pow_cast_right
 
@@ -1326,18 +1317,18 @@ theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n := rfl
 theorem mk_coe_finset {α : Type u} {s : Finset α} : #s = ↑(Finset.card s) := by simp
 #align cardinal.mk_coe_finset Cardinal.mk_coe_finset
 
-theorem mk_finset_of_fintype [Fintype α] : #(Finset α) = 2 ^ℕ Fintype.card α := by
+theorem mk_finset_of_fintype [Fintype α] : #(Finset α) = 2 ^ Fintype.card α := by
   simp [Pow.pow]
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
 
 @[simp]
 theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zero β] :
-    #(α →₀ β) = lift.{u} #β ^ℕ Fintype.card α := by
+    #(α →₀ β) = lift.{u} #β ^ Fintype.card α := by
   simpa using (@Finsupp.equivFunOnFinite α β _ _).cardinal_eq
 #align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintype
 
 theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
-    #(α →₀ β) = #β ^ℕ Fintype.card α := by simp
+    #(α →₀ β) = #β ^ Fintype.card α := by simp
 #align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype
 
 theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
@@ -2086,14 +2077,14 @@ theorem mk_plift_false : #(PLift False) = 0 :=
 #align cardinal.mk_plift_false Cardinal.mk_plift_false
 
 @[simp]
-theorem mk_vector (α : Type u) (n : ℕ) : #(Vector α n) = #α ^ℕ n :=
+theorem mk_vector (α : Type u) (n : ℕ) : #(Vector α n) = #α ^ n :=
   (mk_congr (Equiv.vectorEquivFin α n)).trans <| by simp
 #align cardinal.mk_vector Cardinal.mk_vector
 
-theorem mk_list_eq_sum_pow (α : Type u) : #(List α) = sum fun n : ℕ => #α ^ℕ n :=
+theorem mk_list_eq_sum_pow (α : Type u) : #(List α) = sum fun n : ℕ => #α ^ n :=
   calc
     #(List α) = #(Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
-    _ = sum fun n : ℕ => #α ^ℕ n := by simp
+    _ = sum fun n : ℕ => #α ^ n := by simp
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
 
 theorem mk_quot_le {α : Type u} {r : α → α → Prop} : #(Quot r) ≤ #α :=

chore: bump to v4.3.0-rc2 (#8366)

PR contents

This is the supremum of

#8284
#8056
#8023
#8332
#8226 (already approved)
#7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

switching to using explicit lemmas that have the intended level of application
(config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

40b64f79

Diff

@@ -493,7 +493,7 @@ local infixr:80 " ^' " => @HPow.hPow Cardinal Cardinal Cardinal _
 -- -- mathport name: cardinal.pow.nat
 local infixr:80 " ^ℕ " => @HPow.hPow Cardinal ℕ Cardinal instHPow
 
-theorem power_def (α β) : #α ^ #β = #(β → α) :=
+theorem power_def (α β : Type u) : #α ^ #β = #(β → α) :=
   rfl
 #align cardinal.power_def Cardinal.power_def
 
@@ -508,12 +508,12 @@ theorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = lift.{v} a ^ lift.{
 #align cardinal.lift_power Cardinal.lift_power
 
 @[simp]
-theorem power_zero {a : Cardinal} : a ^ 0 = 1 :=
+theorem power_zero {a : Cardinal} : a ^ (0 : Cardinal) = 1 :=
   inductionOn a fun _ => mk_eq_one _
 #align cardinal.power_zero Cardinal.power_zero
 
 @[simp]
-theorem power_one {a : Cardinal.{u}} : a ^ 1 = a :=
+theorem power_one {a : Cardinal.{u}} : a ^ (1 : Cardinal) = a :=
   inductionOn a fun α => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) α)
 #align cardinal.power_one Cardinal.power_one
 
@@ -538,9 +538,9 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   mul_comm := mul_comm'
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
-  npow n c := c ^ n
+  npow n c := c ^ (n : Cardinal)
   npow_zero := @power_zero
-  npow_succ n c := show c ^ (n + 1 : ℕ) = c * (c ^ n)
+  npow_succ n c := show c ^ (↑(n + 1) : Cardinal) = c * c ^ (↑n : Cardinal)
     by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']
   natCast := (fun n => lift.{u} #(Fin n) : ℕ → Cardinal.{u})
   natCast_zero := rfl
@@ -563,7 +563,7 @@ theorem power_bit1 (a b : Cardinal) : a ^ bit1 b = a ^ b * a ^ b * a := by
 end deprecated
 
 @[simp]
-theorem one_power {a : Cardinal} : 1 ^ a = 1 :=
+theorem one_power {a : Cardinal} : (1 : Cardinal) ^ a = 1 :=
   inductionOn a fun _ => mk_eq_one _
 #align cardinal.one_power Cardinal.one_power
 
@@ -578,7 +578,7 @@ theorem mk_Prop : #Prop = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
 
 @[simp]
-theorem zero_power {a : Cardinal} : a ≠ 0 → 0 ^ a = 0 :=
+theorem zero_power {a : Cardinal} : a ≠ 0 → (0 : Cardinal) ^ a = 0 :=
   inductionOn a fun _ heq =>
     mk_eq_zero_iff.2 <|
       isEmpty_pi.2 <|
@@ -1347,7 +1347,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
 -- Porting note: was `simp`. LHS is not normal form.
 -- @[simp, norm_cast]
 @[norm_cast]
-theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = m ^ n := by
+theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (↑m : Cardinal) ^ (↑n : Cardinal) := by
   induction n <;> simp [pow_succ', power_add, *, Pow.pow]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 
@@ -1898,13 +1898,13 @@ def toPartENat : Cardinal →+ PartENat where
         rw [← Nat.cast_add, toNat_cast, Nat.cast_add]
       · simp_rw [if_neg hy, PartENat.add_top]
         contrapose! hy
-        simp only [ne_eq, ite_eq_right_iff,
-          PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hy
+        simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
+          and_true, not_false_eq_true] at hy
         exact le_add_self.trans_lt hy
     · simp_rw [if_neg hx, PartENat.top_add]
       contrapose! hx
-      simp only [ne_eq, ite_eq_right_iff,
-      PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hx
+      simp only [ne_eq, ite_eq_right_iff, PartENat.natCast_ne_top, not_forall, exists_prop,
+        and_true, not_false_eq_true] at hx
       exact le_self_add.trans_lt hx
 #align cardinal.to_part_enat Cardinal.toPartENat
 
@@ -2422,14 +2422,14 @@ theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z
 
 /-- The function `a ^< b`, defined as the supremum of `a ^ c` for `c < b`. -/
 def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
-  ⨆ c : Iio b, a^c
+  ⨆ c : Iio b, a ^ (c : Cardinal)
 #align cardinal.powerlt Cardinal.powerlt
 
 @[inherit_doc]
 infixl:80 " ^< " => powerlt
 
-theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : a ^ c ≤ a ^< b := by
-  refine le_ciSup (f := fun y : Iio b => a ^ y) ?_ ⟨c, h⟩
+theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by
+  refine le_ciSup (f := fun y : Iio b => a ^ (y : Cardinal)) ?_ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt

feat: When Nat.card is zero (#8202)

and lemmas about injectivity/surjectivity of PLift.map/ULift.map.

52c6af84

Diff

@@ -1615,6 +1615,9 @@ theorem infinite_iff {α : Type u} : Infinite α ↔ ℵ₀ ≤ #α := by
   rw [← not_lt, lt_aleph0_iff_finite, not_finite_iff_infinite]
 #align cardinal.infinite_iff Cardinal.infinite_iff
 
+lemma aleph0_le_mk_iff : ℵ₀ ≤ #α ↔ Infinite α := infinite_iff.symm
+lemma mk_lt_aleph0_iff : #α < ℵ₀ ↔ Finite α := by simp [← not_le, aleph0_le_mk_iff]
+
 @[simp]
 theorem aleph0_le_mk (α : Type u) [Infinite α] : ℵ₀ ≤ #α :=
   infinite_iff.1 ‹_›
@@ -1692,6 +1695,8 @@ theorem ofNat_add_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n)
 theorem aleph0_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ + no_index (OfNat.ofNat n) = ℵ₀ :=
   aleph0_add_nat n
 
+variable {c : Cardinal}
+
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to 0. -/
 def toNat : ZeroHom Cardinal ℕ where
@@ -1703,6 +1708,15 @@ def toNat : ZeroHom Cardinal ℕ where
       Nat.cast_zero]
 #align cardinal.to_nat Cardinal.toNat
 
+@[simp]
+lemma toNat_eq_zero : toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c := by
+  simp only [toNat, ZeroHom.coe_mk, dite_eq_right_iff, or_iff_not_imp_right, not_le]
+  refine' forall_congr' fun h => _
+  rw [←@Nat.cast_eq_zero Cardinal, ← Classical.choose_spec (p := fun n : ℕ ↦ c = n)]
+
+lemma toNat_ne_zero : toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀ := by simp [not_or]
+@[simp] lemma toNat_pos : 0 < toNat c ↔ c ≠ 0 ∧ c < ℵ₀ := pos_iff_ne_zero.trans toNat_ne_zero
+
 theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :
     toNat c = Classical.choose (lt_aleph0.1 h) :=
   dif_pos h

https://github.com/leanprover-community/mathlib4/blob/4055c8b471380825f07416b12cb0cf266da44d84/Mathlib/Tactic/Simps/Basic.lean#L843-L851

style: shorten simps configurations (#8296)

Use .asFn and .lemmasOnly as simps configuration options.

For reference, these are defined here:

e0637173

Diff

@@ -345,7 +345,7 @@ theorem lift_le {a b : Cardinal.{v}} : lift.{u, v} a ≤ lift.{u, v} b ↔ a ≤
 
 -- Porting note: changed `simps` to `simps!` because the linter told to do so.
 /-- `Cardinal.lift` as an `OrderEmbedding`. -/
-@[simps! (config := { fullyApplied := false })]
+@[simps! (config := .asFn)]
 def liftOrderEmbedding : Cardinal.{v} ↪o Cardinal.{max v u} :=
   OrderEmbedding.ofMapLEIff lift.{u, v} fun _ _ => lift_le
 #align cardinal.lift_order_embedding Cardinal.liftOrderEmbedding

style(SetTheory): remove useless parentheses (#8279)

These were caused by a bad notation precedence in mathlib3, where the local version of ^ was given precedence 0. We're not using the local notation at all in Mathlib4 (partly because it was broken, which this PR fixes).

a084bd51

Diff

@@ -65,7 +65,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 * There is an instance `Pow Cardinal`, but this will only fire if Lean already knows that both
   the base and the exponent live in the same universe. As a workaround, you can add
   ```
-    local infixr:0 "^'" => @Pow.pow Cardinal Cardinal Cardinal.instPowCardinal
+    local infixr:80 " ^' " => @HPow.hPow Cardinal Cardinal Cardinal _
   ```
   to a file. This notation will work even if Lean doesn't know yet that the base and the exponent
   live in the same universe (but no exponents in other types can be used).
@@ -489,7 +489,7 @@ instance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=
 -- with `HPow`, but somebody should figure out
 -- if this is still relevant in Lean4.
 -- mathport name: cardinal.pow
-local infixr:0 "^'" => @HPow.hPow Cardinal Cardinal Cardinal.instPowCardinal
+local infixr:80 " ^' " => @HPow.hPow Cardinal Cardinal Cardinal _
 -- -- mathport name: cardinal.pow.nat
 local infixr:80 " ^ℕ " => @HPow.hPow Cardinal ℕ Cardinal instHPow
 
@@ -502,22 +502,22 @@ theorem mk_arrow (α : Type u) (β : Type v) : #(α → β) = (lift.{u} #β^lift
 #align cardinal.mk_arrow Cardinal.mk_arrow
 
 @[simp]
-theorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = ((lift.{v} a) ^ (lift.{v} b)) :=
+theorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = lift.{v} a ^ lift.{v} b :=
   inductionOn₂ a b fun _ _ =>
     mk_congr <| Equiv.ulift.trans (Equiv.ulift.arrowCongr Equiv.ulift).symm
 #align cardinal.lift_power Cardinal.lift_power
 
 @[simp]
-theorem power_zero {a : Cardinal} : (a ^ 0) = 1 :=
+theorem power_zero {a : Cardinal} : a ^ 0 = 1 :=
   inductionOn a fun _ => mk_eq_one _
 #align cardinal.power_zero Cardinal.power_zero
 
 @[simp]
-theorem power_one {a : Cardinal.{u}} : (a ^ 1) = a :=
+theorem power_one {a : Cardinal.{u}} : a ^ 1 = a :=
   inductionOn a fun α => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) α)
 #align cardinal.power_one Cardinal.power_one
 
-theorem power_add {a b c : Cardinal} : (a ^ (b + c)) = (a ^ b) * (a ^ c) :=
+theorem power_add {a b c : Cardinal} : a ^ (b + c) = a ^ b * a ^ c :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumArrowEquivProdArrow β γ α
 #align cardinal.power_add Cardinal.power_add
 
@@ -538,9 +538,9 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   mul_comm := mul_comm'
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
-  npow n c := c^n
+  npow n c := c ^ n
   npow_zero := @power_zero
-  npow_succ n c := show (c ^ (n + 1 : ℕ)) = c * (c ^ n)
+  npow_succ n c := show c ^ (n + 1 : ℕ) = c * (c ^ n)
     by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']
   natCast := (fun n => lift.{u} #(Fin n) : ℕ → Cardinal.{u})
   natCast_zero := rfl
@@ -551,19 +551,19 @@ section deprecated
 set_option linter.deprecated false
 
 @[deprecated]
-theorem power_bit0 (a b : Cardinal) : (a ^ bit0 b) = (a ^ b) * (a ^ b) :=
+theorem power_bit0 (a b : Cardinal) : a ^ bit0 b = a ^ b * a ^ b :=
   power_add
 #align cardinal.power_bit0 Cardinal.power_bit0
 
 @[deprecated]
-theorem power_bit1 (a b : Cardinal) : (a ^ bit1 b) = (a ^ b) * (a ^ b) * a := by
+theorem power_bit1 (a b : Cardinal) : a ^ bit1 b = a ^ b * a ^ b * a := by
   rw [bit1, ← power_bit0, power_add, power_one]
 #align cardinal.power_bit1 Cardinal.power_bit1
 
 end deprecated
 
 @[simp]
-theorem one_power {a : Cardinal} : (1 ^ a) = 1 :=
+theorem one_power {a : Cardinal} : 1 ^ a = 1 :=
   inductionOn a fun _ => mk_eq_one _
 #align cardinal.one_power Cardinal.one_power
 
@@ -578,7 +578,7 @@ theorem mk_Prop : #Prop = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
 
 @[simp]
-theorem zero_power {a : Cardinal} : a ≠ 0 → (0 ^ a) = 0 :=
+theorem zero_power {a : Cardinal} : a ≠ 0 → 0 ^ a = 0 :=
   inductionOn a fun _ heq =>
     mk_eq_zero_iff.2 <|
       isEmpty_pi.2 <|
@@ -586,23 +586,23 @@ theorem zero_power {a : Cardinal} : a ≠ 0 → (0 ^ a) = 0 :=
         ⟨a, inferInstance⟩
 #align cardinal.zero_power Cardinal.zero_power
 
-theorem power_ne_zero {a : Cardinal} (b) : a ≠ 0 → (a ^ b) ≠ 0 :=
+theorem power_ne_zero {a : Cardinal} (b : Cardinal) : a ≠ 0 → a ^ b ≠ 0 :=
   inductionOn₂ a b fun _ _ h =>
     let ⟨a⟩ := mk_ne_zero_iff.1 h
     mk_ne_zero_iff.2 ⟨fun _ => a⟩
 #align cardinal.power_ne_zero Cardinal.power_ne_zero
 
-theorem mul_power {a b c : Cardinal} : ((a * b) ^ c) = (a ^ c) * (b ^ c) :=
+theorem mul_power {a b c : Cardinal} : (a * b) ^ c = a ^ c * b ^ c :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.arrowProdEquivProdArrow α β γ
 #align cardinal.mul_power Cardinal.mul_power
 
-theorem power_mul {a b c : Cardinal} : (a ^ (b * c)) = ((a ^ b) ^ c) := by
+theorem power_mul {a b c : Cardinal} : a ^ (b * c) = (a ^ b) ^ c := by
   rw [mul_comm b c]
   exact inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.curry γ β α
 #align cardinal.power_mul Cardinal.power_mul
 
 @[simp]
-theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : (a^(↑n : Cardinal.{u})) = a ^ℕ n :=
+theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : a ^ (↑n : Cardinal.{u}) = a ^ℕ n :=
   rfl
 #align cardinal.pow_cast_right Cardinal.pow_cast_right
 
@@ -642,16 +642,17 @@ theorem lift_two : lift.{u, v} 2 = 2 := by simp [←one_add_one_eq_two]
 #align cardinal.lift_two Cardinal.lift_two
 
 @[simp]
-theorem mk_set {α : Type u} : #(Set α) = (2 ^ #α) := by simp [←one_add_one_eq_two, Set, mk_arrow]
+theorem mk_set {α : Type u} : #(Set α) = 2 ^ #α := by simp [←one_add_one_eq_two, Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
 
 /-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/
 @[simp]
-theorem mk_powerset {α : Type u} (s : Set α) : #(↥(𝒫 s)) = (2 ^ #(↥s)) :=
+theorem mk_powerset {α : Type u} (s : Set α) : #(↥(𝒫 s)) = 2 ^ #(↥s) :=
   (mk_congr (Equiv.Set.powerset s)).trans mk_set
 #align cardinal.mk_powerset Cardinal.mk_powerset
 
-theorem lift_two_power (a) : lift.{v} (2 ^ a) = (2 ^ lift.{v} a) := by simp [←one_add_one_eq_two]
+theorem lift_two_power (a : Cardinal) : lift.{v} (2 ^ a) = 2 ^ lift.{v} a := by
+  simp [←one_add_one_eq_two]
 #align cardinal.lift_two_power Cardinal.lift_two_power
 
 section OrderProperties
@@ -714,20 +715,20 @@ instance : CommMonoidWithZero Cardinal.{u} :=
 instance : CommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
 
-theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 := by
+theorem zero_power_le (c : Cardinal.{u}) : (0 : Cardinal.{u}) ^ c ≤ 1 := by
   by_cases h : c = 0
   · rw [h, power_zero]
   · rw [zero_power h]
     apply zero_le
 #align cardinal.zero_power_le Cardinal.zero_power_le
 
-theorem power_le_power_left : ∀ {a b c : Cardinal}, a ≠ 0 → b ≤ c → (a^b) ≤ (a^c) := by
+theorem power_le_power_left : ∀ {a b c : Cardinal}, a ≠ 0 → b ≤ c → a ^ b ≤ a ^ c := by
   rintro ⟨α⟩ ⟨β⟩ ⟨γ⟩ hα ⟨e⟩
   let ⟨a⟩ := mk_ne_zero_iff.1 hα
   exact ⟨@Function.Embedding.arrowCongrLeft _ _ _ ⟨a⟩ e⟩
 #align cardinal.power_le_power_left Cardinal.power_le_power_left
 
-theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b) := by
+theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ a ^ b := by
   rcases eq_or_ne a 0 with (rfl | ha)
   · exact zero_le _
   · convert power_le_power_left ha hb
@@ -735,7 +736,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 #align cardinal.self_le_power Cardinal.self_le_power
 
 /-- **Cantor's theorem** -/
-theorem cantor (a : Cardinal.{u}) : a < (2^a) := by
+theorem cantor (a : Cardinal.{u}) : a < 2 ^ a := by
   induction' a using Cardinal.inductionOn with α
   rw [← mk_set]
   refine' ⟨⟨⟨singleton, fun a b => singleton_eq_singleton_iff.1⟩⟩, _⟩
@@ -752,17 +753,17 @@ theorem one_lt_iff_nontrivial {α : Type u} : 1 < #α ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
 
-theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 := by
+theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : a ^ b ≤ max (a ^ c) 1 := by
   by_cases ha : a = 0
   · simp [ha, zero_power_le]
   · exact (power_le_power_left ha h).trans (le_max_left _ _)
 #align cardinal.power_le_max_power_one Cardinal.power_le_max_power_one
 
-theorem power_le_power_right {a b c : Cardinal} : a ≤ b → (a^c) ≤ (b^c) :=
+theorem power_le_power_right {a b c : Cardinal} : a ≤ b → a ^ c ≤ b ^ c :=
   inductionOn₃ a b c fun _ _ _ ⟨e⟩ => ⟨Embedding.arrowCongrRight e⟩
 #align cardinal.power_le_power_right Cardinal.power_le_power_right
 
-theorem power_pos {a : Cardinal} (b) (ha : 0 < a) : 0 < (a^b) :=
+theorem power_pos {a : Cardinal} (b : Cardinal) (ha : 0 < a) : 0 < a ^ b :=
   (power_ne_zero _ ha.ne').bot_lt
 #align cardinal.power_pos Cardinal.power_pos
 
@@ -1051,12 +1052,12 @@ theorem mk_pi {ι : Type u} (α : ι → Type v) : #(∀ i, α i) = prod fun i =
 
 @[simp]
 theorem prod_const (ι : Type u) (a : Cardinal.{v}) :
-    (prod fun _ : ι => a) = (lift.{u} a^lift.{v} #ι) :=
+    (prod fun _ : ι => a) = lift.{u} a ^ lift.{v} #ι :=
   inductionOn a fun _ =>
     mk_congr <| Equiv.piCongr Equiv.ulift.symm fun _ => outMkEquiv.trans Equiv.ulift.symm
 #align cardinal.prod_const Cardinal.prod_const
 
-theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) = (a^#ι) :=
+theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) = a ^ #ι :=
   inductionOn a fun _ => (mk_pi _).symm
 #align cardinal.prod_const' Cardinal.prod_const'
 
@@ -1346,7 +1347,7 @@ theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
 -- Porting note: was `simp`. LHS is not normal form.
 -- @[simp, norm_cast]
 @[norm_cast]
-theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = (m^n) := by
+theorem natCast_pow {m n : ℕ} : (↑(m ^ n) : Cardinal) = m ^ n := by
   induction n <;> simp [pow_succ', power_add, *, Pow.pow]
 #align cardinal.nat_cast_pow Cardinal.natCast_pow
 
@@ -1398,7 +1399,7 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
-theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) := by
+theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a := by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
@@ -1598,7 +1599,7 @@ theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠
     a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by simp [mul_lt_aleph0_iff, ha, hb]
 #align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero
 
-theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (a^b) < ℵ₀ :=
+theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a ^ b < ℵ₀ :=
   match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← natCast_pow]; apply nat_lt_aleph0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
@@ -2413,13 +2414,13 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 @[inherit_doc]
 infixl:80 " ^< " => powerlt
 
-theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by
-  refine le_ciSup (f := fun y : Iio b => a^y) ?_ ⟨c, h⟩
+theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : a ^ c ≤ a ^< b := by
+  refine le_ciSup (f := fun y : Iio b => a ^ y) ?_ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
 
-theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c := by
+theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, a ^ x ≤ c := by
   rw [powerlt, ciSup_le_iff']
   · simp
   · rw [← image_eq_range]
@@ -2433,7 +2434,7 @@ theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b ≤ c) : a ^< b ≤ a
 theorem powerlt_mono_left (a) : Monotone fun c => a ^< c := fun _ _ => powerlt_le_powerlt_left
 #align cardinal.powerlt_mono_left Cardinal.powerlt_mono_left
 
-theorem powerlt_succ {a b : Cardinal} (h : a ≠ 0) : a ^< succ b = (a^b) :=
+theorem powerlt_succ {a b : Cardinal} (h : a ≠ 0) : a ^< succ b = a ^ b :=
   (powerlt_le.2 fun _ h' => power_le_power_left h <| le_of_lt_succ h').antisymm <|
     le_powerlt a (lt_succ b)
 #align cardinal.powerlt_succ Cardinal.powerlt_succ

feat: some simp lemmas to compute more cardinals (#7660)

a577026a

Diff

@@ -1656,6 +1656,16 @@ theorem nat_mul_aleph0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ :=
 theorem aleph0_mul_nat {n : ℕ} (hn : n ≠ 0) : ℵ₀ * n = ℵ₀ := by rw [mul_comm, nat_mul_aleph0 hn]
 #align cardinal.aleph_0_mul_nat Cardinal.aleph0_mul_nat
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem ofNat_mul_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) * ℵ₀ = ℵ₀ :=
+  nat_mul_aleph0 (OfNat.ofNat_ne_zero n)
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem aleph0_mul_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ * no_index (OfNat.ofNat n) = ℵ₀ :=
+  aleph0_mul_nat (OfNat.ofNat_ne_zero n)
+
 @[simp]
 theorem add_le_aleph0 {c₁ c₂ : Cardinal} : c₁ + c₂ ≤ ℵ₀ ↔ c₁ ≤ ℵ₀ ∧ c₂ ≤ ℵ₀ :=
   ⟨fun h => ⟨le_self_add.trans h, le_add_self.trans h⟩, fun h =>
@@ -1671,6 +1681,16 @@ theorem aleph0_add_nat (n : ℕ) : ℵ₀ + n = ℵ₀ :=
 theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ := by rw [add_comm, aleph0_add_nat]
 #align cardinal.nat_add_aleph_0 Cardinal.nat_add_aleph0
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem ofNat_add_aleph0 {n : ℕ} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) + ℵ₀ = ℵ₀ :=
+  nat_add_aleph0 n
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem aleph0_add_ofNat {n : ℕ} [Nat.AtLeastTwo n] : ℵ₀ + no_index (OfNat.ofNat n) = ℵ₀ :=
+  aleph0_add_nat n
+
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to 0. -/
 def toNat : ZeroHom Cardinal ℕ where

chore: rename CanonicallyOrderedAddMonoid to ..AddCommMonoid (#7503)

Renames:

CanonicallyOrderedMonoid -> CanonicallyOrderedCommMonoid

CanonicallyOrderedAddMonoid -> CanonicallyOrderedAddCommMonoid

CanonicallyLinearOrderedMonoid -> CanonicallyLinearOrderedCommMonoid

CanonicallyLinearOrderedAddMonoid -> CanonicallyLinearOrderedAddCommMonoid

f3bc24c3

Diff

@@ -692,11 +692,11 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardina
       inductionOn₂ a b fun α β => by
         simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }
 
-instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
+instance : CanonicallyLinearOrderedAddCommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
 
 -- Computable instance to prevent a non-computable one being found via the one above
-instance : CanonicallyOrderedAddMonoid Cardinal.{u} :=
+instance : CanonicallyOrderedAddCommMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring with }
 
 instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=

chore: only four spaces for subsequent lines (#7286)

Co-authored-by: Moritz Firsching <firsching@google.com>

0c7d6fa5

Diff

@@ -1901,7 +1901,7 @@ theorem toPartENat_surjective : Surjective toPartENat := fun x =>
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 
 theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} :
-  toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
+    toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
   cases lt_or_ge c ℵ₀ with
   | inl hc =>
     simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]
@@ -1909,7 +1909,7 @@ theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} :
 #align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0
 
 lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
-  toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
+    toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
   cases lt_or_ge c ℵ₀ with
   | inl hc =>
     rw [toPartENat_apply_of_lt_aleph0 hc]
@@ -1928,7 +1928,7 @@ lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
 #align to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_of_le_aleph0
 
 lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
-  toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
+    toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
   cases lt_or_ge c ℵ₀ with
   | inl hc =>
     rw [toPartENat_apply_of_lt_aleph0 hc]
@@ -1942,13 +1942,13 @@ lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
 #align to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_of_lt_aleph0
 
 lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :
-  toPartENat c = toPartENat c' ↔ c = c' := by
+    toPartENat c = toPartENat c' ↔ c = c' := by
   rw [le_antisymm_iff, le_antisymm_iff, toPartENat_le_iff_of_le_aleph0 hc,
     toPartENat_le_iff_of_le_aleph0 hc']
 #align to_part_enat_eq_iff_eq_of_le_aleph_0 Cardinal.toPartENat_eq_iff_of_le_aleph0
 
 theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') :
-  toPartENat c ≤ toPartENat c' := by
+    toPartENat c ≤ toPartENat c' := by
   cases lt_or_ge c ℵ₀ with
   | inl hc =>
     rw [toPartENat_apply_of_lt_aleph0 hc]

chore: replace anonymous morphism constructors with named fields (#7015)

This makes it easier to refactor the order or inheritance structure of morphisms without having to change all of the anonymous constructors.

This is far from exhaustive.

2bffb702

Diff

@@ -1673,12 +1673,13 @@ theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ := by rw [add_comm, al
 
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to 0. -/
-def toNat : ZeroHom Cardinal ℕ :=
-  ⟨fun c => if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0, by
+def toNat : ZeroHom Cardinal ℕ where
+  toFun c := if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0
+  map_zero' := by
     have h : 0 < ℵ₀ := nat_lt_aleph0 0
     dsimp only
     rw [dif_pos h, ← Cardinal.natCast_inj, ← Classical.choose_spec (lt_aleph0.1 h),
-      Nat.cast_zero]⟩
+      Nat.cast_zero]
 #align cardinal.to_nat Cardinal.toNat
 
 theorem toNat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) :

feat: patch for new alias command (#6172)

19e0ba92

Diff

@@ -163,7 +163,7 @@ theorem mk_congr (e : α ≃ β) : #α = #β :=
   Quot.sound ⟨e⟩
 #align cardinal.mk_congr Cardinal.mk_congr
 
-alias mk_congr ← _root_.Equiv.cardinal_eq
+alias _root_.Equiv.cardinal_eq := mk_congr
 #align equiv.cardinal_eq Equiv.cardinal_eq
 
 /-- Lift a function between `Type*`s to a function between `Cardinal`s. -/
@@ -428,7 +428,7 @@ theorem mk_le_one_iff_set_subsingleton {s : Set α} : #s ≤ 1 ↔ s.Subsingleto
   le_one_iff_subsingleton.trans s.subsingleton_coe
 #align cardinal.mk_le_one_iff_set_subsingleton Cardinal.mk_le_one_iff_set_subsingleton
 
-alias mk_le_one_iff_set_subsingleton ↔ _ _root_.Set.Subsingleton.cardinal_mk_le_one
+alias ⟨_, _root_.Set.Subsingleton.cardinal_mk_le_one⟩ := mk_le_one_iff_set_subsingleton
 #align set.subsingleton.cardinal_mk_le_one Set.Subsingleton.cardinal_mk_le_one
 
 instance : Add Cardinal.{u} :=
@@ -1491,7 +1491,7 @@ theorem lt_aleph0_iff_set_finite {S : Set α} : #S < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
 
-alias lt_aleph0_iff_set_finite ↔ _ _root_.Set.Finite.lt_aleph0
+alias ⟨_, _root_.Set.Finite.lt_aleph0⟩ := lt_aleph0_iff_set_finite
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
 @[simp]
@@ -1514,7 +1514,7 @@ theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable
   rw [mk_le_aleph0_iff, countable_coe_iff]
 #align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable
 
-alias le_aleph0_iff_set_countable ↔ _ _root_.Set.Countable.le_aleph0
+alias ⟨_, _root_.Set.Countable.le_aleph0⟩ := le_aleph0_iff_set_countable
 #align set.countable.le_aleph_0 Set.Countable.le_aleph0
 
 @[simp]

chore(SetTheory): add missing pp_with_univ to Cardinal.lift and Ordinal.lift (#6683)

The docstrings telling you to enable pp.universes are no longer relevant.

4d4baabf

Diff

@@ -185,6 +185,7 @@ def map₂ (f : Type u → Type v → Type w) (hf : ∀ α β γ δ, α ≃ β 
 
 /-- The universe lift operation on cardinals. You can specify the universes explicitly with
   `lift.{u v} : Cardinal.{v} → Cardinal.{max v u}` -/
+@[pp_with_univ]
 def lift (c : Cardinal.{v}) : Cardinal.{max v u} :=
   map ULift.{u, v} (fun _ _ e => Equiv.ulift.trans <| e.trans Equiv.ulift.symm) c
 #align cardinal.lift Cardinal.lift
@@ -196,8 +197,7 @@ theorem mk_uLift (α) : #(ULift.{v, u} α) = lift.{v} #α :=
 
 -- Porting note : simpNF is not happy with universe levels, but this is needed as simp lemma
 -- further down in this file
-/-- `lift.{(max u v) u}` equals `lift.{v u}`. Using `set_option pp.universes true` will make it much
-    easier to understand what's happening when using this lemma. -/
+/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/
 @[simp, nolint simpNF]
 theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
   funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
@@ -205,8 +205,7 @@ theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
 
 -- Porting note : simpNF is not happy with universe levels, but this is needed as simp lemma
 -- further down in this file
-/-- `lift.{(max v u) u}` equals `lift.{v u}`. Using `set_option pp.universes true` will make it much
-    easier to understand what's happening when using this lemma. -/
+/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/
 @[simp, nolint simpNF]
 theorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=
   lift_umax

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -166,7 +166,7 @@ theorem mk_congr (e : α ≃ β) : #α = #β :=
 alias mk_congr ← _root_.Equiv.cardinal_eq
 #align equiv.cardinal_eq Equiv.cardinal_eq
 
-/-- Lift a function between `Type _`s to a function between `Cardinal`s. -/
+/-- Lift a function between `Type*`s to a function between `Cardinal`s. -/
 def map (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f β) : Cardinal.{u} → Cardinal.{v} :=
   Quotient.map f fun α β ⟨e⟩ => ⟨hf α β e⟩
 #align cardinal.map Cardinal.map
@@ -177,7 +177,7 @@ theorem map_mk (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f
   rfl
 #align cardinal.map_mk Cardinal.map_mk
 
-/-- Lift a binary operation `Type _ → Type _ → Type _` to a binary operation on `Cardinal`s. -/
+/-- Lift a binary operation `Type* → Type* → Type*` to a binary operation on `Cardinal`s. -/
 def map₂ (f : Type u → Type v → Type w) (hf : ∀ α β γ δ, α ≃ β → γ ≃ δ → f α γ ≃ f β δ) :
     Cardinal.{u} → Cardinal.{v} → Cardinal.{w} :=
   Quotient.map₂ f fun α β ⟨e₁⟩ γ δ ⟨e₂⟩ => ⟨hf α β γ δ e₁ e₂⟩
@@ -871,7 +871,7 @@ theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
 #align cardinal.le_sum Cardinal.le_sum
 
 @[simp]
-theorem mk_sigma {ι} (f : ι → Type _) : #(Σ i, f i) = sum fun i => #(f i) :=
+theorem mk_sigma {ι} (f : ι → Type*) : #(Σ i, f i) = sum fun i => #(f i) :=
   mk_congr <| Equiv.sigmaCongrRight fun _ => outMkEquiv.symm
 #align cardinal.mk_sigma Cardinal.mk_sigma
 
@@ -1604,7 +1604,7 @@ theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← natCast_pow]; apply nat_lt_aleph0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
 
-theorem eq_one_iff_unique {α : Type _} : #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
+theorem eq_one_iff_unique {α : Type*} : #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
   calc
     #α = 1 ↔ #α ≤ 1 ∧ 1 ≤ #α := le_antisymm_iff
     _ ↔ Subsingleton α ∧ Nonempty α :=
@@ -1621,7 +1621,7 @@ theorem aleph0_le_mk (α : Type u) [Infinite α] : ℵ₀ ≤ #α :=
 #align cardinal.aleph_0_le_mk Cardinal.aleph0_le_mk
 
 @[simp]
-theorem mk_eq_aleph0 (α : Type _) [Countable α] [Infinite α] : #α = ℵ₀ :=
+theorem mk_eq_aleph0 (α : Type*) [Countable α] [Infinite α] : #α = ℵ₀ :=
   mk_le_aleph0.antisymm <| aleph0_le_mk _
 #align cardinal.mk_eq_aleph_0 Cardinal.mk_eq_aleph0
 
@@ -1789,7 +1789,7 @@ theorem toNat_eq_one {c : Cardinal} : toNat c = 1 ↔ c = 1 := by
   rw [toNat_eq_iff one_ne_zero, Nat.cast_one]
 #align cardinal.to_nat_eq_one Cardinal.toNat_eq_one
 
-theorem toNat_eq_one_iff_unique {α : Type _} : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
+theorem toNat_eq_one_iff_unique {α : Type*} : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
   toNat_eq_one.trans eq_one_iff_unique
 #align cardinal.to_nat_eq_one_iff_unique Cardinal.toNat_eq_one_iff_unique
 
@@ -2368,7 +2368,7 @@ theorem mk_eq_two_iff' (x : α) : #α = 2 ↔ ∃! y, y ≠ x := by
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'
 
-theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length < #α) :
+theorem exists_not_mem_of_length_lt {α : Type*} (l : List α) (h : ↑l.length < #α) :
     ∃ z : α, z ∉ l := by
   contrapose! h
   calc
@@ -2378,7 +2378,7 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
     _ ≤ l.length := Cardinal.natCast_le.mpr (List.toFinset_card_le l)
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
-theorem three_le {α : Type _} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
+theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
   have : ↑(3 : ℕ) ≤ #α; simpa using h
   have : ↑(2 : ℕ) < #α; rwa [← succ_le_iff, ← Cardinal.nat_succ]
   have := exists_not_mem_of_length_lt [x, y] this

feat: cardinality of a set with a countable cover (#6351)

Assume that a set t is eventually covered by a countable family of sets, all with cardinality ≤ a. Then t itself has cardinality at most a.

76efa470

Diff

@@ -293,6 +293,15 @@ theorem mk_set_le (s : Set α) : #s ≤ #α :=
   mk_subtype_le s
 #align cardinal.mk_set_le Cardinal.mk_set_le
 
+@[simp]
+lemma mk_preimage_down {s : Set α} : #(ULift.down.{v} ⁻¹' s) = lift.{v} (#s) := by
+  rw [← mk_uLift, Cardinal.eq]
+  constructor
+  let f : ULift.down ⁻¹' s → ULift s := fun x ↦ ULift.up (restrictPreimage s ULift.down x)
+  have : Function.Bijective f :=
+    ULift.up_bijective.comp (restrictPreimage_bijective _ (ULift.down_bijective))
+  exact Equiv.ofBijective f this
+
 theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.out) := by
   trans
   · rw [← Quotient.out_eq c, ← Quotient.out_eq c']
@@ -2112,6 +2121,11 @@ theorem mk_range_eq_of_injective {α : Type u} {β : Type v} {f : α → β} (hf
   lift_mk_eq'.mpr ⟨(Equiv.ofInjective f hf).symm⟩
 #align cardinal.mk_range_eq_of_injective Cardinal.mk_range_eq_of_injective
 
+lemma lift_mk_le_lift_mk_of_injective {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
+    Cardinal.lift.{v} (#α) ≤ Cardinal.lift.{u} (#β) := by
+  rw [← Cardinal.mk_range_eq_of_injective hf]
+  exact Cardinal.lift_le.2 (Cardinal.mk_set_le _)
+
 theorem mk_range_eq_lift {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
     lift.{max u w} #(range f) = lift.{max v w} #α :=
   lift_mk_eq.{v,u,w}.mpr ⟨(Equiv.ofInjective f hf).symm⟩
@@ -2207,6 +2221,17 @@ theorem mk_le_mk_of_subset {α} {s t : Set α} (h : s ⊆ t) : #s ≤ #t :=
   ⟨Set.embeddingOfSubset s t h⟩
 #align cardinal.mk_le_mk_of_subset Cardinal.mk_le_mk_of_subset
 
+theorem mk_le_iff_forall_finset_subset_card_le {α : Type u} {n : ℕ} {t : Set α} :
+    #t ≤ n ↔ ∀ s : Finset α, (s : Set α) ⊆ t → s.card ≤ n := by
+  refine ⟨fun H s hs ↦ by simpa using (mk_le_mk_of_subset hs).trans H, fun H ↦ ?_⟩
+  apply card_le_of (fun s ↦ ?_)
+  let u : Finset α := s.image Subtype.val
+  have : u.card = s.card :=
+    Finset.card_image_of_injOn (injOn_of_injective Subtype.coe_injective _)
+  rw [← this]
+  apply H
+  simp only [Finset.coe_image, image_subset_iff, Subtype.coe_preimage_self, subset_univ]
+
 theorem mk_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) :
     #{ x // p x } ≤ #{ x // q x } :=
   ⟨embeddingOfSubset _ _ h⟩
@@ -2413,6 +2438,15 @@ theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 := by
   exact Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt
 #align cardinal.powerlt_zero Cardinal.powerlt_zero
 
+/-- The cardinality of a nontrivial module over a ring is at least the cardinality of the ring if
+there are no zero divisors (for instance if the ring is a field) -/
+theorem mk_le_of_module (R : Type u) (E : Type v)
+    [AddCommGroup E] [Ring R] [Module R E] [Nontrivial E] [NoZeroSMulDivisors R E] :
+    Cardinal.lift.{v} (#R) ≤ Cardinal.lift.{u} (#E) := by
+  obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0
+  have : Injective (fun k ↦ k • x) := smul_left_injective R hx
+  exact lift_mk_le_lift_mk_of_injective this
+
 end Cardinal
 
 -- namespace Tactic

feat: change definition of natCast in Cardinal to lift #(Fin n) (#6384)

The new definition is conceptually more satisfactory.

By also changing the definition of the Zero and One instances, we get ((0 : ℕ) : Cardinal) = 0 and ((1 : ℕ) : Cardinal) = 1 definitionally (which wasn't true before), which is in practice more convenient than definitional equality with PEmpty or PUnit.

dc0a7a96

Diff

@@ -364,18 +364,18 @@ theorem lift_monotone : Monotone lift :=
 #align cardinal.lift_monotone Cardinal.lift_monotone
 
 instance : Zero Cardinal.{u} :=
-  ⟨#PEmpty⟩
+  -- `PEmpty` might be more canonical, but this is convenient for defeq with natCast
+  ⟨lift #(Fin 0)⟩
 
 instance : Inhabited Cardinal.{u} :=
   ⟨0⟩
 
 theorem mk_eq_zero (α : Type u) [IsEmpty α] : #α = 0 :=
-  (Equiv.equivPEmpty α).cardinal_eq
+  (Equiv.equivOfIsEmpty α (ULift (Fin 0))).cardinal_eq
 #align cardinal.mk_eq_zero Cardinal.mk_eq_zero
 
 @[simp]
-theorem lift_zero : lift 0 = 0 :=
-  mk_congr (Equiv.equivPEmpty _)
+theorem lift_zero : lift 0 = 0 := mk_eq_zero _
 #align cardinal.lift_zero Cardinal.lift_zero
 
 @[simp]
@@ -400,18 +400,19 @@ theorem mk_ne_zero (α : Type u) [Nonempty α] : #α ≠ 0 :=
 #align cardinal.mk_ne_zero Cardinal.mk_ne_zero
 
 instance : One Cardinal.{u} :=
-  ⟨#PUnit⟩
+  -- `PUnit` might be more canonical, but this is convenient for defeq with natCast
+  ⟨lift #(Fin 1)⟩
 
 instance : Nontrivial Cardinal.{u} :=
   ⟨⟨1, 0, mk_ne_zero _⟩⟩
 
 theorem mk_eq_one (α : Type u) [Unique α] : #α = 1 :=
-  (Equiv.equivPUnit α).cardinal_eq
+  (Equiv.equivOfUnique α (ULift (Fin 1))).cardinal_eq
 #align cardinal.mk_eq_one Cardinal.mk_eq_one
 
 theorem le_one_iff_subsingleton {α : Type u} : #α ≤ 1 ↔ Subsingleton α :=
   ⟨fun ⟨f⟩ => ⟨fun _ _ => f.injective (Subsingleton.elim _ _)⟩, fun ⟨h⟩ =>
-    ⟨⟨fun _ => PUnit.unit, fun _ _ _ => h _ _⟩⟩⟩
+    ⟨fun _ => ULift.up 0, fun _ _ _ => h _ _⟩⟩
 #align cardinal.le_one_iff_subsingleton Cardinal.le_one_iff_subsingleton
 
 @[simp]
@@ -429,10 +430,8 @@ theorem add_def (α β : Type u) : #α + #β = #(Sum α β) :=
   rfl
 #align cardinal.add_def Cardinal.add_def
 
--- Porting note: Should this be changed to
--- `⟨fun n => lift #(Fin n)⟩` in the future?
 instance : NatCast Cardinal.{u} :=
-  ⟨Nat.unaryCast⟩
+  ⟨fun n => lift #(Fin n)⟩
 
 @[simp]
 theorem mk_sum (α : Type u) (β : Type v) : #(α ⊕ β) = lift.{v, u} #α + lift.{u, v} #β :=
@@ -440,8 +439,8 @@ theorem mk_sum (α : Type u) (β : Type v) : #(α ⊕ β) = lift.{v, u} #α + li
 #align cardinal.mk_sum Cardinal.mk_sum
 
 @[simp]
-theorem mk_option {α : Type u} : #(Option α) = #α + 1 :=
-  (Equiv.optionEquivSumPUnit α).cardinal_eq
+theorem mk_option {α : Type u} : #(Option α) = #α + 1 := by
+  rw [(Equiv.optionEquivSumPUnit.{u, u} α).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id]
 #align cardinal.mk_option Cardinal.mk_option
 
 @[simp]
@@ -450,16 +449,13 @@ theorem mk_psum (α : Type u) (β : Type v) : #(PSum α β) = lift.{v} #α + lif
 #align cardinal.mk_psum Cardinal.mk_psum
 
 @[simp]
-theorem mk_fintype (α : Type u) [h : Fintype α] : #α = Fintype.card α := by
-  refine Fintype.induction_empty_option ?_ ?_ ?_ α (h_fintype := h)
-  · intro α β h e hα
-    letI := Fintype.ofEquiv β e.symm
-    rwa [mk_congr e, Fintype.card_congr e] at hα
-  · rfl
-  · intro α h hα
-    simp [hα]
-    rfl
-#align cardinal.mk_fintype Cardinal.mk_fintype
+theorem mk_fintype (α : Type u) [h : Fintype α] : #α = Fintype.card α :=
+  mk_congr (Fintype.equivOfCardEq (by simp))
+
+protected theorem cast_succ (n : ℕ) : ((n + 1 : ℕ) : Cardinal.{u}) = n + 1 := by
+  change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1
+  rw [← mk_option, mk_fintype, mk_fintype]
+  simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option]
 
 instance : Mul Cardinal.{u} :=
   ⟨map₂ Prod fun _ _ _ _ => Equiv.prodCongr⟩
@@ -505,12 +501,12 @@ theorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = ((lift.{v} a) ^ (li
 
 @[simp]
 theorem power_zero {a : Cardinal} : (a ^ 0) = 1 :=
-  inductionOn a fun α => mk_congr <| Equiv.pemptyArrowEquivPUnit α
+  inductionOn a fun _ => mk_eq_one _
 #align cardinal.power_zero Cardinal.power_zero
 
 @[simp]
-theorem power_one {a : Cardinal} : (a ^ 1) = a :=
-  inductionOn a fun α => mk_congr <| Equiv.punitArrowEquiv α
+theorem power_one {a : Cardinal.{u}} : (a ^ 1) = a :=
+  inductionOn a fun α => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) α)
 #align cardinal.power_one Cardinal.power_one
 
 theorem power_add {a b c : Cardinal} : (a ^ (b + c)) = (a ^ b) * (a ^ c) :=
@@ -522,21 +518,25 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   one := 1
   add := (· + ·)
   mul := (· * ·)
-  zero_add a := inductionOn a fun α => mk_congr <| Equiv.emptySum PEmpty α
-  add_zero a := inductionOn a fun α => mk_congr <| Equiv.sumEmpty α PEmpty
+  zero_add a := inductionOn a fun α => mk_congr <| Equiv.emptySum (ULift (Fin 0)) α
+  add_zero a := inductionOn a fun α => mk_congr <| Equiv.sumEmpty α (ULift (Fin 0))
   add_assoc a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumAssoc α β γ
   add_comm a b := inductionOn₂ a b fun α β => mk_congr <| Equiv.sumComm α β
-  zero_mul a := inductionOn a fun α => mk_congr <| Equiv.pemptyProd α
-  mul_zero a := inductionOn a fun α => mk_congr <| Equiv.prodPEmpty α
-  one_mul a := inductionOn a fun α => mk_congr <| Equiv.punitProd α
-  mul_one a := inductionOn a fun α => mk_congr <| Equiv.prodPUnit α
+  zero_mul a := inductionOn a fun α => mk_eq_zero _
+  mul_zero a := inductionOn a fun α => mk_eq_zero _
+  one_mul a := inductionOn a fun α => mk_congr <| Equiv.uniqueProd α (ULift (Fin 1))
+  mul_one a := inductionOn a fun α => mk_congr <| Equiv.prodUnique α (ULift (Fin 1))
   mul_assoc a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodAssoc α β γ
   mul_comm := mul_comm'
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
   npow n c := c^n
   npow_zero := @power_zero
-  npow_succ n c := show (c ^ (n + 1)) = c * (c ^ n) by rw [power_add, power_one, mul_comm']
+  npow_succ n c := show (c ^ (n + 1 : ℕ)) = c * (c ^ n)
+    by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']
+  natCast := (fun n => lift.{u} #(Fin n) : ℕ → Cardinal.{u})
+  natCast_zero := rfl
+  natCast_succ := Cardinal.cast_succ
 
 /-! Porting note: Deprecated section. Remove. -/
 section deprecated
@@ -556,7 +556,7 @@ end deprecated
 
 @[simp]
 theorem one_power {a : Cardinal} : (1 ^ a) = 1 :=
-  inductionOn a fun α => (Equiv.arrowPUnitEquivPUnit α).cardinal_eq
+  inductionOn a fun _ => mk_eq_one _
 #align cardinal.one_power Cardinal.one_power
 
 -- Porting note : simp can prove this
@@ -599,8 +599,7 @@ theorem pow_cast_right (a : Cardinal.{u}) (n : ℕ) : (a^(↑n : Cardinal.{u}))
 #align cardinal.pow_cast_right Cardinal.pow_cast_right
 
 @[simp]
-theorem lift_one : lift 1 = 1 :=
-  mk_congr <| Equiv.ulift.trans Equiv.punitEquivPUnit
+theorem lift_one : lift 1 = 1 := mk_eq_one _
 #align cardinal.lift_one Cardinal.lift_one
 
 @[simp]
@@ -1312,7 +1311,7 @@ theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} : n < lift.{v} a ↔ n < a
   rw [← lift_natCast.{v,u}, lift_lt]
 #align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
 
-theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n := by simp
+theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n := rfl
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
 
 theorem mk_coe_finset {α : Type u} {s : Finset α} : #s = ↑(Finset.card s) := by simp
@@ -1372,9 +1371,11 @@ theorem natCast_injective : Injective ((↑) : ℕ → Cardinal) :=
 #align cardinal.nat_cast_injective Cardinal.natCast_injective
 
 @[simp, norm_cast]
-theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n :=
-  (add_one_le_succ _).antisymm (succ_le_of_lt <| natCast_lt.2 <| Nat.lt_succ_self _)
-#align cardinal.nat_succ Cardinal.nat_succ
+theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n := by
+  rw [Nat.cast_succ]
+  refine (add_one_le_succ _).antisymm (succ_le_of_lt ?_)
+  rw [← Nat.cast_succ]
+  exact natCast_lt.2 (Nat.lt_succ_self _)
 
 @[simp]
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast

feat(SetTheory/Cardinal): add ofNat lemmas (#6362)

90907eb3

Diff

@@ -1769,6 +1769,11 @@ theorem toNat_eq_iff {c : Cardinal} {n : ℕ} (hn : n ≠ 0) : toNat c = n ↔ c
     fun h => (congr_arg toNat h).trans (toNat_cast n)⟩
 #align cardinal.to_nat_eq_iff Cardinal.toNat_eq_iff
 
+/-- A version of `toNat_eq_iff` for literals -/
+theorem toNat_eq_ofNat {c : Cardinal} {n : ℕ} [Nat.AtLeastTwo n] :
+    toNat c = OfNat.ofNat n ↔ c = OfNat.ofNat n :=
+  toNat_eq_iff <| Nat.cast_ne_zero.mpr <| OfNat.ofNat_ne_zero n
+
 @[simp]
 theorem toNat_eq_one {c : Cardinal} : toNat c = 1 ↔ c = 1 := by
   rw [toNat_eq_iff one_ne_zero, Nat.cast_one]

feat: let convert infer instances from goal and allow tc failure (#6041)

Changes the way the term is elaborated so that typeclass inference is tolerated and so that instances are allowed to come from the goal without re-inferring them.

90689c42

Diff

@@ -2402,10 +2402,9 @@ theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 := by
 #align cardinal.zero_powerlt Cardinal.zero_powerlt
 
 @[simp]
-theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 :=
-  -- Porting note: used to expect that `convert` would leave an instance argument as a goal
-  @Cardinal.iSup_of_empty _ _
-    (Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt)
+theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 := by
+  convert Cardinal.iSup_of_empty _
+  exact Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt
 #align cardinal.powerlt_zero Cardinal.powerlt_zero
 
 end Cardinal

doc: add missing word (#6006)

eb706c16

Diff

@@ -23,7 +23,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 
 ## Main definitions
 
-* `Cardinal` the type of cardinal numbers (in a given universe).
+* `Cardinal` is the type of cardinal numbers (in a given universe).
 * `Cardinal.mk α` or `#α` is the cardinality of `α`. The notation `#` lives in the locale
   `Cardinal`.
 * Addition `c₁ + c₂` is defined by `Cardinal.add_def α β : #α + #β = #(α ⊕ β)`.

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-
-! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 3ff3f2d6a3118b8711063de7111a0d77a53219a8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.Finsupp.Defs
@@ -19,6 +14,8 @@ import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.Tactic.Positivity
 import Mathlib.Tactic.PPWithUniv
 
+#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
+
 /-!
 # Cardinal Numbers

chore: cleanup whitespace (#5988)

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

12343aa5

Diff

@@ -1099,7 +1099,7 @@ theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{
 theorem lift_sInf (s : Set Cardinal) : lift.{u,v} (sInf s) = sInf (lift.{u,v} '' s) := by
   rcases eq_empty_or_nonempty s with (rfl | hs)
   · simp
-  · exact  lift_monotone.map_csInf hs
+  · exact lift_monotone.map_csInf hs
 #align cardinal.lift_Inf Cardinal.lift_sInf
 
 -- Porting note: Inserted .{u,v} below

feat:add correct SHA to earlier PR (#5799)

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

5d83d804

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 9bb28972724354ac0574e2b318be896ec252025f
+! leanprover-community/mathlib commit 3ff3f2d6a3118b8711063de7111a0d77a53219a8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/

refactor: make pp_with_univ be an attribute (#5633)

99ec98e2

Diff

@@ -107,10 +107,10 @@ instance Cardinal.isEquivalent : Setoid (Type u) where
 /-- `Cardinal.{u}` is the type of cardinal numbers in `Type u`,
   defined as the quotient of `Type u` by existence of an equivalence
   (a bijection with explicit inverse). -/
+@[pp_with_univ]
 def Cardinal : Type (u + 1) :=
   Quotient Cardinal.isEquivalent
 #align cardinal Cardinal
-pp_with_univ Cardinal
 
 namespace Cardinal

chore: fix focusing dots (#5708)

This PR is the result of running

find . -type f -name "*.lean" -exec sed -i -E 's/^( +)\. /\1· /' {} \;
find . -type f -name "*.lean" -exec sed -i -E 'N;s/^( +·)\n +(.*)$/\1 \2/;P;D' {} \;

which firstly replaces . focusing dots with · and secondly removes isolated instances of such dots, unifying them with the following line. A new rule is placed in the style linter to verify this.

cb02d09e

Diff

@@ -1958,7 +1958,7 @@ theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPart
     simp only [toNat_lift]
     rw [lift_lt_aleph0]
     exact hc
-  . rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]
+  · rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]
     rw [aleph0_le_lift]
     exact hc
 #align cardinal.to_part_enat_lift Cardinal.toPartENat_lift

fix: precedence of # (#5623)

d450fcd7

Diff

@@ -120,40 +120,40 @@ def mk : Type u → Cardinal :=
 #align cardinal.mk Cardinal.mk
 
 @[inherit_doc]
-scoped prefix:0 "#" => Cardinal.mk
+scoped prefix:max "#" => Cardinal.mk
 
 instance canLiftCardinalType : CanLift Cardinal.{u} (Type u) mk fun _ => True :=
   ⟨fun c _ => Quot.inductionOn c fun α => ⟨α, rfl⟩⟩
 #align cardinal.can_lift_cardinal_Type Cardinal.canLiftCardinalType
 
 @[elab_as_elim]
-theorem inductionOn {p : Cardinal → Prop} (c : Cardinal) (h : ∀ α, p (#α)) : p c :=
+theorem inductionOn {p : Cardinal → Prop} (c : Cardinal) (h : ∀ α, p #α) : p c :=
   Quotient.inductionOn c h
 #align cardinal.induction_on Cardinal.inductionOn
 
 @[elab_as_elim]
 theorem inductionOn₂ {p : Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
-    (h : ∀ α β, p (#α) (#β)) : p c₁ c₂ :=
+    (h : ∀ α β, p #α #β) : p c₁ c₂ :=
   Quotient.inductionOn₂ c₁ c₂ h
 #align cardinal.induction_on₂ Cardinal.inductionOn₂
 
 @[elab_as_elim]
 theorem inductionOn₃ {p : Cardinal → Cardinal → Cardinal → Prop} (c₁ : Cardinal) (c₂ : Cardinal)
-    (c₃ : Cardinal) (h : ∀ α β γ, p (#α) (#β) (#γ)) : p c₁ c₂ c₃ :=
+    (c₃ : Cardinal) (h : ∀ α β γ, p #α #β #γ) : p c₁ c₂ c₃ :=
   Quotient.inductionOn₃ c₁ c₂ c₃ h
 #align cardinal.induction_on₃ Cardinal.inductionOn₃
 
-protected theorem eq : (#α) = (#β) ↔ Nonempty (α ≃ β) :=
+protected theorem eq : #α = #β ↔ Nonempty (α ≃ β) :=
   Quotient.eq'
 #align cardinal.eq Cardinal.eq
 
 @[simp]
-theorem mk'_def (α : Type u) : @Eq Cardinal ⟦α⟧ (#α) :=
+theorem mk'_def (α : Type u) : @Eq Cardinal ⟦α⟧ #α :=
   rfl
 #align cardinal.mk_def Cardinal.mk'_def
 
 @[simp]
-theorem mk_out (c : Cardinal) : (#c.out) = c :=
+theorem mk_out (c : Cardinal) : #c.out = c :=
   Quotient.out_eq _
 #align cardinal.mk_out Cardinal.mk_out
 
@@ -162,7 +162,7 @@ def outMkEquiv {α : Type v} : (#α).out ≃ α :=
   Nonempty.some <| Cardinal.eq.mp (by simp)
 #align cardinal.out_mk_equiv Cardinal.outMkEquiv
 
-theorem mk_congr (e : α ≃ β) : (#α) = (#β) :=
+theorem mk_congr (e : α ≃ β) : #α = #β :=
   Quot.sound ⟨e⟩
 #align cardinal.mk_congr Cardinal.mk_congr
 
@@ -176,7 +176,7 @@ def map (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f β) :
 
 @[simp]
 theorem map_mk (f : Type u → Type v) (hf : ∀ α β, α ≃ β → f α ≃ f β) (α : Type u) :
-    map f hf (#α) = (#f α) :=
+    map f hf #α = #(f α) :=
   rfl
 #align cardinal.map_mk Cardinal.map_mk
 
@@ -193,7 +193,7 @@ def lift (c : Cardinal.{v}) : Cardinal.{max v u} :=
 #align cardinal.lift Cardinal.lift
 
 @[simp]
-theorem mk_uLift (α) : (#ULift.{v, u} α) = lift.{v} (#α) :=
+theorem mk_uLift (α) : #(ULift.{v, u} α) = lift.{v} #α :=
   rfl
 #align cardinal.mk_ulift Cardinal.mk_uLift
 
@@ -267,32 +267,32 @@ instance linearOrder : LinearOrder Cardinal.{u} :=
       apply Embedding.total
     decidableLE := Classical.decRel _ }
 
-theorem le_def (α β : Type u) : (#α) ≤ (#β) ↔ Nonempty (α ↪ β) :=
+theorem le_def (α β : Type u) : #α ≤ #β ↔ Nonempty (α ↪ β) :=
   Iff.rfl
 #align cardinal.le_def Cardinal.le_def
 
-theorem mk_le_of_injective {α β : Type u} {f : α → β} (hf : Injective f) : (#α) ≤ (#β) :=
+theorem mk_le_of_injective {α β : Type u} {f : α → β} (hf : Injective f) : #α ≤ #β :=
   ⟨⟨f, hf⟩⟩
 #align cardinal.mk_le_of_injective Cardinal.mk_le_of_injective
 
-theorem _root_.Function.Embedding.cardinal_le {α β : Type u} (f : α ↪ β) : (#α) ≤ (#β) :=
+theorem _root_.Function.Embedding.cardinal_le {α β : Type u} (f : α ↪ β) : #α ≤ #β :=
   ⟨f⟩
 #align function.embedding.cardinal_le Function.Embedding.cardinal_le
 
-theorem mk_le_of_surjective {α β : Type u} {f : α → β} (hf : Surjective f) : (#β) ≤ (#α) :=
+theorem mk_le_of_surjective {α β : Type u} {f : α → β} (hf : Surjective f) : #β ≤ #α :=
   ⟨Embedding.ofSurjective f hf⟩
 #align cardinal.mk_le_of_surjective Cardinal.mk_le_of_surjective
 
-theorem le_mk_iff_exists_set {c : Cardinal} {α : Type u} : c ≤ (#α) ↔ ∃ p : Set α, (#p) = c :=
+theorem le_mk_iff_exists_set {c : Cardinal} {α : Type u} : c ≤ #α ↔ ∃ p : Set α, #p = c :=
   ⟨inductionOn c fun _ ⟨⟨f, hf⟩⟩ => ⟨Set.range f, (Equiv.ofInjective f hf).cardinal_eq.symm⟩,
     fun ⟨_, e⟩ => e ▸ ⟨⟨Subtype.val, fun _ _ => Subtype.eq⟩⟩⟩
 #align cardinal.le_mk_iff_exists_set Cardinal.le_mk_iff_exists_set
 
-theorem mk_subtype_le {α : Type u} (p : α → Prop) : (#Subtype p) ≤ (#α) :=
+theorem mk_subtype_le {α : Type u} (p : α → Prop) : #(Subtype p) ≤ #α :=
   ⟨Embedding.subtype p⟩
 #align cardinal.mk_subtype_le Cardinal.mk_subtype_le
 
-theorem mk_set_le (s : Set α) : (#s) ≤ (#α) :=
+theorem mk_set_le (s : Set α) : #s ≤ #α :=
   mk_subtype_le s
 #align cardinal.mk_set_le Cardinal.mk_set_le
 
@@ -303,7 +303,7 @@ theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.ou
 #align cardinal.out_embedding Cardinal.out_embedding
 
 theorem lift_mk_le {α : Type v} {β : Type w} :
-    lift.{max u w} (#α) ≤ lift.{max u v} (#β) ↔ Nonempty (α ↪ β) :=
+    lift.{max u w} #α ≤ lift.{max u v} #β ↔ Nonempty (α ↪ β) :=
   ⟨fun ⟨f⟩ => ⟨Embedding.congr Equiv.ulift Equiv.ulift f⟩, fun ⟨f⟩ =>
     ⟨Embedding.congr Equiv.ulift.symm Equiv.ulift.symm f⟩⟩
 #align cardinal.lift_mk_le Cardinal.lift_mk_le
@@ -312,12 +312,12 @@ theorem lift_mk_le {α : Type v} {β : Type w} :
 Because Lean often can not realize it should use this specialization itself,
 we provide this statement separately so you don't have to solve the specialization problem either.
 -/
-theorem lift_mk_le' {α : Type u} {β : Type v} : lift.{v} (#α) ≤ lift.{u} (#β) ↔ Nonempty (α ↪ β) :=
+theorem lift_mk_le' {α : Type u} {β : Type v} : lift.{v} #α ≤ lift.{u} #β ↔ Nonempty (α ↪ β) :=
   lift_mk_le.{0}
 #align cardinal.lift_mk_le' Cardinal.lift_mk_le'
 
 theorem lift_mk_eq {α : Type u} {β : Type v} :
-    lift.{max v w} (#α) = lift.{max u w} (#β) ↔ Nonempty (α ≃ β) :=
+    lift.{max v w} #α = lift.{max u w} #β ↔ Nonempty (α ≃ β) :=
   Quotient.eq'.trans
     ⟨fun ⟨f⟩ => ⟨Equiv.ulift.symm.trans <| f.trans Equiv.ulift⟩, fun ⟨f⟩ =>
       ⟨Equiv.ulift.trans <| f.trans Equiv.ulift.symm⟩⟩
@@ -327,7 +327,7 @@ theorem lift_mk_eq {α : Type u} {β : Type v} :
 Because Lean often can not realize it should use this specialization itself,
 we provide this statement separately so you don't have to solve the specialization problem either.
 -/
-theorem lift_mk_eq' {α : Type u} {β : Type v} : lift.{v} (#α) = lift.{u} (#β) ↔ Nonempty (α ≃ β) :=
+theorem lift_mk_eq' {α : Type u} {β : Type v} : lift.{v} #α = lift.{u} #β ↔ Nonempty (α ≃ β) :=
   lift_mk_eq.{u, v, 0}
 #align cardinal.lift_mk_eq' Cardinal.lift_mk_eq'
 
@@ -372,7 +372,7 @@ instance : Zero Cardinal.{u} :=
 instance : Inhabited Cardinal.{u} :=
   ⟨0⟩
 
-theorem mk_eq_zero (α : Type u) [IsEmpty α] : (#α) = 0 :=
+theorem mk_eq_zero (α : Type u) [IsEmpty α] : #α = 0 :=
   (Equiv.equivPEmpty α).cardinal_eq
 #align cardinal.mk_eq_zero Cardinal.mk_eq_zero
 
@@ -386,19 +386,19 @@ theorem lift_eq_zero {a : Cardinal.{v}} : lift.{u} a = 0 ↔ a = 0 :=
   lift_injective.eq_iff' lift_zero
 #align cardinal.lift_eq_zero Cardinal.lift_eq_zero
 
-theorem mk_eq_zero_iff {α : Type u} : (#α) = 0 ↔ IsEmpty α :=
+theorem mk_eq_zero_iff {α : Type u} : #α = 0 ↔ IsEmpty α :=
   ⟨fun e =>
     let ⟨h⟩ := Quotient.exact e
     h.isEmpty,
     @mk_eq_zero α⟩
 #align cardinal.mk_eq_zero_iff Cardinal.mk_eq_zero_iff
 
-theorem mk_ne_zero_iff {α : Type u} : (#α) ≠ 0 ↔ Nonempty α :=
+theorem mk_ne_zero_iff {α : Type u} : #α ≠ 0 ↔ Nonempty α :=
   (not_iff_not.2 mk_eq_zero_iff).trans not_isEmpty_iff
 #align cardinal.mk_ne_zero_iff Cardinal.mk_ne_zero_iff
 
 @[simp]
-theorem mk_ne_zero (α : Type u) [Nonempty α] : (#α) ≠ 0 :=
+theorem mk_ne_zero (α : Type u) [Nonempty α] : #α ≠ 0 :=
   mk_ne_zero_iff.2 ‹_›
 #align cardinal.mk_ne_zero Cardinal.mk_ne_zero
 
@@ -408,17 +408,17 @@ instance : One Cardinal.{u} :=
 instance : Nontrivial Cardinal.{u} :=
   ⟨⟨1, 0, mk_ne_zero _⟩⟩
 
-theorem mk_eq_one (α : Type u) [Unique α] : (#α) = 1 :=
+theorem mk_eq_one (α : Type u) [Unique α] : #α = 1 :=
   (Equiv.equivPUnit α).cardinal_eq
 #align cardinal.mk_eq_one Cardinal.mk_eq_one
 
-theorem le_one_iff_subsingleton {α : Type u} : (#α) ≤ 1 ↔ Subsingleton α :=
+theorem le_one_iff_subsingleton {α : Type u} : #α ≤ 1 ↔ Subsingleton α :=
   ⟨fun ⟨f⟩ => ⟨fun _ _ => f.injective (Subsingleton.elim _ _)⟩, fun ⟨h⟩ =>
     ⟨⟨fun _ => PUnit.unit, fun _ _ _ => h _ _⟩⟩⟩
 #align cardinal.le_one_iff_subsingleton Cardinal.le_one_iff_subsingleton
 
 @[simp]
-theorem mk_le_one_iff_set_subsingleton {s : Set α} : (#s) ≤ 1 ↔ s.Subsingleton :=
+theorem mk_le_one_iff_set_subsingleton {s : Set α} : #s ≤ 1 ↔ s.Subsingleton :=
   le_one_iff_subsingleton.trans s.subsingleton_coe
 #align cardinal.mk_le_one_iff_set_subsingleton Cardinal.mk_le_one_iff_set_subsingleton
 
@@ -428,32 +428,32 @@ alias mk_le_one_iff_set_subsingleton ↔ _ _root_.Set.Subsingleton.cardinal_mk_l
 instance : Add Cardinal.{u} :=
   ⟨map₂ Sum fun _ _ _ _ => Equiv.sumCongr⟩
 
-theorem add_def (α β : Type u) : (#α) + (#β) = (#Sum α β) :=
+theorem add_def (α β : Type u) : #α + #β = #(Sum α β) :=
   rfl
 #align cardinal.add_def Cardinal.add_def
 
 -- Porting note: Should this be changed to
--- `⟨fun n => lift (#(Fin n))⟩` in the future?
+-- `⟨fun n => lift #(Fin n)⟩` in the future?
 instance : NatCast Cardinal.{u} :=
   ⟨Nat.unaryCast⟩
 
 @[simp]
-theorem mk_sum (α : Type u) (β : Type v) : (#Sum α β) = lift.{v, u} (#α) + lift.{u, v} (#β) :=
+theorem mk_sum (α : Type u) (β : Type v) : #(α ⊕ β) = lift.{v, u} #α + lift.{u, v} #β :=
   mk_congr (Equiv.ulift.symm.sumCongr Equiv.ulift.symm)
 #align cardinal.mk_sum Cardinal.mk_sum
 
 @[simp]
-theorem mk_option {α : Type u} : (#Option α) = (#α) + 1 :=
+theorem mk_option {α : Type u} : #(Option α) = #α + 1 :=
   (Equiv.optionEquivSumPUnit α).cardinal_eq
 #align cardinal.mk_option Cardinal.mk_option
 
 @[simp]
-theorem mk_psum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
+theorem mk_psum (α : Type u) (β : Type v) : #(PSum α β) = lift.{v} #α + lift.{u} #β :=
   (mk_congr (Equiv.psumEquivSum α β)).trans (mk_sum α β)
 #align cardinal.mk_psum Cardinal.mk_psum
 
 @[simp]
-theorem mk_fintype (α : Type u) [h : Fintype α] : (#α) = Fintype.card α := by
+theorem mk_fintype (α : Type u) [h : Fintype α] : #α = Fintype.card α := by
   refine Fintype.induction_empty_option ?_ ?_ ?_ α (h_fintype := h)
   · intro α β h e hα
     letI := Fintype.ofEquiv β e.symm
@@ -467,12 +467,12 @@ theorem mk_fintype (α : Type u) [h : Fintype α] : (#α) = Fintype.card α := b
 instance : Mul Cardinal.{u} :=
   ⟨map₂ Prod fun _ _ _ _ => Equiv.prodCongr⟩
 
-theorem mul_def (α β : Type u) : (#α) * (#β) = (#α × β) :=
+theorem mul_def (α β : Type u) : #α * #β = #(α × β) :=
   rfl
 #align cardinal.mul_def Cardinal.mul_def
 
 @[simp]
-theorem mk_prod (α : Type u) (β : Type v) : (#α × β) = lift.{v, u} (#α) * lift.{u, v} (#β) :=
+theorem mk_prod (α : Type u) (β : Type v) : #(α × β) = lift.{v, u} #α * lift.{u, v} #β :=
   mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)
 #align cardinal.mk_prod Cardinal.mk_prod
 
@@ -492,11 +492,11 @@ local infixr:0 "^'" => @HPow.hPow Cardinal Cardinal Cardinal.instPowCardinal
 -- -- mathport name: cardinal.pow.nat
 local infixr:80 " ^ℕ " => @HPow.hPow Cardinal ℕ Cardinal instHPow
 
-theorem power_def (α β) : ((#α) ^ (#β)) = (#β → α) :=
+theorem power_def (α β) : #α ^ #β = #(β → α) :=
   rfl
 #align cardinal.power_def Cardinal.power_def
 
-theorem mk_arrow (α : Type u) (β : Type v) : (#α → β) = (lift.{u} (#β)^lift.{v} (#α)) :=
+theorem mk_arrow (α : Type u) (β : Type v) : #(α → β) = (lift.{u} #β^lift.{v} #α) :=
   mk_congr (Equiv.ulift.symm.arrowCongr Equiv.ulift.symm)
 #align cardinal.mk_arrow Cardinal.mk_arrow
 
@@ -564,12 +564,12 @@ theorem one_power {a : Cardinal} : (1 ^ a) = 1 :=
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_bool : (#Bool) = 2 := by simp
+theorem mk_bool : #Bool = 2 := by simp
 #align cardinal.mk_bool Cardinal.mk_bool
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_Prop : (#Prop) = 2 := by simp
+theorem mk_Prop : #Prop = 2 := by simp
 #align cardinal.mk_Prop Cardinal.mk_Prop
 
 @[simp]
@@ -638,12 +638,12 @@ theorem lift_two : lift.{u, v} 2 = 2 := by simp [←one_add_one_eq_two]
 #align cardinal.lift_two Cardinal.lift_two
 
 @[simp]
-theorem mk_set {α : Type u} : (#Set α) = (2 ^ (#α)) := by simp [←one_add_one_eq_two, Set, mk_arrow]
+theorem mk_set {α : Type u} : #(Set α) = (2 ^ #α) := by simp [←one_add_one_eq_two, Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
 
 /-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/
 @[simp]
-theorem mk_powerset {α : Type u} (s : Set α) : (#↥(𝒫 s)) = (2 ^ (#↥s)) :=
+theorem mk_powerset {α : Type u} (s : Set α) : #(↥(𝒫 s)) = (2 ^ #(↥s)) :=
   (mk_congr (Equiv.Set.powerset s)).trans mk_set
 #align cardinal.mk_powerset Cardinal.mk_powerset
 
@@ -682,7 +682,7 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardina
         have : Sum α ((range f)ᶜ : Set β) ≃ β :=
           (Equiv.sumCongr (Equiv.ofInjective f hf) (Equiv.refl _)).trans <|
             Equiv.Set.sumCompl (range f)
-        ⟨#↥(range f)ᶜ, mk_congr this.symm⟩
+        ⟨#(↥(range f)ᶜ), mk_congr this.symm⟩
     le_self_add := fun a b => (add_zero a).ge.trans <| add_le_add_left (Cardinal.zero_le _) _
     eq_zero_or_eq_zero_of_mul_eq_zero := fun {a b} =>
       inductionOn₂ a b fun α β => by
@@ -744,7 +744,7 @@ instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := inferInstance
 
-theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
+theorem one_lt_iff_nontrivial {α : Type u} : 1 < #α ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
 
@@ -773,7 +773,7 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       obtain ⟨⟨c : Cardinal, hc : ¬Acc (· < ·) c⟩, ⟨h_1 : ∀ j, (f ⟨c, hc⟩).out ↪ (f j).out⟩⟩ :=
         Embedding.min_injective fun i => (f i).out
       refine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_)
-      have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
+      have : #_ ≤ #_ := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
 
@@ -826,8 +826,8 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
   simp only [Surjective, not_forall] at this
   rcases this with ⟨b, hb⟩
   calc
-    (#γ) + 1 = (#Option γ) := mk_option.symm
-    _ ≤ (#β) := (f.optionElim b hb).cardinal_le
+    #γ + 1 = #(Option γ) := mk_option.symm
+    _ ≤ #β := (f.optionElim b hb).cardinal_le
 #align cardinal.add_one_le_succ Cardinal.add_one_le_succ
 
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
@@ -866,21 +866,21 @@ theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
 #align cardinal.le_sum Cardinal.le_sum
 
 @[simp]
-theorem mk_sigma {ι} (f : ι → Type _) : (#Σi, f i) = sum fun i => #f i :=
+theorem mk_sigma {ι} (f : ι → Type _) : #(Σ i, f i) = sum fun i => #(f i) :=
   mk_congr <| Equiv.sigmaCongrRight fun _ => outMkEquiv.symm
 #align cardinal.mk_sigma Cardinal.mk_sigma
 
 @[simp]
 theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
-    (sum fun _ : ι => a) = lift.{v} (#ι) * lift.{u} a :=
+    (sum fun _ : ι => a) = lift.{v} #ι * lift.{u} a :=
   inductionOn a fun α =>
     mk_congr <|
       calc
-        (Σ _ : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
+        (Σ _ : ι, Quotient.out #α) ≃ ι × Quotient.out #α := Equiv.sigmaEquivProd _ _
         _ ≃ ULift ι × ULift α := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)
 #align cardinal.sum_const Cardinal.sum_const
 
-theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = (#ι) * a := by simp
+theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = #ι * a := by simp
 #align cardinal.sum_const' Cardinal.sum_const'
 
 @[simp]
@@ -911,14 +911,14 @@ theorem sum_le_sum {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : sum f
       Classical.choice <| by have := H i; rwa [← Quot.out_eq (f i), ← Quot.out_eq (g i)] at this⟩
 #align cardinal.sum_le_sum Cardinal.sum_le_sum
 
-theorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : α → β) (hf : ∀ b : β, (#f ⁻¹' {b}) ≤ c) :
-    (#α) ≤ (#β) * c := by
+theorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : α → β) (hf : ∀ b : β, #(f ⁻¹' {b}) ≤ c) :
+    #α ≤ #β * c := by
   simpa only [← mk_congr (@Equiv.sigmaFiberEquiv α β f), mk_sigma, ← sum_const'] using
     sum_le_sum _ _ hf
 #align cardinal.mk_le_mk_mul_of_mk_preimage_le Cardinal.mk_le_mk_mul_of_mk_preimage_le
 
 theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v} {c : Cardinal}
-    (f : α → β) (hf : ∀ b : β, lift.{v} (#f ⁻¹' {b}) ≤ c) : lift.{v} (#α) ≤ lift.{u} (#β) * c :=
+    (f : α → β) (hf : ∀ b : β, lift.{v} #(f ⁻¹' {b}) ≤ c) : lift.{v} #α ≤ lift.{u} #β * c :=
   (mk_le_mk_mul_of_mk_preimage_le fun x : ULift.{v} α => ULift.up.{u} (f x.1)) <|
     ULift.forall.2 fun b =>
       (mk_congr <|
@@ -947,7 +947,7 @@ theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove
 
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by
   rw [← mk_out a]
-  apply @small_of_surjective (Set a.out) (Iic (#a.out)) _ fun x => ⟨#x, mk_set_le x⟩
+  apply @small_of_surjective (Set a.out) (Iic #a.out) _ fun x => ⟨#x, mk_set_le x⟩
   rintro ⟨x, hx⟩
   simpa using le_mk_iff_exists_set.1 hx
 
@@ -991,13 +991,13 @@ theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
 
 -- Porting note: Added universe hint .{v,_} below
 theorem sum_le_iSup_lift {ι : Type u}
-    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} (#ι) * iSup f := by
+    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} #ι * iSup f := by
   rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
   exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)
 #align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift
 
-theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f := by
-  rw [← lift_id (#ι)]
+theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ #ι * iSup f := by
+  rw [← lift_id #ι]
   exact sum_le_iSup_lift f
 #align cardinal.sum_le_supr Cardinal.sum_le_iSup
 
@@ -1017,42 +1017,42 @@ protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f
 -- Portin note: simpNF is not happy with universe levels.
 @[simp, nolint simpNF]
 theorem lift_mk_shrink (α : Type u) [Small.{v} α] :
-    Cardinal.lift.{max u w} (#Shrink.{v} α) = Cardinal.lift.{max v w} (#α) :=
+    Cardinal.lift.{max u w} #(Shrink.{v} α) = Cardinal.lift.{max v w} #α :=
 -- Porting note: Added .{v,u,w} universe hint below
   lift_mk_eq.{v,u,w}.2 ⟨(equivShrink α).symm⟩
 #align cardinal.lift_mk_shrink Cardinal.lift_mk_shrink
 
 @[simp]
 theorem lift_mk_shrink' (α : Type u) [Small.{v} α] :
-    Cardinal.lift.{u} (#Shrink.{v} α) = Cardinal.lift.{v} (#α) :=
+    Cardinal.lift.{u} #(Shrink.{v} α) = Cardinal.lift.{v} #α :=
   lift_mk_shrink.{u, v, 0} α
 #align cardinal.lift_mk_shrink' Cardinal.lift_mk_shrink'
 
 @[simp]
 theorem lift_mk_shrink'' (α : Type max u v) [Small.{v} α] :
-    Cardinal.lift.{u} (#Shrink.{v} α) = (#α) := by
+    Cardinal.lift.{u} #(Shrink.{v} α) = #α := by
   rw [← lift_umax', lift_mk_shrink.{max u v, v, 0} α, ← lift_umax, lift_id]
 #align cardinal.lift_mk_shrink'' Cardinal.lift_mk_shrink''
 
 /-- The indexed product of cardinals is the cardinality of the Pi type
   (dependent product). -/
 def prod {ι : Type u} (f : ι → Cardinal) : Cardinal :=
-  #∀ i, (f i).out
+  #(∀ i, (f i).out)
 #align cardinal.prod Cardinal.prod
 
 @[simp]
-theorem mk_pi {ι : Type u} (α : ι → Type v) : (#∀ i, α i) = prod fun i => #α i :=
+theorem mk_pi {ι : Type u} (α : ι → Type v) : #(∀ i, α i) = prod fun i => #(α i) :=
   mk_congr <| Equiv.piCongrRight fun _ => outMkEquiv.symm
 #align cardinal.mk_pi Cardinal.mk_pi
 
 @[simp]
 theorem prod_const (ι : Type u) (a : Cardinal.{v}) :
-    (prod fun _ : ι => a) = (lift.{u} a^lift.{v} (#ι)) :=
+    (prod fun _ : ι => a) = (lift.{u} a^lift.{v} #ι) :=
   inductionOn a fun _ =>
     mk_congr <| Equiv.piCongr Equiv.ulift.symm fun _ => outMkEquiv.trans Equiv.ulift.symm
 #align cardinal.prod_const Cardinal.prod_const
 
-theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) = (a^(#ι)) :=
+theorem prod_const' (ι : Type u) (a : Cardinal.{u}) : (prod fun _ : ι => a) = (a^#ι) :=
   inductionOn a fun _ => (mk_pi _).symm
 #align cardinal.prod_const' Cardinal.prod_const'
 
@@ -1113,9 +1113,9 @@ theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u,v} (iInf f) = ⨅ i, lift
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b :=
   inductionOn₂ a b fun α β => by
-    rw [← lift_id (#β), ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}]
+    rw [← lift_id #β, ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}]
     exact fun ⟨f⟩ =>
-      ⟨#Set.range f,
+      ⟨#(Set.range f),
         Eq.symm <| lift_mk_eq.{_, _, v}.2
           ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)
               fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩
@@ -1229,14 +1229,14 @@ theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardin
 
 /-- `ℵ₀` is the smallest infinite cardinal. -/
 def aleph0 : Cardinal.{u} :=
-  lift (#ℕ)
+  lift #ℕ
 #align cardinal.aleph_0 Cardinal.aleph0
 
 -- mathport name: cardinal.aleph_0
 @[inherit_doc]
 scoped notation "ℵ₀" => Cardinal.aleph0
 
-theorem mk_nat : (#ℕ) = ℵ₀ :=
+theorem mk_nat : #ℕ = ℵ₀ :=
   (lift_id _).symm
 #align cardinal.mk_nat Cardinal.mk_nat
 
@@ -1277,7 +1277,7 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_fin (n : ℕ) : (#Fin n) = n := by simp
+theorem mk_fin (n : ℕ) : #(Fin n) = n := by simp
 #align cardinal.mk_fin Cardinal.mk_fin
 
 @[simp]
@@ -1315,27 +1315,27 @@ theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} : n < lift.{v} a ↔ n < a
   rw [← lift_natCast.{v,u}, lift_lt]
 #align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
 
-theorem lift_mk_fin (n : ℕ) : lift (#Fin n) = n := by simp
+theorem lift_mk_fin (n : ℕ) : lift #(Fin n) = n := by simp
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
 
-theorem mk_coe_finset {α : Type u} {s : Finset α} : (#s) = ↑(Finset.card s) := by simp
+theorem mk_coe_finset {α : Type u} {s : Finset α} : #s = ↑(Finset.card s) := by simp
 #align cardinal.mk_coe_finset Cardinal.mk_coe_finset
 
-theorem mk_finset_of_fintype [Fintype α] : (#Finset α) = 2 ^ℕ Fintype.card α := by
+theorem mk_finset_of_fintype [Fintype α] : #(Finset α) = 2 ^ℕ Fintype.card α := by
   simp [Pow.pow]
 #align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype
 
 @[simp]
 theorem mk_finsupp_lift_of_fintype (α : Type u) (β : Type v) [Fintype α] [Zero β] :
-    (#α →₀ β) = lift.{u} (#β) ^ℕ Fintype.card α := by
+    #(α →₀ β) = lift.{u} #β ^ℕ Fintype.card α := by
   simpa using (@Finsupp.equivFunOnFinite α β _ _).cardinal_eq
 #align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintype
 
 theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
-    (#α →₀ β) = (#β) ^ℕ Fintype.card α := by simp
+    #(α →₀ β) = #β ^ℕ Fintype.card α := by simp
 #align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype
 
-theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ (#α) :=
+theorem card_le_of_finset {α} (s : Finset α) : (s.card : Cardinal) ≤ #α :=
   @mk_coe_finset _ s ▸ mk_set_le _
 #align cardinal.card_le_of_finset Cardinal.card_le_of_finset
 
@@ -1383,7 +1383,7 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n :=
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
 
-theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n := by
+theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : #α ≤ n := by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
   rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn
   cases' hn with f
@@ -1463,25 +1463,25 @@ theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ :=
   ext fun x => by simp only [mem_Iio, mem_range, eq_comm, lt_aleph0]
 #align cardinal.range_nat_cast Cardinal.range_natCast
 
-theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : (#α) = n ↔ Nonempty (α ≃ Fin n) := by
-  rw [← lift_mk_fin, ← lift_uzero (#α), lift_mk_eq']
+theorem mk_eq_nat_iff {α : Type u} {n : ℕ} : #α = n ↔ Nonempty (α ≃ Fin n) := by
+  rw [← lift_mk_fin, ← lift_uzero #α, lift_mk_eq']
 #align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff
 
-theorem lt_aleph0_iff_finite {α : Type u} : (#α) < ℵ₀ ↔ Finite α := by
+theorem lt_aleph0_iff_finite {α : Type u} : #α < ℵ₀ ↔ Finite α := by
   simp only [lt_aleph0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]
 #align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite
 
-theorem lt_aleph0_iff_fintype {α : Type u} : (#α) < ℵ₀ ↔ Nonempty (Fintype α) :=
+theorem lt_aleph0_iff_fintype {α : Type u} : #α < ℵ₀ ↔ Nonempty (Fintype α) :=
   lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)
 #align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype
 
-theorem lt_aleph0_of_finite (α : Type u) [Finite α] : (#α) < ℵ₀ :=
+theorem lt_aleph0_of_finite (α : Type u) [Finite α] : #α < ℵ₀ :=
   lt_aleph0_iff_finite.2 ‹_›
 #align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem lt_aleph0_iff_set_finite {S : Set α} : (#S) < ℵ₀ ↔ S.Finite :=
+theorem lt_aleph0_iff_set_finite {S : Set α} : #S < ℵ₀ ↔ S.Finite :=
   lt_aleph0_iff_finite.trans finite_coe_iff
 #align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite
 
@@ -1489,22 +1489,22 @@ alias lt_aleph0_iff_set_finite ↔ _ _root_.Set.Finite.lt_aleph0
 #align set.finite.lt_aleph_0 Set.Finite.lt_aleph0
 
 @[simp]
-theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : (#{ x // p x }) < ℵ₀ ↔ { x | p x }.Finite :=
+theorem lt_aleph0_iff_subtype_finite {p : α → Prop} : #{ x // p x } < ℵ₀ ↔ { x | p x }.Finite :=
   lt_aleph0_iff_set_finite
 #align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite
 
-theorem mk_le_aleph0_iff : (#α) ≤ ℵ₀ ↔ Countable α := by
-  rw [countable_iff_nonempty_embedding, aleph0, ← lift_uzero (#α), lift_mk_le']
+theorem mk_le_aleph0_iff : #α ≤ ℵ₀ ↔ Countable α := by
+  rw [countable_iff_nonempty_embedding, aleph0, ← lift_uzero #α, lift_mk_le']
 #align cardinal.mk_le_aleph_0_iff Cardinal.mk_le_aleph0_iff
 
 @[simp]
-theorem mk_le_aleph0 [Countable α] : (#α) ≤ ℵ₀ :=
+theorem mk_le_aleph0 [Countable α] : #α ≤ ℵ₀ :=
   mk_le_aleph0_iff.mpr ‹_›
 #align cardinal.mk_le_aleph_0 Cardinal.mk_le_aleph0
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem le_aleph0_iff_set_countable {s : Set α} : (#s) ≤ ℵ₀ ↔ s.Countable := by
+theorem le_aleph0_iff_set_countable {s : Set α} : #s ≤ ℵ₀ ↔ s.Countable := by
   rw [mk_le_aleph0_iff, countable_coe_iff]
 #align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable
 
@@ -1513,7 +1513,7 @@ alias le_aleph0_iff_set_countable ↔ _ _root_.Set.Countable.le_aleph0
 
 @[simp]
 theorem le_aleph0_iff_subtype_countable {p : α → Prop} :
-    (#{ x // p x }) ≤ ℵ₀ ↔ { x | p x }.Countable :=
+    #{ x // p x } ≤ ℵ₀ ↔ { x | p x }.Countable :=
   le_aleph0_iff_set_countable
 #align cardinal.le_aleph_0_iff_subtype_countable Cardinal.le_aleph0_iff_subtype_countable
 
@@ -1597,28 +1597,28 @@ theorem power_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : (
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← natCast_pow]; apply nat_lt_aleph0
 #align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0
 
-theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
+theorem eq_one_iff_unique {α : Type _} : #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
   calc
-    (#α) = 1 ↔ (#α) ≤ 1 ∧ 1 ≤ (#α) := le_antisymm_iff
+    #α = 1 ↔ #α ≤ 1 ∧ 1 ≤ #α := le_antisymm_iff
     _ ↔ Subsingleton α ∧ Nonempty α :=
       le_one_iff_subsingleton.and (one_le_iff_ne_zero.trans mk_ne_zero_iff)
 #align cardinal.eq_one_iff_unique Cardinal.eq_one_iff_unique
 
-theorem infinite_iff {α : Type u} : Infinite α ↔ ℵ₀ ≤ (#α) := by
+theorem infinite_iff {α : Type u} : Infinite α ↔ ℵ₀ ≤ #α := by
   rw [← not_lt, lt_aleph0_iff_finite, not_finite_iff_infinite]
 #align cardinal.infinite_iff Cardinal.infinite_iff
 
 @[simp]
-theorem aleph0_le_mk (α : Type u) [Infinite α] : ℵ₀ ≤ (#α) :=
+theorem aleph0_le_mk (α : Type u) [Infinite α] : ℵ₀ ≤ #α :=
   infinite_iff.1 ‹_›
 #align cardinal.aleph_0_le_mk Cardinal.aleph0_le_mk
 
 @[simp]
-theorem mk_eq_aleph0 (α : Type _) [Countable α] [Infinite α] : (#α) = ℵ₀ :=
+theorem mk_eq_aleph0 (α : Type _) [Countable α] [Infinite α] : #α = ℵ₀ :=
   mk_le_aleph0.antisymm <| aleph0_le_mk _
 #align cardinal.mk_eq_aleph_0 Cardinal.mk_eq_aleph0
 
-theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ (#α) = ℵ₀ :=
+theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ #α = ℵ₀ :=
   ⟨fun ⟨h⟩ => mk_congr ((@Denumerable.eqv α h).trans Equiv.ulift.symm), fun h => by
     cases' Quotient.exact h with f
     exact ⟨Denumerable.mk' <| f.trans Equiv.ulift⟩⟩
@@ -1626,7 +1626,7 @@ theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ (#α) = 
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_denumerable (α : Type u) [Denumerable α] : (#α) = ℵ₀ :=
+theorem mk_denumerable (α : Type u) [Denumerable α] : #α = ℵ₀ :=
   denumerable_iff.1 ⟨‹_›⟩
 #align cardinal.mk_denumerable Cardinal.mk_denumerable
 
@@ -1743,7 +1743,7 @@ theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m,
 #align cardinal.exists_nat_eq_of_le_nat Cardinal.exists_nat_eq_of_le_nat
 
 @[simp]
-theorem mk_toNat_of_infinite [h : Infinite α] : toNat (#α) = 0 :=
+theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 :=
   dif_neg (infinite_iff.1 h).not_lt
 #align cardinal.mk_to_nat_of_infinite Cardinal.mk_toNat_of_infinite
 
@@ -1752,7 +1752,7 @@ theorem aleph0_toNat : toNat ℵ₀ = 0 :=
   toNat_apply_of_aleph0_le le_rfl
 #align cardinal.aleph_0_to_nat Cardinal.aleph0_toNat
 
-theorem mk_toNat_eq_card [Fintype α] : toNat (#α) = Fintype.card α := by simp
+theorem mk_toNat_eq_card [Fintype α] : toNat #α = Fintype.card α := by simp
 #align cardinal.mk_to_nat_eq_card Cardinal.mk_toNat_eq_card
 
 -- Porting note : simp can prove this
@@ -1777,7 +1777,7 @@ theorem toNat_eq_one {c : Cardinal} : toNat c = 1 ↔ c = 1 := by
   rw [toNat_eq_iff one_ne_zero, Nat.cast_one]
 #align cardinal.to_nat_eq_one Cardinal.toNat_eq_one
 
-theorem toNat_eq_one_iff_unique {α : Type _} : toNat (#α) = 1 ↔ Subsingleton α ∧ Nonempty α :=
+theorem toNat_eq_one_iff_unique {α : Type _} : toNat #α = 1 ↔ Subsingleton α ∧ Nonempty α :=
   toNat_eq_one.trans eq_one_iff_unique
 #align cardinal.to_nat_eq_one_iff_unique Cardinal.toNat_eq_one_iff_unique
 
@@ -1791,7 +1791,7 @@ theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by
     rwa [aleph0_le_lift]
 #align cardinal.to_nat_lift Cardinal.toNat_lift
 
-theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat (#α) = toNat (#β) := by
+theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat #α = toNat #β := by
   -- Porting note: Inserted universe hint below
   rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]
 #align cardinal.to_nat_congr Cardinal.toNat_congr
@@ -1875,7 +1875,7 @@ theorem toPartENat_cast (n : ℕ) : toPartENat n = n := by
 #align cardinal.to_part_enat_cast Cardinal.toPartENat_cast
 
 @[simp]
-theorem mk_toPartENat_of_infinite [h : Infinite α] : toPartENat (#α) = ⊤ :=
+theorem mk_toPartENat_of_infinite [h : Infinite α] : toPartENat #α = ⊤ :=
   toPartENat_apply_of_aleph0_le (infinite_iff.1 h)
 #align cardinal.mk_to_part_enat_of_infinite Cardinal.mk_toPartENat_of_infinite
 
@@ -1963,18 +1963,18 @@ theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPart
     exact hc
 #align cardinal.to_part_enat_lift Cardinal.toPartENat_lift
 
-theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat (#α) = toPartENat (#β) := by
+theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat #α = toPartENat #β := by
   rw [← toPartENat_lift, lift_mk_eq.{_, _,v}.mpr ⟨e⟩, toPartENat_lift]
 #align cardinal.to_part_enat_congr Cardinal.toPartENat_congr
 
-theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat (#α) = Fintype.card α := by simp
+theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat #α = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card
 
-theorem mk_int : (#ℤ) = ℵ₀ :=
+theorem mk_int : #ℤ = ℵ₀ :=
   mk_denumerable ℤ
 #align cardinal.mk_int Cardinal.mk_int
 
-theorem mk_pNat : (#ℕ+) = ℵ₀ :=
+theorem mk_pNat : #ℕ+ = ℵ₀ :=
   mk_denumerable ℕ+
 #align cardinal.mk_pnat Cardinal.mk_pNat
 
@@ -2000,75 +2000,75 @@ theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_empty : (#Empty) = 0 :=
+theorem mk_empty : #Empty = 0 :=
   mk_eq_zero _
 #align cardinal.mk_empty Cardinal.mk_empty
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_pempty : (#PEmpty) = 0 :=
+theorem mk_pempty : #PEmpty = 0 :=
   mk_eq_zero _
 #align cardinal.mk_pempty Cardinal.mk_pempty
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_punit : (#PUnit) = 1 :=
+theorem mk_punit : #PUnit = 1 :=
   mk_eq_one PUnit
 #align cardinal.mk_punit Cardinal.mk_punit
 
-theorem mk_unit : (#Unit) = 1 :=
+theorem mk_unit : #Unit = 1 :=
   mk_punit
 #align cardinal.mk_unit Cardinal.mk_unit
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_singleton {α : Type u} (x : α) : (#({x} : Set α)) = 1 :=
+theorem mk_singleton {α : Type u} (x : α) : #({x} : Set α) = 1 :=
   mk_eq_one _
 #align cardinal.mk_singleton Cardinal.mk_singleton
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_plift_true : (#PLift True) = 1 :=
+theorem mk_plift_true : #(PLift True) = 1 :=
   mk_eq_one _
 #align cardinal.mk_plift_true Cardinal.mk_plift_true
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_plift_false : (#PLift False) = 0 :=
+theorem mk_plift_false : #(PLift False) = 0 :=
   mk_eq_zero _
 #align cardinal.mk_plift_false Cardinal.mk_plift_false
 
 @[simp]
-theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
+theorem mk_vector (α : Type u) (n : ℕ) : #(Vector α n) = #α ^ℕ n :=
   (mk_congr (Equiv.vectorEquivFin α n)).trans <| by simp
 #align cardinal.mk_vector Cardinal.mk_vector
 
-theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α) ^ℕ n :=
+theorem mk_list_eq_sum_pow (α : Type u) : #(List α) = sum fun n : ℕ => #α ^ℕ n :=
   calc
-    (#List α) = (#Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
-    _ = sum fun n : ℕ => (#α) ^ℕ n := by simp
+    #(List α) = #(Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
+    _ = sum fun n : ℕ => #α ^ℕ n := by simp
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
 
-theorem mk_quot_le {α : Type u} {r : α → α → Prop} : (#Quot r) ≤ (#α) :=
+theorem mk_quot_le {α : Type u} {r : α → α → Prop} : #(Quot r) ≤ #α :=
   mk_le_of_surjective Quot.exists_rep
 #align cardinal.mk_quot_le Cardinal.mk_quot_le
 
-theorem mk_quotient_le {α : Type u} {s : Setoid α} : (#Quotient s) ≤ (#α) :=
+theorem mk_quotient_le {α : Type u} {s : Setoid α} : #(Quotient s) ≤ #α :=
   mk_quot_le
 #align cardinal.mk_quotient_le Cardinal.mk_quotient_le
 
 theorem mk_subtype_le_of_subset {α : Type u} {p q : α → Prop} (h : ∀ ⦃x⦄, p x → q x) :
-    (#Subtype p) ≤ (#Subtype q) :=
+    #(Subtype p) ≤ #(Subtype q) :=
   ⟨Embedding.subtypeMap (Embedding.refl α) h⟩
 #align cardinal.mk_subtype_le_of_subset Cardinal.mk_subtype_le_of_subset
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_emptyCollection (α : Type u) : (#(∅ : Set α)) = 0 :=
+theorem mk_emptyCollection (α : Type u) : #(∅ : Set α) = 0 :=
   mk_eq_zero _
 #align cardinal.mk_emptyc Cardinal.mk_emptyCollection
 
-theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅ := by
+theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : #s = 0 ↔ s = ∅ := by
   constructor
   · intro h
     rw [mk_eq_zero_iff] at h
@@ -2078,80 +2078,80 @@ theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅
 #align cardinal.mk_emptyc_iff Cardinal.mk_emptyCollection_iff
 
 @[simp]
-theorem mk_univ {α : Type u} : (#@univ α) = (#α) :=
+theorem mk_univ {α : Type u} : #(@univ α) = #α :=
   mk_congr (Equiv.Set.univ α)
 #align cardinal.mk_univ Cardinal.mk_univ
 
-theorem mk_image_le {α β : Type u} {f : α → β} {s : Set α} : (#f '' s) ≤ (#s) :=
+theorem mk_image_le {α β : Type u} {f : α → β} {s : Set α} : #(f '' s) ≤ #s :=
   mk_le_of_surjective surjective_onto_image
 #align cardinal.mk_image_le Cardinal.mk_image_le
 
 theorem mk_image_le_lift {α : Type u} {β : Type v} {f : α → β} {s : Set α} :
-    lift.{u} (#f '' s) ≤ lift.{v} (#s) :=
+    lift.{u} #(f '' s) ≤ lift.{v} #s :=
   lift_mk_le.{0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_image⟩
 #align cardinal.mk_image_le_lift Cardinal.mk_image_le_lift
 
-theorem mk_range_le {α β : Type u} {f : α → β} : (#range f) ≤ (#α) :=
+theorem mk_range_le {α β : Type u} {f : α → β} : #(range f) ≤ #α :=
   mk_le_of_surjective surjective_onto_range
 #align cardinal.mk_range_le Cardinal.mk_range_le
 
 theorem mk_range_le_lift {α : Type u} {β : Type v} {f : α → β} :
-    lift.{u} (#range f) ≤ lift.{v} (#α) :=
+    lift.{u} #(range f) ≤ lift.{v} #α :=
   lift_mk_le.{0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_range⟩
 #align cardinal.mk_range_le_lift Cardinal.mk_range_le_lift
 
-theorem mk_range_eq (f : α → β) (h : Injective f) : (#range f) = (#α) :=
+theorem mk_range_eq (f : α → β) (h : Injective f) : #(range f) = #α :=
   mk_congr (Equiv.ofInjective f h).symm
 #align cardinal.mk_range_eq Cardinal.mk_range_eq
 
 theorem mk_range_eq_of_injective {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
-    lift.{u} (#range f) = lift.{v} (#α) :=
+    lift.{u} #(range f) = lift.{v} #α :=
   lift_mk_eq'.mpr ⟨(Equiv.ofInjective f hf).symm⟩
 #align cardinal.mk_range_eq_of_injective Cardinal.mk_range_eq_of_injective
 
 theorem mk_range_eq_lift {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :
-    lift.{max u w} (#range f) = lift.{max v w} (#α) :=
+    lift.{max u w} #(range f) = lift.{max v w} #α :=
   lift_mk_eq.{v,u,w}.mpr ⟨(Equiv.ofInjective f hf).symm⟩
 #align cardinal.mk_range_eq_lift Cardinal.mk_range_eq_lift
 
-theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injective f) : (#f '' s) = (#s) :=
+theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injective f) : #(f '' s) = #s :=
   mk_congr (Equiv.Set.image f s hf).symm
 #align cardinal.mk_image_eq Cardinal.mk_image_eq
 
-theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
+theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : #(⋃ i, f i) ≤ sum fun i => #(f i) :=
   calc
-    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
-    _ = sum fun i => #f i := mk_sigma _
+    #(⋃ i, f i) ≤ #(Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
+    _ = sum fun i => #(f i) := mk_sigma _
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 
 theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
-    (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
+    (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : #(⋃ i, f i) = sum fun i => #(f i) :=
   calc
-    (#⋃ i, f i) = (#Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
-    _ = sum fun i => #f i := mk_sigma _
+    #(⋃ i, f i) = #(Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
+    _ = sum fun i => #(f i) := mk_sigma _
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
-theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, (#f i) :=
+theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : #(⋃ i, f i) ≤ #ι * ⨆ i, #(f i) :=
   mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
 #align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
-theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, (#s) := by
+theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : #(⋃₀ A) ≤ #A * ⨆ s : A, #s := by
   rw [sUnion_eq_iUnion]
   apply mk_iUnion_le
 #align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
 theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
-    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, (#A x.1) := by
+    #(⋃ x ∈ s, A x) ≤ #s * ⨆ x : s, #(A x.1) := by
   rw [biUnion_eq_iUnion]
   apply mk_iUnion_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
-theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
+theorem finset_card_lt_aleph0 (s : Finset α) : #(↑s : Set α) < ℵ₀ :=
   lt_aleph0_of_finite _
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
 
 theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
-    (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n := by
+    #s = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n := by
   constructor
   · intro h
     lift s to Finset α using lt_aleph0_iff_set_finite.1 (h.symm ▸ nat_lt_aleph0 n)
@@ -2161,11 +2161,11 @@ theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
 #align cardinal.mk_set_eq_nat_iff_finset Cardinal.mk_set_eq_nat_iff_finset
 
 theorem mk_eq_nat_iff_finset {n : ℕ} :
-    (#α) = n ↔ ∃ t : Finset α, (t : Set α) = univ ∧ t.card = n :=
+    #α = n ↔ ∃ t : Finset α, (t : Set α) = univ ∧ t.card = n :=
   by rw [← mk_univ, mk_set_eq_nat_iff_finset]
 #align cardinal.mk_eq_nat_iff_finset Cardinal.mk_eq_nat_iff_finset
 
-theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fintype.card α h = n := by
+theorem mk_eq_nat_iff_fintype {n : ℕ} : #α = n ↔ ∃ h : Fintype α, @Fintype.card α h = n := by
   rw [mk_eq_nat_iff_finset]
   constructor
   · rintro ⟨t, ht, hn⟩
@@ -2175,81 +2175,81 @@ theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fin
 #align cardinal.mk_eq_nat_iff_fintype Cardinal.mk_eq_nat_iff_fintype
 
 theorem mk_union_add_mk_inter {α : Type u} {S T : Set α} :
-    (#(S ∪ T : Set α)) + (#(S ∩ T : Set α)) = (#S) + (#T) :=
+    #(S ∪ T : Set α) + #(S ∩ T : Set α) = #S + #T :=
   Quot.sound ⟨Equiv.Set.unionSumInter S T⟩
 #align cardinal.mk_union_add_mk_inter Cardinal.mk_union_add_mk_inter
 
 /-- The cardinality of a union is at most the sum of the cardinalities
 of the two sets. -/
-theorem mk_union_le {α : Type u} (S T : Set α) : (#(S ∪ T : Set α)) ≤ (#S) + (#T) :=
-  @mk_union_add_mk_inter α S T ▸ self_le_add_right (#(S ∪ T : Set α)) (#(S ∩ T : Set α))
+theorem mk_union_le {α : Type u} (S T : Set α) : #(S ∪ T : Set α) ≤ #S + #T :=
+  @mk_union_add_mk_inter α S T ▸ self_le_add_right #(S ∪ T : Set α) #(S ∩ T : Set α)
 #align cardinal.mk_union_le Cardinal.mk_union_le
 
 theorem mk_union_of_disjoint {α : Type u} {S T : Set α} (H : Disjoint S T) :
-    (#(S ∪ T : Set α)) = (#S) + (#T) :=
+    #(S ∪ T : Set α) = #S + #T :=
   Quot.sound ⟨Equiv.Set.union H.le_bot⟩
 #align cardinal.mk_union_of_disjoint Cardinal.mk_union_of_disjoint
 
 theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
-    (#(insert a s : Set α)) = (#s) + 1 := by
+    #(insert a s : Set α) = #s + 1 := by
   rw [← union_singleton, mk_union_of_disjoint, mk_singleton]
   simpa
 #align cardinal.mk_insert Cardinal.mk_insert
 
-theorem mk_sum_compl {α} (s : Set α) : (#s) + (#(sᶜ : Set α)) = (#α) :=
+theorem mk_sum_compl {α} (s : Set α) : #s + #(sᶜ : Set α) = #α :=
   mk_congr (Equiv.Set.sumCompl s)
 #align cardinal.mk_sum_compl Cardinal.mk_sum_compl
 
-theorem mk_le_mk_of_subset {α} {s t : Set α} (h : s ⊆ t) : (#s) ≤ (#t) :=
+theorem mk_le_mk_of_subset {α} {s t : Set α} (h : s ⊆ t) : #s ≤ #t :=
   ⟨Set.embeddingOfSubset s t h⟩
 #align cardinal.mk_le_mk_of_subset Cardinal.mk_le_mk_of_subset
 
 theorem mk_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) :
-    (#{ x // p x }) ≤ (#{ x // q x }) :=
+    #{ x // p x } ≤ #{ x // q x } :=
   ⟨embeddingOfSubset _ _ h⟩
 #align cardinal.mk_subtype_mono Cardinal.mk_subtype_mono
 
-theorem le_mk_diff_add_mk (S T : Set α) : (#S) ≤ (#(S \ T : Set α)) + (#T) :=
+theorem le_mk_diff_add_mk (S T : Set α) : #S ≤ #(S \ T : Set α) + #T :=
   (mk_le_mk_of_subset <| subset_diff_union _ _).trans <| mk_union_le _ _
 #align cardinal.le_mk_diff_add_mk Cardinal.le_mk_diff_add_mk
 
-theorem mk_diff_add_mk {S T : Set α} (h : T ⊆ S) : (#(S \ T : Set α)) + (#T) = (#S) := by
+theorem mk_diff_add_mk {S T : Set α} (h : T ⊆ S) : #(S \ T : Set α) + #T = #S := by
   refine (mk_union_of_disjoint <| ?_).symm.trans <| by rw [diff_union_of_subset h]
   exact disjoint_sdiff_self_left
 #align cardinal.mk_diff_add_mk Cardinal.mk_diff_add_mk
 
 theorem mk_union_le_aleph0 {α} {P Q : Set α} :
-    (#(P ∪ Q : Set α)) ≤ ℵ₀ ↔ (#P) ≤ ℵ₀ ∧ (#Q) ≤ ℵ₀ := by
+    #(P ∪ Q : Set α) ≤ ℵ₀ ↔ #P ≤ ℵ₀ ∧ #Q ≤ ℵ₀ := by
   simp only [le_aleph0_iff_subtype_countable, mem_union, setOf_mem_eq, Set.union_def,
     ← countable_union]
 #align cardinal.mk_union_le_aleph_0 Cardinal.mk_union_le_aleph0
 
 theorem mk_image_eq_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α) (h : Injective f) :
-    lift.{u} (#f '' s) = lift.{v} (#s) :=
+    lift.{u} #(f '' s) = lift.{v} #s :=
   lift_mk_eq.{v, u, 0}.mpr ⟨(Equiv.Set.image f s h).symm⟩
 #align cardinal.mk_image_eq_lift Cardinal.mk_image_eq_lift
 
 theorem mk_image_eq_of_injOn_lift {α : Type u} {β : Type v} (f : α → β) (s : Set α)
-    (h : InjOn f s) : lift.{u} (#f '' s) = lift.{v} (#s) :=
+    (h : InjOn f s) : lift.{u} #(f '' s) = lift.{v} #s :=
   lift_mk_eq.{v, u, 0}.mpr ⟨(Equiv.Set.imageOfInjOn f s h).symm⟩
 #align cardinal.mk_image_eq_of_inj_on_lift Cardinal.mk_image_eq_of_injOn_lift
 
 theorem mk_image_eq_of_injOn {α β : Type u} (f : α → β) (s : Set α) (h : InjOn f s) :
-    (#f '' s) = (#s) :=
+    #(f '' s) = #s :=
   mk_congr (Equiv.Set.imageOfInjOn f s h).symm
 #align cardinal.mk_image_eq_of_inj_on Cardinal.mk_image_eq_of_injOn
 
 theorem mk_subtype_of_equiv {α β : Type u} (p : β → Prop) (e : α ≃ β) :
-    (#{ a : α // p (e a) }) = (#{ b : β // p b }) :=
+    #{ a : α // p (e a) } = #{ b : β // p b } :=
   mk_congr (Equiv.subtypeEquivOfSubtype e)
 #align cardinal.mk_subtype_of_equiv Cardinal.mk_subtype_of_equiv
 
-theorem mk_sep (s : Set α) (t : α → Prop) : (#({ x ∈ s | t x } : Set α)) = (#{ x : s | t x.1 }) :=
+theorem mk_sep (s : Set α) (t : α → Prop) : #({ x ∈ s | t x } : Set α) = #{ x : s | t x.1 } :=
   mk_congr (Equiv.Set.sep s t)
 #align cardinal.mk_sep Cardinal.mk_sep
 
 theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
-    (h : Injective f) : lift.{v} (#f ⁻¹' s) ≤ lift.{u} (#s) := by
+    (h : Injective f) : lift.{v} #(f ⁻¹' s) ≤ lift.{u} #s := by
   rw [lift_mk_le.{0}]
   -- Porting note: Needed to insert `mem_preimage.mp` below
   use Subtype.coind (fun x => f x.1) fun x => mem_preimage.mp x.2
@@ -2257,7 +2257,7 @@ theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β
 #align cardinal.mk_preimage_of_injective_lift Cardinal.mk_preimage_of_injective_lift
 
 theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
-    (h : s ⊆ range f) : lift.{u} (#s) ≤ lift.{v} (#f ⁻¹' s) := by
+    (h : s ⊆ range f) : lift.{u} #s ≤ lift.{v} #(f ⁻¹' s) := by
   rw [lift_mk_le.{0}]
   refine' ⟨⟨_, _⟩⟩
   · rintro ⟨y, hy⟩
@@ -2270,29 +2270,29 @@ theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α →
 #align cardinal.mk_preimage_of_subset_range_lift Cardinal.mk_preimage_of_subset_range_lift
 
 theorem mk_preimage_of_injective_of_subset_range_lift {β : Type v} (f : α → β) (s : Set β)
-    (h : Injective f) (h2 : s ⊆ range f) : lift.{v} (#f ⁻¹' s) = lift.{u} (#s) :=
+    (h : Injective f) (h2 : s ⊆ range f) : lift.{v} #(f ⁻¹' s) = lift.{u} #s :=
   le_antisymm (mk_preimage_of_injective_lift f s h) (mk_preimage_of_subset_range_lift f s h2)
 #align cardinal.mk_preimage_of_injective_of_subset_range_lift Cardinal.mk_preimage_of_injective_of_subset_range_lift
 
 theorem mk_preimage_of_injective (f : α → β) (s : Set β) (h : Injective f) :
-    (#f ⁻¹' s) ≤ (#s) := by
-  rw [← lift_id (#↑(f ⁻¹' s)), ← lift_id (#↑(s))]
+    #(f ⁻¹' s) ≤ #s := by
+  rw [← lift_id #(↑(f ⁻¹' s)), ← lift_id #(↑s)]
   exact mk_preimage_of_injective_lift f s h
 #align cardinal.mk_preimage_of_injective Cardinal.mk_preimage_of_injective
 
 theorem mk_preimage_of_subset_range (f : α → β) (s : Set β) (h : s ⊆ range f) :
-    (#s) ≤ (#f ⁻¹' s) := by
-  rw [← lift_id (#↑(f ⁻¹' s)), ← lift_id (#↑(s))]
+    #s ≤ #(f ⁻¹' s) := by
+  rw [← lift_id #(↑(f ⁻¹' s)), ← lift_id #(↑s)]
   exact mk_preimage_of_subset_range_lift f s h
 #align cardinal.mk_preimage_of_subset_range Cardinal.mk_preimage_of_subset_range
 
 theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h : Injective f)
-    (h2 : s ⊆ range f) : (#f ⁻¹' s) = (#s) := by
+    (h2 : s ⊆ range f) : #(f ⁻¹' s) = #s := by
   convert mk_preimage_of_injective_of_subset_range_lift.{u, u} f s h h2 using 1 <;> rw [lift_id]
 #align cardinal.mk_preimage_of_injective_of_subset_range Cardinal.mk_preimage_of_injective_of_subset_range
 
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
-    {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) := by
+    {t : Set β} (h : t ⊆ f '' s) : lift.{u} #t ≤ lift.{v} #({ x ∈ s | f x ∈ t } : Set α) := by
   rw [image_eq_range] at h
   convert mk_preimage_of_subset_range_lift _ _ h using 1
   rw [mk_sep]
@@ -2300,7 +2300,7 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 #align cardinal.mk_subset_ge_of_subset_image_lift Cardinal.mk_subset_ge_of_subset_image_lift
 
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
-    (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) := by
+    #t ≤ #({ x ∈ s | f x ∈ t } : Set α) := by
   rw [image_eq_range] at h
   convert mk_preimage_of_subset_range _ _ h using 1
   rw [mk_sep]
@@ -2308,20 +2308,20 @@ theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (
 #align cardinal.mk_subset_ge_of_subset_image Cardinal.mk_subset_ge_of_subset_image
 
 theorem le_mk_iff_exists_subset {c : Cardinal} {α : Type u} {s : Set α} :
-    c ≤ (#s) ↔ ∃ p : Set α, p ⊆ s ∧ (#p) = c := by
+    c ≤ #s ↔ ∃ p : Set α, p ⊆ s ∧ #p = c := by
   rw [le_mk_iff_exists_set, ← Subtype.exists_set_subtype]
   apply exists_congr; intro t; rw [mk_image_eq]; apply Subtype.val_injective
 #align cardinal.le_mk_iff_exists_subset Cardinal.le_mk_iff_exists_subset
 
-theorem two_le_iff : (2 : Cardinal) ≤ (#α) ↔ ∃ x y : α, x ≠ y := by
+theorem two_le_iff : (2 : Cardinal) ≤ #α ↔ ∃ x y : α, x ≠ y := by
   rw [← Nat.cast_two, nat_succ, succ_le_iff, Nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff]
 #align cardinal.two_le_iff Cardinal.two_le_iff
 
-theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ (#α) ↔ ∃ y : α, y ≠ x := by
+theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ #α ↔ ∃ y : α, y ≠ x := by
   rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
 #align cardinal.two_le_iff' Cardinal.two_le_iff'
 
-theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ := by
+theorem mk_eq_two_iff : #α = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ := by
   simp only [← @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]
   constructor
   · rintro ⟨t, ht, x, y, hne, rfl⟩
@@ -2330,7 +2330,7 @@ theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α
     exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩
 #align cardinal.mk_eq_two_iff Cardinal.mk_eq_two_iff
 
-theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x := by
+theorem mk_eq_two_iff' (x : α) : #α = 2 ↔ ∃! y, y ≠ x := by
   rw [mk_eq_two_iff]; constructor
   · rintro ⟨a, b, hne, h⟩
     simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
@@ -2340,19 +2340,19 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x := by
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'
 
-theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length < (#α)) :
+theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length < #α) :
     ∃ z : α, z ∉ l := by
   contrapose! h
   calc
-    (#α) = (#(Set.univ : Set α)) := mk_univ.symm
-    _ ≤ (#l.toFinset) := mk_le_mk_of_subset fun x _ => List.mem_toFinset.mpr (h x)
+    #α = #(Set.univ : Set α) := mk_univ.symm
+    _ ≤ #l.toFinset := mk_le_mk_of_subset fun x _ => List.mem_toFinset.mpr (h x)
     _ = l.toFinset.card := Cardinal.mk_coe_finset
     _ ≤ l.length := Cardinal.natCast_le.mpr (List.toFinset_card_le l)
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
-theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
-  have : ↑(3 : ℕ) ≤ (#α); simpa using h
-  have : ↑(2 : ℕ) < (#α); rwa [← succ_le_iff, ← Cardinal.nat_succ]
+theorem three_le {α : Type _} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
+  have : ↑(3 : ℕ) ≤ #α; simpa using h
+  have : ↑(2 : ℕ) < #α; rwa [← succ_le_iff, ← Cardinal.nat_succ]
   have := exists_not_mem_of_length_lt [x, y] this
   simpa [not_or] using this
 #align cardinal.three_le Cardinal.three_le

fix: precedences of ⨆⋃⋂⨅ (#5614)

e3c8f983

Diff

@@ -2131,17 +2131,17 @@ theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     _ = sum fun i => #f i := mk_sigma _
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
-theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
+theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, (#f i) :=
   mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
 #align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
-theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
+theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, (#s) := by
   rw [sUnion_eq_iUnion]
   apply mk_iUnion_le
 #align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
 theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
-    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by
+    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, (#A x.1) := by
   rw [biUnion_eq_iUnion]
   apply mk_iUnion_le
 #align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le

fix: change compl precedence (#5586)

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

4d4f0f25

Diff

@@ -679,10 +679,10 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardina
     add_le_add_left := fun a b => add_le_add_left
     exists_add_of_le := fun {a b} =>
       inductionOn₂ a b fun α β ⟨⟨f, hf⟩⟩ =>
-        have : Sum α (range fᶜ : Set β) ≃ β :=
+        have : Sum α ((range f)ᶜ : Set β) ≃ β :=
           (Equiv.sumCongr (Equiv.ofInjective f hf) (Equiv.refl _)).trans <|
             Equiv.Set.sumCompl (range f)
-        ⟨#↥(range fᶜ), mk_congr this.symm⟩
+        ⟨#↥(range f)ᶜ, mk_congr this.symm⟩
     le_self_add := fun a b => (add_zero a).ge.trans <| add_le_add_left (Cardinal.zero_le _) _
     eq_zero_or_eq_zero_of_mul_eq_zero := fun {a b} =>
       inductionOn₂ a b fun α β => by

feat: pretty-print Ordinal.{u} (#5564)

Adds a pp_with_univ command that pretty-prints the universe parameters for the specified constant by default. Just like Type u, we should include the universe parameter in Ordinal.{u} since you would otherwise have to guess it (and several lemmas require level inequalities, so it's important to know whether you have an Ordinal.{u} or merely an Ordinal.{max u v}).

It might be cute to pretty-print Category.{v} as well, but that is much more involved since the pretty-printer for constants is hardcoded.

08133abc

Diff

@@ -17,6 +17,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
 import Mathlib.Order.SuccPred.Limit
 import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.Tactic.Positivity
+import Mathlib.Tactic.PPWithUniv
 
 /-!
 # Cardinal Numbers
@@ -109,6 +110,7 @@ instance Cardinal.isEquivalent : Setoid (Type u) where
 def Cardinal : Type (u + 1) :=
   Quotient Cardinal.isEquivalent
 #align cardinal Cardinal
+pp_with_univ Cardinal
 
 namespace Cardinal

feat(SetTheory/Cardinal/Finite): prove lemmas about PartENat.card (#5307)

Prove lemmas to handle PartENat.card Inspired from the similar lemmas for Nat.card This is a mathlib4 companion to the PR #19198 of mathlib3

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

9f3a6c79

Diff

@@ -1690,6 +1690,12 @@ theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c)
   rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]
 #align cardinal.cast_to_nat_of_aleph_0_le Cardinal.cast_toNat_of_aleph0_le
 
+/-- Two finite cardinals are equal iff they are equal their to_nat are equal -/
+theorem toNat_eq_iff_eq_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
+    toNat c = toNat d ↔ c = d := by
+  rw [← natCast_inj, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
+#align cardinal.to_nat_eq_iff_eq_of_lt_aleph_0 Cardinal.toNat_eq_iff_eq_of_lt_aleph0
+
 theorem toNat_le_iff_le_of_lt_aleph0 {c d : Cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
     toNat c ≤ toNat d ↔ c ≤ d := by
   rw [← natCast_le, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]
@@ -1880,6 +1886,85 @@ theorem toPartENat_surjective : Surjective toPartENat := fun x =>
   PartENat.casesOn x ⟨ℵ₀, toPartENat_apply_of_aleph0_le le_rfl⟩ fun n => ⟨n, toPartENat_cast n⟩
 #align cardinal.to_part_enat_surjective Cardinal.toPartENat_surjective
 
+theorem toPartENat_eq_top_iff_le_aleph0 {c : Cardinal} :
+  toPartENat c = ⊤ ↔ ℵ₀ ≤ c := by
+  cases lt_or_ge c ℵ₀ with
+  | inl hc =>
+    simp only [toPartENat_apply_of_lt_aleph0 hc, PartENat.natCast_ne_top, false_iff, not_le, hc]
+  | inr hc => simp only [toPartENat_apply_of_aleph0_le hc, eq_self_iff_true, true_iff]; exact hc
+#align to_part_enat_eq_top_iff_le_aleph_0 Cardinal.toPartENat_eq_top_iff_le_aleph0
+
+lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :
+  toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
+  cases lt_or_ge c ℵ₀ with
+  | inl hc =>
+    rw [toPartENat_apply_of_lt_aleph0 hc]
+    cases lt_or_ge c' ℵ₀ with
+    | inl hc' =>
+      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
+      exact toNat_le_iff_le_of_lt_aleph0 hc hc'
+    | inr hc' =>
+      simp only [toPartENat_apply_of_aleph0_le hc',
+      le_top, true_iff]
+      exact le_trans h hc'
+  | inr hc =>
+    rw [toPartENat_apply_of_aleph0_le hc]
+    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0,
+    le_antisymm h hc]
+#align to_part_enat_le_iff_le_of_le_aleph_0 Cardinal.toPartENat_le_iff_of_le_aleph0
+
+lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :
+  toPartENat c ≤ toPartENat c' ↔ c ≤ c' := by
+  cases lt_or_ge c ℵ₀ with
+  | inl hc =>
+    rw [toPartENat_apply_of_lt_aleph0 hc]
+    rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
+    exact toNat_le_iff_le_of_lt_aleph0 hc hc'
+  | inr hc =>
+    rw [toPartENat_apply_of_aleph0_le hc]
+    simp only [top_le_iff, toPartENat_eq_top_iff_le_aleph0]
+    rw [← not_iff_not, not_le, not_le]
+    simp only [hc', lt_of_lt_of_le hc' hc]
+#align to_part_enat_le_iff_le_of_lt_aleph_0 Cardinal.toPartENat_le_iff_of_lt_aleph0
+
+lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :
+  toPartENat c = toPartENat c' ↔ c = c' := by
+  rw [le_antisymm_iff, le_antisymm_iff, toPartENat_le_iff_of_le_aleph0 hc,
+    toPartENat_le_iff_of_le_aleph0 hc']
+#align to_part_enat_eq_iff_eq_of_le_aleph_0 Cardinal.toPartENat_eq_iff_of_le_aleph0
+
+theorem toPartENat_mono {c c' : Cardinal} (h : c ≤ c') :
+  toPartENat c ≤ toPartENat c' := by
+  cases lt_or_ge c ℵ₀ with
+  | inl hc =>
+    rw [toPartENat_apply_of_lt_aleph0 hc]
+    cases lt_or_ge c' ℵ₀ with
+    | inl hc' =>
+      rw [toPartENat_apply_of_lt_aleph0 hc', PartENat.coe_le_coe]
+      exact toNat_le_of_le_of_lt_aleph0 hc' h
+    | inr hc' =>
+        rw [toPartENat_apply_of_aleph0_le hc']
+        exact le_top
+  | inr hc =>
+      rw [toPartENat_apply_of_aleph0_le hc,
+      toPartENat_apply_of_aleph0_le (le_trans hc h)]
+#align cardinal.to_part_enat_mono Cardinal.toPartENat_mono
+
+theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c := by
+  cases' lt_or_ge c ℵ₀ with hc hc
+  · rw [toPartENat_apply_of_lt_aleph0 hc, Cardinal.toPartENat_apply_of_lt_aleph0 _]
+    simp only [toNat_lift]
+    rw [lift_lt_aleph0]
+    exact hc
+  . rw [toPartENat_apply_of_aleph0_le hc, toPartENat_apply_of_aleph0_le _]
+    rw [aleph0_le_lift]
+    exact hc
+#align cardinal.to_part_enat_lift Cardinal.toPartENat_lift
+
+theorem toPartENat_congr {β : Type v} (e : α ≃ β) : toPartENat (#α) = toPartENat (#β) := by
+  rw [← toPartENat_lift, lift_mk_eq.{_, _,v}.mpr ⟨e⟩, toPartENat_lift]
+#align cardinal.to_part_enat_congr Cardinal.toPartENat_congr
+
 theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat (#α) = Fintype.card α := by simp
 #align cardinal.mk_to_part_enat_eq_coe_card Cardinal.mk_toPartENat_eq_coe_card

chore: reorder universe variables in Cardinal.lift_le and Cardinal.lift_mk_le (#5325)

Cardinal.lift_le and Cardinal.lift_mk_le have their universes out of order, in the sense that persistently through the rest of the library we need to specify the 2nd universe (resp 3rd), while the others are solved by unification.

This PR reorders the universes so it's easier to specify the thing you need to specify!

(This PR doesn't get rid of all the occurrences of \.\{_, in the library, but I'd like to do that later.)

I do have a hidden agenda here, which is that I've been experimenting with solutions to the dreaded "Can't solve max u v = max v ?u" universe unification issue (which is making life hellish forward porting https://github.com/leanprover-community/mathlib/pull/19153), and my favourite (but still hacky) solution doesn't like some of the occasions where we reference a lemma filling in some of its universe arguments with _ but then fully specify a later one. (e.g. rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{_, _, v}] in ModelTheory/Skolem.lean). Hence the cleanup proposed in this PR makes my life easier working on these experiments. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

719a2170

Diff

@@ -300,8 +300,8 @@ theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.ou
   · rw [mk'_def, mk'_def, le_def]
 #align cardinal.out_embedding Cardinal.out_embedding
 
-theorem lift_mk_le {α : Type u} {β : Type v} :
-    lift.{max v w} (#α) ≤ lift.{max u w} (#β) ↔ Nonempty (α ↪ β) :=
+theorem lift_mk_le {α : Type v} {β : Type w} :
+    lift.{max u w} (#α) ≤ lift.{max u v} (#β) ↔ Nonempty (α ↪ β) :=
   ⟨fun ⟨f⟩ => ⟨Embedding.congr Equiv.ulift Equiv.ulift f⟩, fun ⟨f⟩ =>
     ⟨Embedding.congr Equiv.ulift.symm Equiv.ulift.symm f⟩⟩
 #align cardinal.lift_mk_le Cardinal.lift_mk_le
@@ -311,7 +311,7 @@ Because Lean often can not realize it should use this specialization itself,
 we provide this statement separately so you don't have to solve the specialization problem either.
 -/
 theorem lift_mk_le' {α : Type u} {β : Type v} : lift.{v} (#α) ≤ lift.{u} (#β) ↔ Nonempty (α ↪ β) :=
-  lift_mk_le.{u, v, 0}
+  lift_mk_le.{0}
 #align cardinal.lift_mk_le' Cardinal.lift_mk_le'
 
 theorem lift_mk_eq {α : Type u} {β : Type v} :
@@ -330,10 +330,10 @@ theorem lift_mk_eq' {α : Type u} {β : Type v} : lift.{v} (#α) = lift.{u} (#β
 #align cardinal.lift_mk_eq' Cardinal.lift_mk_eq'
 
 @[simp]
-theorem lift_le {a b : Cardinal.{u}} : lift.{v, u} a ≤ lift.{v, u} b ↔ a ≤ b :=
+theorem lift_le {a b : Cardinal.{v}} : lift.{u, v} a ≤ lift.{u, v} b ↔ a ≤ b :=
   inductionOn₂ a b fun α β => by
     rw [← lift_umax]
-    exact lift_mk_le.{u, u, v}
+    exact lift_mk_le.{u}
 #align cardinal.lift_le Cardinal.lift_le
 
 -- Porting note: changed `simps` to `simps!` because the linter told to do so.
@@ -1111,7 +1111,7 @@ theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u,v} (iInf f) = ⨅ i, lift
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b :=
   inductionOn₂ a b fun α β => by
-    rw [← lift_id (#β), ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{_,_,v}]
+    rw [← lift_id (#β), ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}]
     exact fun ⟨f⟩ =>
       ⟨#Set.range f,
         Eq.symm <| lift_mk_eq.{_, _, v}.2
@@ -1406,7 +1406,7 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by
 theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
   succ_le_iff.1
     (by
-      rw [← nat_succ, ← lift_mk_fin, aleph0, lift_mk_le.{0, 0, u}]
+      rw [← nat_succ, ← lift_mk_fin, aleph0, lift_mk_le.{u}]
       exact ⟨⟨(↑), fun a b => Fin.ext⟩⟩)
 #align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0
 
@@ -2001,7 +2001,7 @@ theorem mk_image_le {α β : Type u} {f : α → β} {s : Set α} : (#f '' s) 
 
 theorem mk_image_le_lift {α : Type u} {β : Type v} {f : α → β} {s : Set α} :
     lift.{u} (#f '' s) ≤ lift.{v} (#s) :=
-  lift_mk_le.{v, u, 0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_image⟩
+  lift_mk_le.{0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_image⟩
 #align cardinal.mk_image_le_lift Cardinal.mk_image_le_lift
 
 theorem mk_range_le {α β : Type u} {f : α → β} : (#range f) ≤ (#α) :=
@@ -2010,7 +2010,7 @@ theorem mk_range_le {α β : Type u} {f : α → β} : (#range f) ≤ (#α) :=
 
 theorem mk_range_le_lift {α : Type u} {β : Type v} {f : α → β} :
     lift.{u} (#range f) ≤ lift.{v} (#α) :=
-  lift_mk_le.{v, u, 0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_range⟩
+  lift_mk_le.{0}.mpr ⟨Embedding.ofSurjective _ surjective_onto_range⟩
 #align cardinal.mk_range_le_lift Cardinal.mk_range_le_lift
 
 theorem mk_range_eq (f : α → β) (h : Injective f) : (#range f) = (#α) :=
@@ -2163,7 +2163,7 @@ theorem mk_sep (s : Set α) (t : α → Prop) : (#({ x ∈ s | t x } : Set α))
 
 theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
     (h : Injective f) : lift.{v} (#f ⁻¹' s) ≤ lift.{u} (#s) := by
-  rw [lift_mk_le.{u, v, 0}]
+  rw [lift_mk_le.{0}]
   -- Porting note: Needed to insert `mem_preimage.mp` below
   use Subtype.coind (fun x => f x.1) fun x => mem_preimage.mp x.2
   apply Subtype.coind_injective; exact h.comp Subtype.val_injective
@@ -2171,7 +2171,7 @@ theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β
 
 theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
     (h : s ⊆ range f) : lift.{u} (#s) ≤ lift.{v} (#f ⁻¹' s) := by
-  rw [lift_mk_le.{v, u, 0}]
+  rw [lift_mk_le.{0}]
   refine' ⟨⟨_, _⟩⟩
   · rintro ⟨y, hy⟩
     rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩

chore: fix grammar 3/3 (#5003)

Part 3 of #5001

df58f20b

Diff

@@ -952,7 +952,7 @@ instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by
 instance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=
   small_subset Iio_subset_Iic_self
 
-/-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to an usual ZFC set. -/
+/-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to a usual ZFC set. -/
 theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :=
   ⟨fun ⟨a, ha⟩ => @small_subset _ (Iic a) s (fun x h => ha h) _, by
     rintro ⟨ι, ⟨e⟩⟩

chore: formatting issues (#4947)

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

5068808d

Diff

@@ -433,7 +433,7 @@ theorem add_def (α β : Type u) : (#α) + (#β) = (#Sum α β) :=
 -- Porting note: Should this be changed to
 -- `⟨fun n => lift (#(Fin n))⟩` in the future?
 instance : NatCast Cardinal.{u} :=
-⟨Nat.unaryCast⟩
+  ⟨Nat.unaryCast⟩
 
 @[simp]
 theorem mk_sum (α : Type u) (β : Type v) : (#Sum α β) = lift.{v, u} (#α) + lift.{u, v} (#β) :=
@@ -1784,7 +1784,7 @@ theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by
 #align cardinal.to_nat_lift Cardinal.toNat_lift
 
 theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat (#α) = toNat (#β) := by
--- Porting note: Inserted universe hint below
+  -- Porting note: Inserted universe hint below
   rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]
 #align cardinal.to_nat_congr Cardinal.toNat_congr

chore: fix many typos (#4967)

These are all doc fixes

6f9e51c3

Diff

@@ -44,7 +44,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 
 ## Main instances
 
-* Cardinals form a `CanonicallyOrderedCommCemiring` with the aforementioned sum and product.
+* Cardinals form a `CanonicallyOrderedCommSemiring` with the aforementioned sum and product.
 * Cardinals form a `SuccOrder`. Use `Order.succ c` for the smallest cardinal greater than `c`.
 * The less than relation on cardinals forms a well-order.
 * Cardinals form a `ConditionallyCompleteLinearOrderBot`. Bounded sets for cardinals in universe

chore: add space after exacts (#4945)

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

bf799bb9

Diff

@@ -2248,7 +2248,7 @@ theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x := by
   · rintro ⟨a, b, hne, h⟩
     simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
     rcases h x with (rfl | rfl)
-    exacts[⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
+    exacts [⟨b, hne.symm, fun z => (h z).resolve_left⟩, ⟨a, hne, fun z => (h z).resolve_right⟩]
   · rintro ⟨y, hne, hy⟩
     exact ⟨x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)⟩
 #align cardinal.mk_eq_two_iff' Cardinal.mk_eq_two_iff'

chore: fix many typos (#4535)

Run codespell Mathlib and keep some suggestions.

92f5c622

Diff

@@ -155,7 +155,7 @@ theorem mk_out (c : Cardinal) : (#c.out) = c :=
   Quotient.out_eq _
 #align cardinal.mk_out Cardinal.mk_out
 
-/-- The representative of the cardinal of a type is equivalent ot the original type. -/
+/-- The representative of the cardinal of a type is equivalent to the original type. -/
 def outMkEquiv {α : Type v} : (#α).out ≃ α :=
   Nonempty.some <| Cardinal.eq.mp (by simp)
 #align cardinal.out_mk_equiv Cardinal.outMkEquiv

feat: port Algebra.Quaternion (#3776)

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

8c6ccafa

Diff

@@ -1716,6 +1716,10 @@ theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n := by
   exact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm
 #align cardinal.to_nat_cast Cardinal.toNat_cast
 
+@[simp]
+theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] : Cardinal.toNat (OfNat.ofNat n) = OfNat.ofNat n :=
+  toNat_cast n
+
 /-- `toNat` has a right-inverse: coercion. -/
 theorem toNat_rightInverse : Function.RightInverse ((↑) : ℕ → Cardinal) toNat :=
   toNat_cast

fix: correct names of LinearOrder decidable fields (#4006)

This renames

decidable_eq to decidableEq
decidable_lt to decidableLT
decidable_le to decidableLE
decidableLT_of_decidableLE to decidableLTOfDecidableLE
decidableEq_of_decidableLE to decidableEqOfDecidableLE

These fields are data not proofs, so they should be lowerCamelCased.

e7280432

Diff

@@ -263,7 +263,7 @@ instance linearOrder : LinearOrder Cardinal.{u} :=
     le_total := by
       rintro ⟨α⟩ ⟨β⟩
       apply Embedding.total
-    decidable_le := Classical.decRel _ }
+    decidableLE := Classical.decRel _ }
 
 theorem le_def (α β : Type u) : (#α) ≤ (#β) ↔ Nonempty (α ↪ β) :=
   Iff.rfl

chore: Rename to sSup/iSup (#3938)

As discussed on Zulip

Renames

supₛ → sSup
infₛ → sInf
supᵢ → iSup
infᵢ → iInf
bsupₛ → bsSup
binfₛ → bsInf
bsupᵢ → biSup
binfᵢ → biInf
csupₛ → csSup
cinfₛ → csInf
csupᵢ → ciSup
cinfᵢ → ciInf
unionₛ → sUnion
interₛ → sInter
unionᵢ → iUnion
interᵢ → iInter
bunionₛ → bsUnion
binterₛ → bsInter
bunionᵢ → biUnion
binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

d1eb6264

Diff

@@ -49,7 +49,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 * The less than relation on cardinals forms a well-order.
 * Cardinals form a `ConditionallyCompleteLinearOrderBot`. Bounded sets for cardinals in universe
   `u` are precisely the sets indexed by some type in universe `u`, see
-  `Cardinal.bddAbove_iff_small`. One can use `supₛ` for the cardinal supremum, and `infₛ` for the
+  `Cardinal.bddAbove_iff_small`. One can use `sSup` for the cardinal supremum, and `sInf` for the
   minimum of a set of cardinals.
 
 ## Main Statements
@@ -789,19 +789,19 @@ instance : ConditionallyCompleteLinearOrderBot Cardinal :=
   IsWellOrder.conditionallyCompleteLinearOrderBot _
 
 @[simp]
-theorem infₛ_empty : infₛ (∅ : Set Cardinal.{u}) = 0 :=
+theorem sInf_empty : sInf (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg Set.not_nonempty_empty
-#align cardinal.Inf_empty Cardinal.infₛ_empty
+#align cardinal.Inf_empty Cardinal.sInf_empty
 
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
-  SuccOrder.ofSuccLeIff (fun c => infₛ { c' | c < c' })
+  SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' })
     -- Porting note: Needed to insert `by apply` in the next line
-    ⟨by apply lt_of_lt_of_le <| cinfₛ_mem <| exists_gt _,
-    -- Porting note used to be just `cinfₛ_le'`
-    fun h ↦ cinfₛ_le' h⟩
+    ⟨by apply lt_of_lt_of_le <| csInf_mem <| exists_gt _,
+    -- Porting note used to be just `csInf_le'`
+    fun h ↦ csInf_le' h⟩
 
-theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
+theorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
 
@@ -816,7 +816,7 @@ theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
 theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
   -- Porting note: rewrote the next three lines to avoid defeq abuse.
   have : Set.Nonempty { c' | c < c' } := exists_gt c
-  simp_rw [succ_def, le_cinfₛ_iff'' this, mem_setOf]
+  simp_rw [succ_def, le_csInf_iff'' this, mem_setOf]
   intro b hlt
   rcases b, c with ⟨⟨β⟩, ⟨γ⟩⟩
   cases' le_of_lt hlt with f
@@ -983,21 +983,21 @@ theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbo
   exact bddAbove_image.{v,w} g hf
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
 
-theorem supᵢ_le_sum {ι} (f : ι → Cardinal) : supᵢ f ≤ sum f :=
-  csupᵢ_le' <| le_sum.{u_2,u_1} _
-#align cardinal.supr_le_sum Cardinal.supᵢ_le_sum
+theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
+  ciSup_le' <| le_sum.{u_2,u_1} _
+#align cardinal.supr_le_sum Cardinal.iSup_le_sum
 
 -- Porting note: Added universe hint .{v,_} below
-theorem sum_le_supᵢ_lift {ι : Type u}
-    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} (#ι) * supᵢ f := by
-  rw [← (supᵢ f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
-  exact sum_le_sum _ _ (le_csupᵢ <| bddAbove_range.{u, v} f)
-#align cardinal.sum_le_supr_lift Cardinal.sum_le_supᵢ_lift
+theorem sum_le_iSup_lift {ι : Type u}
+    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} (#ι) * iSup f := by
+  rw [← (iSup f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
+  exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)
+#align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift
 
-theorem sum_le_supᵢ {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * supᵢ f := by
+theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * iSup f := by
   rw [← lift_id (#ι)]
-  exact sum_le_supᵢ_lift f
-#align cardinal.sum_le_supr Cardinal.sum_le_supᵢ
+  exact sum_le_iSup_lift f
+#align cardinal.sum_le_supr Cardinal.sum_le_iSup
 
 theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
     Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by
@@ -1007,10 +1007,10 @@ theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
 
 -- Porting note: LFS is not in normal form.
 -- @[simp]
-/-- A variant of `csupᵢ_of_empty` but with `0` on the RHS for convenience -/
-protected theorem supᵢ_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : supᵢ f = 0 :=
-  csupᵢ_of_empty f
-#align cardinal.supr_of_empty Cardinal.supᵢ_of_empty
+/-- A variant of `ciSup_of_empty` but with `0` on the RHS for convenience -/
+protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
+  ciSup_of_empty f
+#align cardinal.supr_of_empty Cardinal.iSup_of_empty
 
 -- Portin note: simpNF is not happy with universe levels.
 @[simp, nolint simpNF]
@@ -1094,19 +1094,19 @@ theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{
 
 -- Porting note: Inserted .{u,v} below
 @[simp]
-theorem lift_infₛ (s : Set Cardinal) : lift.{u,v} (infₛ s) = infₛ (lift.{u,v} '' s) := by
+theorem lift_sInf (s : Set Cardinal) : lift.{u,v} (sInf s) = sInf (lift.{u,v} '' s) := by
   rcases eq_empty_or_nonempty s with (rfl | hs)
   · simp
-  · exact  lift_monotone.map_cinfₛ hs
-#align cardinal.lift_Inf Cardinal.lift_infₛ
+  · exact  lift_monotone.map_csInf hs
+#align cardinal.lift_Inf Cardinal.lift_sInf
 
 -- Porting note: Inserted .{u,v} below
 @[simp]
-theorem lift_infᵢ {ι} (f : ι → Cardinal) : lift.{u,v} (infᵢ f) = ⨅ i, lift.{u,v} (f i) := by
-  unfold infᵢ
-  convert lift_infₛ (range f)
+theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u,v} (iInf f) = ⨅ i, lift.{u,v} (f i) := by
+  unfold iInf
+  convert lift_sInf (range f)
   simp_rw [←comp_apply (f := lift), range_comp]
-#align cardinal.lift_infi Cardinal.lift_infᵢ
+#align cardinal.lift_infi Cardinal.lift_iInf
 
 theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
     b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b :=
@@ -1169,40 +1169,40 @@ theorem lift_max {a b : Cardinal} : lift.{u,v} (max a b) = max (lift.{u,v} a) (l
 #align cardinal.lift_max Cardinal.lift_max
 
 /-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) :
-    lift.{u} (supₛ s) = supₛ (lift.{u} '' s) := by
-  apply ((le_csupₛ_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csupₛ_le' _)
+theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :
+    lift.{u} (sSup s) = sSup (lift.{u} '' s) := by
+  apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
   · intro c hc
     by_contra h
     obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
     simp_rw [lift_le] at h hc
-    rw [csupₛ_le_iff' hs] at h
+    rw [csSup_le_iff' hs] at h
     exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
   · rintro i ⟨j, hj, rfl⟩
-    exact lift_le.2 (le_csupₛ hs hj)
-#align cardinal.lift_Sup Cardinal.lift_supₛ
+    exact lift_le.2 (le_csSup hs hj)
+#align cardinal.lift_Sup Cardinal.lift_sSup
 
 /-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_supᵢ {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
-    lift.{u} (supᵢ f) = ⨆ i, lift.{u} (f i) := by
-  rw [supᵢ, supᵢ, lift_supₛ hf, ← range_comp]
+theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
+    lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by
+  rw [iSup, iSup, lift_sSup hf, ← range_comp]
   simp [Function.comp]
-#align cardinal.lift_supr Cardinal.lift_supᵢ
+#align cardinal.lift_supr Cardinal.lift_iSup
 
 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
-theorem lift_supᵢ_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
-    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (supᵢ f) ≤ t := by
-  rw [lift_supᵢ hf]
-  exact csupᵢ_le' w
-#align cardinal.lift_supr_le Cardinal.lift_supᵢ_le
+theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
+    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by
+  rw [lift_iSup hf]
+  exact ciSup_le' w
+#align cardinal.lift_supr_le Cardinal.lift_iSup_le
 
 @[simp]
-theorem lift_supᵢ_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
-    {t : Cardinal} : lift.{u} (supᵢ f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by
-  rw [lift_supᵢ hf]
-  exact csupᵢ_le_iff' (bddAbove_range_comp.{_,_,u} hf _)
-#align cardinal.lift_supr_le_iff Cardinal.lift_supᵢ_le_iff
+theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
+    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by
+  rw [lift_iSup hf]
+  exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _)
+#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff
 
 universe v' w'
 
@@ -1210,20 +1210,20 @@ universe v' w'
 it suffices to show that the lift of each cardinal from the smaller supremum
 if bounded by the lift of some cardinal from the larger supremum.
 -/
-theorem lift_supᵢ_le_lift_supᵢ {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
+theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
     {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
-    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (supᵢ f) ≤ lift.{w} (supᵢ f') := by
-  rw [lift_supᵢ hf, lift_supᵢ hf']
-  exact csupᵢ_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩
-#align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢ
+    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (iSup f) ≤ lift.{w} (iSup f') := by
+  rw [lift_iSup hf, lift_iSup hf']
+  exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩
+#align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup
 
-/-- A variant of `lift_supᵢ_le_lift_supᵢ` with universes specialized via `w = v` and `w' = v'`.
+/-- A variant of `lift_iSup_le_lift_iSup` with universes specialized via `w = v` and `w' = v'`.
 This is sometimes necessary to avoid universe unification issues. -/
-theorem lift_supᵢ_le_lift_supᵢ' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
+theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
     {f' : ι' → Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : ι → ι')
-    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (supᵢ f) ≤ lift.{v} (supᵢ f') :=
-  lift_supᵢ_le_lift_supᵢ hf hf' h
-#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_supᵢ_le_lift_supᵢ'
+    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') :=
+  lift_iSup_le_lift_iSup hf hf' h
+#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'
 
 /-- `ℵ₀` is the smallest infinite cardinal. -/
 def aleph0 : Cardinal.{u} :=
@@ -2027,33 +2027,33 @@ theorem mk_image_eq {α β : Type u} {f : α → β} {s : Set α} (hf : Injectiv
   mk_congr (Equiv.Set.image f s hf).symm
 #align cardinal.mk_image_eq Cardinal.mk_image_eq
 
-theorem mk_unionᵢ_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
+theorem mk_iUnion_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f i) ≤ sum fun i => #f i :=
   calc
-    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToUnionᵢ_surjective f)
+    (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)
     _ = sum fun i => #f i := mk_sigma _
-#align cardinal.mk_Union_le_sum_mk Cardinal.mk_unionᵢ_le_sum_mk
+#align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk
 
-theorem mk_unionᵢ_eq_sum_mk {α ι : Type u} {f : ι → Set α}
+theorem mk_iUnion_eq_sum_mk {α ι : Type u} {f : ι → Set α}
     (h : ∀ i j, i ≠ j → Disjoint (f i) (f j)) : (#⋃ i, f i) = sum fun i => #f i :=
   calc
     (#⋃ i, f i) = (#Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
-#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_unionᵢ_eq_sum_mk
+#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk
 
-theorem mk_unionᵢ_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
-  mk_unionᵢ_le_sum_mk.trans (sum_le_supᵢ _)
-#align cardinal.mk_Union_le Cardinal.mk_unionᵢ_le
+theorem mk_iUnion_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
+  mk_iUnion_le_sum_mk.trans (sum_le_iSup _)
+#align cardinal.mk_Union_le Cardinal.mk_iUnion_le
 
-theorem mk_unionₛ_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
-  rw [unionₛ_eq_unionᵢ]
-  apply mk_unionᵢ_le
-#align cardinal.mk_sUnion_le Cardinal.mk_unionₛ_le
+theorem mk_sUnion_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
+  rw [sUnion_eq_iUnion]
+  apply mk_iUnion_le
+#align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le
 
-theorem mk_bunionᵢ_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
+theorem mk_biUnion_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
     (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by
-  rw [bunionᵢ_eq_unionᵢ]
-  apply mk_unionᵢ_le
-#align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_le
+  rw [biUnion_eq_iUnion]
+  apply mk_iUnion_le
+#align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le
 
 theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
   lt_aleph0_of_finite _
@@ -2275,13 +2275,13 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 infixl:80 " ^< " => powerlt
 
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by
-  refine le_csupᵢ (f := fun y : Iio b => a^y) ?_ ⟨c, h⟩
+  refine le_ciSup (f := fun y : Iio b => a^y) ?_ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
 
 theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c := by
-  rw [powerlt, csupᵢ_le_iff']
+  rw [powerlt, ciSup_le_iff']
   · simp
   · rw [← image_eq_range]
     exact bddAbove_image.{u, u} _ bddAbove_Iio
@@ -2316,7 +2316,7 @@ theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 := by
 @[simp]
 theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 :=
   -- Porting note: used to expect that `convert` would leave an instance argument as a goal
-  @Cardinal.supᵢ_of_empty _ _
+  @Cardinal.iSup_of_empty _ _
     (Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt)
 #align cardinal.powerlt_zero Cardinal.powerlt_zero

closes #3680, see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Stepping.20through.20simp_rw/near/326712986

feat: implement intermediate state for simp_rw (#3738)

ff334843

Diff

@@ -1842,7 +1842,7 @@ def toPartENat : Cardinal →+ PartENat where
         simp only [ne_eq, ite_eq_right_iff,
           PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hy
         exact le_add_self.trans_lt hy
-    · simp_rw [if_neg hx, if_neg, PartENat.top_add]
+    · simp_rw [if_neg hx, PartENat.top_add]
       contrapose! hx
       simp only [ne_eq, ite_eq_right_iff,
       PartENat.natCast_ne_top, not_forall, exists_prop, and_true] at hx

chore: fix #align lines (#3640)

This PR fixes two things:

Most align statements for definitions and theorems and instances that are separated by two newlines from the relevant declaration (s/\n\n#align/\n#align). This is often seen in the mathport output after ending calc blocks.
All remaining more-than-one-line #align statements. (This was needed for a script I wrote for #3630.)

2b42dc7c

Diff

@@ -2181,9 +2181,7 @@ theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α →
 theorem mk_preimage_of_injective_of_subset_range_lift {β : Type v} (f : α → β) (s : Set β)
     (h : Injective f) (h2 : s ⊆ range f) : lift.{v} (#f ⁻¹' s) = lift.{u} (#s) :=
   le_antisymm (mk_preimage_of_injective_lift f s h) (mk_preimage_of_subset_range_lift f s h2)
-#align
-  cardinal.mk_preimage_of_injective_of_subset_range_lift
-  Cardinal.mk_preimage_of_injective_of_subset_range_lift
+#align cardinal.mk_preimage_of_injective_of_subset_range_lift Cardinal.mk_preimage_of_injective_of_subset_range_lift
 
 theorem mk_preimage_of_injective (f : α → β) (s : Set β) (h : Injective f) :
     (#f ⁻¹' s) ≤ (#s) := by
@@ -2200,9 +2198,7 @@ theorem mk_preimage_of_subset_range (f : α → β) (s : Set β) (h : s ⊆ rang
 theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h : Injective f)
     (h2 : s ⊆ range f) : (#f ⁻¹' s) = (#s) := by
   convert mk_preimage_of_injective_of_subset_range_lift.{u, u} f s h h2 using 1 <;> rw [lift_id]
-#align
-  cardinal.mk_preimage_of_injective_of_subset_range
-  Cardinal.mk_preimage_of_injective_of_subset_range
+#align cardinal.mk_preimage_of_injective_of_subset_range Cardinal.mk_preimage_of_injective_of_subset_range
 
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
     {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) := by

chore: tidy various files (#3483)

dcd5622e

Diff

@@ -45,7 +45,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 ## Main instances
 
 * Cardinals form a `CanonicallyOrderedCommCemiring` with the aforementioned sum and product.
-* Cardinals form a `succ_order`. Use `Order.succ c` for the smallest cardinal greater than `c`.
+* Cardinals form a `SuccOrder`. Use `Order.succ c` for the smallest cardinal greater than `c`.
 * The less than relation on cardinals forms a well-order.
 * Cardinals form a `ConditionallyCompleteLinearOrderBot`. Bounded sets for cardinals in universe
   `u` are precisely the sets indexed by some type in universe `u`, see
@@ -84,9 +84,7 @@ Cantor's theorem, König's theorem, Konig's theorem
 -/
 
 
-open Function Set Order
-
-open BigOperators Classical
+open Function Set Order BigOperators Classical
 
 noncomputable section
 
@@ -119,7 +117,6 @@ def mk : Type u → Cardinal :=
   Quotient.mk'
 #align cardinal.mk Cardinal.mk
 
--- mathport name: cardinal.mk
 @[inherit_doc]
 scoped prefix:0 "#" => Cardinal.mk
 
@@ -299,8 +296,8 @@ theorem mk_set_le (s : Set α) : (#s) ≤ (#α) :=
 
 theorem out_embedding {c c' : Cardinal} : c ≤ c' ↔ Nonempty (c.out ↪ c'.out) := by
   trans
-  rw [← Quotient.out_eq c, ← Quotient.out_eq c']
-  rfl
+  · rw [← Quotient.out_eq c, ← Quotient.out_eq c']
+  · rw [mk'_def, mk'_def, le_def]
 #align cardinal.out_embedding Cardinal.out_embedding
 
 theorem lift_mk_le {α : Type u} {β : Type v} :
@@ -449,9 +446,9 @@ theorem mk_option {α : Type u} : (#Option α) = (#α) + 1 :=
 #align cardinal.mk_option Cardinal.mk_option
 
 @[simp]
-theorem mk_pSum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
+theorem mk_psum (α : Type u) (β : Type v) : (#PSum α β) = lift.{v} (#α) + lift.{u} (#β) :=
   (mk_congr (Equiv.psumEquivSum α β)).trans (mk_sum α β)
-#align cardinal.mk_psum Cardinal.mk_pSum
+#align cardinal.mk_psum Cardinal.mk_psum
 
 @[simp]
 theorem mk_fintype (α : Type u) [h : Fintype α] : (#α) = Fintype.card α := by
@@ -479,7 +476,6 @@ theorem mk_prod (α : Type u) (β : Type v) : (#α × β) = lift.{v, u} (#α) *
 
 private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
   inductionOn₂ a b fun α β => mk_congr <| Equiv.prodComm α β
--- #align cardinal.mul_comm' Cardinal.mul_comm'
 
 /-- The cardinal exponential. `#α ^ #β` is the cardinal of `β → α`. -/
 instance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=
@@ -643,7 +639,7 @@ theorem lift_two : lift.{u, v} 2 = 2 := by simp [←one_add_one_eq_two]
 theorem mk_set {α : Type u} : (#Set α) = (2 ^ (#α)) := by simp [←one_add_one_eq_two, Set, mk_arrow]
 #align cardinal.mk_set Cardinal.mk_set
 
-/-- A variant of `cardinal.mk_set` expressed in terms of a `set` instead of a `Type`. -/
+/-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/
 @[simp]
 theorem mk_powerset {α : Type u} (s : Set α) : (#↥(𝒫 s)) = (2 ^ (#↥s)) :=
   (mk_congr (Equiv.Set.powerset s)).trans mk_set
@@ -679,17 +675,15 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardina
     bot := 0
     bot_le := Cardinal.zero_le
     add_le_add_left := fun a b => add_le_add_left
-    exists_add_of_le := by
-      intro a b
-      exact inductionOn₂ a b fun α β ⟨⟨f, hf⟩⟩ =>
+    exists_add_of_le := fun {a b} =>
+      inductionOn₂ a b fun α β ⟨⟨f, hf⟩⟩ =>
         have : Sum α (range fᶜ : Set β) ≃ β :=
           (Equiv.sumCongr (Equiv.ofInjective f hf) (Equiv.refl _)).trans <|
             Equiv.Set.sumCompl (range f)
         ⟨#↥(range fᶜ), mk_congr this.symm⟩
     le_self_add := fun a b => (add_zero a).ge.trans <| add_le_add_left (Cardinal.zero_le _) _
-    eq_zero_or_eq_zero_of_mul_eq_zero := by
-      intro a b
-      exact inductionOn₂ a b fun α β => by
+    eq_zero_or_eq_zero_of_mul_eq_zero := fun {a b} =>
+      inductionOn₂ a b fun α β => by
         simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }
 
 instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
@@ -776,9 +770,7 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
       haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩
       obtain ⟨⟨c : Cardinal, hc : ¬Acc (· < ·) c⟩, ⟨h_1 : ∀ j, (f ⟨c, hc⟩).out ↪ (f j).out⟩⟩ :=
         Embedding.min_injective fun i => (f i).out
-      apply hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 _)
-      -- Porting note: Needed to add this intro
-      intro j _ hj
+      refine hc (Acc.intro _ fun j h' => byContradiction fun hj => h'.2 ?_)
       have : (#_) ≤ (#_) := ⟨h_1 ⟨j, hj⟩⟩
       simpa only [mk_out] using this⟩
 #align cardinal.lt_wf Cardinal.lt_wf
@@ -807,7 +799,7 @@ instance : SuccOrder Cardinal :=
     -- Porting note: Needed to insert `by apply` in the next line
     ⟨by apply lt_of_lt_of_le <| cinfₛ_mem <| exists_gt _,
     -- Porting note used to be just `cinfₛ_le'`
-    fun h ↦ by apply cinfₛ_le'; exact h⟩
+    fun h ↦ cinfₛ_le' h⟩
 
 theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
   rfl
@@ -839,7 +831,7 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
   cardinal by this definition, but `0` isn't.
 
-  Use `isSuccLimit` if you want to include the `c = 0` case. -/
+  Use `IsSuccLimit` if you want to include the `c = 0` case. -/
 def IsLimit (c : Cardinal) : Prop :=
   c ≠ 0 ∧ IsSuccLimit c
 #align cardinal.is_limit Cardinal.IsLimit
@@ -936,7 +928,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
                    behaviour for Equivs -/
                   have : FunLike.coe (Equiv.symm (Equiv.ulift (α := α))) = ULift.up (α := α) := rfl
                   rw [this]
-                  simp [Set.preimage]
+                  simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf]
                   exact Equiv.refl _)
                 Equiv.ulift.symm)).trans_le
         (hf b)
@@ -1835,8 +1827,7 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
 
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to `⊤`. -/
-def toPartENat : Cardinal →+ PartENat
-    where
+def toPartENat : Cardinal →+ PartENat where
   toFun c := if c < ℵ₀ then toNat c else ⊤
   map_zero' := by simp [if_pos (zero_lt_one.trans one_lt_aleph0)]
   map_add' x y := by
@@ -1912,9 +1903,8 @@ theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f
         refine' fun h => (H i).not_le _
         rw [← mk_out (f i), ← mk_out (g i)]
         exact ⟨Embedding.ofSurjective _ h⟩
-    exact
-      let ⟨⟨i, a⟩, h⟩ := sG C
-      hc i a (congr_fun h _)
+    let ⟨⟨i, a⟩, h⟩ := sG C
+    exact hc i a (congr_fun h _)
 #align cardinal.sum_lt_prod Cardinal.sum_lt_prod
 
 -- Porting note : simp can prove this
@@ -1925,18 +1915,18 @@ theorem mk_empty : (#Empty) = 0 :=
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_pEmpty : (#PEmpty) = 0 :=
+theorem mk_pempty : (#PEmpty) = 0 :=
   mk_eq_zero _
-#align cardinal.mk_pempty Cardinal.mk_pEmpty
+#align cardinal.mk_pempty Cardinal.mk_pempty
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_pUnit : (#PUnit) = 1 :=
+theorem mk_punit : (#PUnit) = 1 :=
   mk_eq_one PUnit
-#align cardinal.mk_punit Cardinal.mk_pUnit
+#align cardinal.mk_punit Cardinal.mk_punit
 
 theorem mk_unit : (#Unit) = 1 :=
-  mk_pUnit
+  mk_punit
 #align cardinal.mk_unit Cardinal.mk_unit
 
 -- Porting note : simp can prove this
@@ -1947,15 +1937,15 @@ theorem mk_singleton {α : Type u} (x : α) : (#({x} : Set α)) = 1 :=
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_pLift_true : (#PLift True) = 1 :=
+theorem mk_plift_true : (#PLift True) = 1 :=
   mk_eq_one _
-#align cardinal.mk_plift_true Cardinal.mk_pLift_true
+#align cardinal.mk_plift_true Cardinal.mk_plift_true
 
 -- Porting note : simp can prove this
 -- @[simp]
-theorem mk_pLift_false : (#PLift False) = 0 :=
+theorem mk_plift_false : (#PLift False) = 0 :=
   mk_eq_zero _
-#align cardinal.mk_plift_false Cardinal.mk_pLift_false
+#align cardinal.mk_plift_false Cardinal.mk_plift_false
 
 @[simp]
 theorem mk_vector (α : Type u) (n : ℕ) : (#Vector α n) = (#α) ^ℕ n :=
@@ -2134,8 +2124,7 @@ theorem le_mk_diff_add_mk (S T : Set α) : (#S) ≤ (#(S \ T : Set α)) + (#T) :
 
 theorem mk_diff_add_mk {S T : Set α} (h : T ⊆ S) : (#(S \ T : Set α)) + (#T) = (#S) := by
   refine (mk_union_of_disjoint <| ?_).symm.trans <| by rw [diff_union_of_subset h]
-  -- Porting note: `apply` works here, `exact` does not
-  apply disjoint_sdiff_self_left
+  exact disjoint_sdiff_self_left
 #align cardinal.mk_diff_add_mk Cardinal.mk_diff_add_mk
 
 theorem mk_union_le_aleph0 {α} {P Q : Set α} :
@@ -2170,8 +2159,9 @@ theorem mk_sep (s : Set α) (t : α → Prop) : (#({ x ∈ s | t x } : Set α))
 
 theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
     (h : Injective f) : lift.{v} (#f ⁻¹' s) ≤ lift.{u} (#s) := by
--- Porting note: Needed to insert `by exact` below
-  rw [lift_mk_le.{u, v, 0}]; use Subtype.coind (fun x => f x.1) fun x => by exact x.2
+  rw [lift_mk_le.{u, v, 0}]
+  -- Porting note: Needed to insert `mem_preimage.mp` below
+  use Subtype.coind (fun x => f x.1) fun x => mem_preimage.mp x.2
   apply Subtype.coind_injective; exact h.comp Subtype.val_injective
 #align cardinal.mk_preimage_of_injective_lift Cardinal.mk_preimage_of_injective_lift
 
@@ -2285,12 +2275,11 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
   ⨆ c : Iio b, a^c
 #align cardinal.powerlt Cardinal.powerlt
 
--- mathport name: «expr ^< »
 @[inherit_doc]
 infixl:80 " ^< " => powerlt
 
 theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by
-  apply @le_csupᵢ _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
+  refine le_csupᵢ (f := fun y : Iio b => a^y) ?_ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt

feat: port LinearAlgebra.FreeModule.Rank (#3377)

This PR also fixes the order of universes in Cardinal.lift_lift.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: ChrisHughes24 <chrishughes24@gmail.com>

2870d41a

Diff

@@ -238,7 +238,7 @@ theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=
 #align cardinal.lift_uzero Cardinal.lift_uzero
 
 @[simp]
-theorem lift_lift (a : Cardinal) : lift.{w} (lift.{v} a) = lift.{max v w} a :=
+theorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a :=
   inductionOn a fun _ => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
 #align cardinal.lift_lift Cardinal.lift_lift
 
@@ -1161,7 +1161,7 @@ theorem lift_succ (a) : lift.{v,u} (succ a) = succ (lift.{v,u} a) :=
 @[simp, nolint simpNF]
 theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
     lift.{max v w} a = lift.{max u w} b ↔ lift.{v} a = lift.{u} b := by
-  rw [← lift_lift.{u,v,w}, ← lift_lift.{v,u,w}, lift_inj]
+  rw [← lift_lift.{v, w, u}, ← lift_lift.{u, w, v}, lift_inj]
 #align cardinal.lift_umax_eq Cardinal.lift_umax_eq
 
 -- Porting note: Inserted .{u,v} below

chore: forward port mathlib#18668, 18781, 18783, 18792, 18794 (#3410)

set_theory.cardinal.basic: forward-ports leanprover-community/mathlib#18781, which backports some ad-hoc changes in #3247 in a more principled way
ring_theory.int.basic: these changes were already forward-ported in #3278, but we forgot the SHA.
linear_algebra.dimension: forward-ports leanprover-community/mathlib#18792
linear_algebra.matrix.dot_product: forward-ports leanprover-community/mathlib#18783
linear_algebra.finrank: forward-ports leanprover-community/mathlib#18794 which is itself a backport of #3378, hence no changes to the file

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

902ce181

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit e05ead7993520a432bec94ac504842d90707ad63
+! leanprover-community/mathlib commit 9bb28972724354ac0574e2b318be896ec252025f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -543,9 +543,6 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   npow_zero := @power_zero
   npow_succ n c := show (c ^ (n + 1)) = c * (c ^ n) by rw [power_add, power_one, mul_comm']
 
--- Porting note: this ensures a computable instance.
-instance commMonoid : CommMonoid Cardinal.{u} := CommSemiring.toCommMonoid
-
 /-! Porting note: Deprecated section. Remove. -/
 section deprecated
 set_option linter.deprecated false
@@ -698,12 +695,25 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardina
 instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
   { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : CanonicallyOrderedAddMonoid Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring with }
+
 instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=
   { Cardinal.commSemiring,
     Cardinal.linearOrder with
     mul_le_mul_left := @mul_le_mul_left' _ _ _ _
     zero_le_one := zero_le _ }
 
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : CommMonoidWithZero Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring with }
+
+-- porting note: new
+-- Computable instance to prevent a non-computable one being found via the one above
+instance : CommMonoid Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring with }
+
 theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 := by
   by_cases h : c = 0
   · rw [h, power_zero]

e05ead79

chore: update SHA for mathlib#18268, #18771, RelIso/Basic (#3362)

algebra.group.ulift: this is a change that was already backported in leanprover-community/mathlib#18268 to match mathlib4, only the SHA needs to be updated.
set_theory.cardinal.basic: the changes from leanprover-community/mathlib#18771 were already forward-ported in #3343, but the sha was not updated. Here I've also inlined all bys and generally tidied up the formatting.
order.rel_iso.basic (leanprover-community/mathlib#18699, leanprover-community/mathlib#18758) is now fully forward-ported too.

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

5558d43c

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 9dba31df156d9d65b9d78db449542ca73d147c68
+! leanprover-community/mathlib commit e05ead7993520a432bec94ac504842d90707ad63
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -717,8 +717,7 @@ theorem power_le_power_left : ∀ {a b c : Cardinal}, a ≠ 0 → b ≤ c → (a
   exact ⟨@Function.Embedding.arrowCongrLeft _ _ _ ⟨a⟩ e⟩
 #align cardinal.power_le_power_left Cardinal.power_le_power_left
 
-theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b) :=
-  by
+theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b) := by
   rcases eq_or_ne a 0 with (rfl | ha)
   · exact zero_le _
   · convert power_le_power_left ha hb
@@ -726,8 +725,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ (a^b)
 #align cardinal.self_le_power Cardinal.self_le_power
 
 /-- **Cantor's theorem** -/
-theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
-  by
+theorem cantor (a : Cardinal.{u}) : a < (2^a) := by
   induction' a using Cardinal.inductionOn with α
   rw [← mk_set]
   refine' ⟨⟨⟨singleton, fun a b => singleton_eq_singleton_iff.1⟩⟩, _⟩
@@ -744,8 +742,7 @@ theorem one_lt_iff_nontrivial {α : Type u} : 1 < (#α) ↔ Nontrivial α := by
   rw [← not_le, le_one_iff_subsingleton, ← not_nontrivial_iff_subsingleton, Classical.not_not]
 #align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial
 
-theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 :=
-  by
+theorem power_le_max_power_one {a b c : Cardinal} (h : b ≤ c) : (a^b) ≤ max (a^c) 1 := by
   by_cases ha : a = 0
   · simp [ha, zero_power_le]
   · exact (power_le_power_left ha h).trans (le_max_left _ _)
@@ -827,13 +824,12 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
   calc
     (#γ) + 1 = (#Option γ) := mk_option.symm
     _ ≤ (#β) := (f.optionElim b hb).cardinal_le
-
 #align cardinal.add_one_le_succ Cardinal.add_one_le_succ
 
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
   cardinal by this definition, but `0` isn't.
 
-  Use `is_succ_limit` if you want to include the `c = 0` case. -/
+  Use `isSuccLimit` if you want to include the `c = 0` case. -/
 def IsLimit (c : Cardinal) : Prop :=
   c ≠ 0 ∧ IsSuccLimit c
 #align cardinal.is_limit Cardinal.IsLimit
@@ -878,7 +874,6 @@ theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
       calc
         (Σ _ : ι, Quotient.out (#α)) ≃ ι × Quotient.out (#α) := Equiv.sigmaEquivProd _ _
         _ ≃ ULift ι × ULift α := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)
-
 #align cardinal.sum_const Cardinal.sum_const
 
 theorem sum_const' (ι : Type u) (a : Cardinal.{u}) : (sum fun _ : ι => a) = (#ι) * a := by simp
@@ -935,9 +930,7 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
                   exact Equiv.refl _)
                 Equiv.ulift.symm)).trans_le
         (hf b)
-#align
-  cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
-  Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
+#align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le
 
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
     inputs, is always bounded above. -/
@@ -948,8 +941,7 @@ theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove
     exact le_sum.{v,u} f i⟩
 #align cardinal.bdd_above_range Cardinal.bddAbove_range
 
-instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) :=
-  by
+instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by
   rw [← mk_out a]
   apply @small_of_surjective (Set a.out) (Iic (#a.out)) _ fun x => ⟨#x, mk_set_le x⟩
   rintro ⟨x, hx⟩
@@ -960,11 +952,9 @@ instance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=
 
 /-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to an usual ZFC set. -/
 theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :=
-  ⟨fun ⟨a, ha⟩ => @small_subset _ (Iic a) s (fun x h => ha h) _,
-    by
+  ⟨fun ⟨a, ha⟩ => @small_subset _ (Iic a) s (fun x h => ha h) _, by
     rintro ⟨ι, ⟨e⟩⟩
-    suffices (range fun x : ι => (e.symm x).1) = s
-      by
+    suffices (range fun x : ι => (e.symm x).1) = s by
       rw [← this]
       apply bddAbove_range.{u, u}
     ext x
@@ -986,8 +976,7 @@ theorem bddAbove_image (f : Cardinal.{u} → Cardinal.{max u v}) {s : Set Cardin
 #align cardinal.bdd_above_image Cardinal.bddAbove_image
 
 theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbove (range f))
-    (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) :=
-  by
+    (g : Cardinal.{v} → Cardinal.{max v w}) : BddAbove (range (g ∘ f)) := by
   rw [range_comp]
   exact bddAbove_image.{v,w} g hf
 #align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp
@@ -998,21 +987,18 @@ theorem supᵢ_le_sum {ι} (f : ι → Cardinal) : supᵢ f ≤ sum f :=
 
 -- Porting note: Added universe hint .{v,_} below
 theorem sum_le_supᵢ_lift {ι : Type u}
-    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} (#ι) * supᵢ f :=
-  by
+    (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift.{v,_} (#ι) * supᵢ f := by
   rw [← (supᵢ f).lift_id, ← lift_umax, lift_umax.{max u v, u}, ← sum_const]
   exact sum_le_sum _ _ (le_csupᵢ <| bddAbove_range.{u, v} f)
 #align cardinal.sum_le_supr_lift Cardinal.sum_le_supᵢ_lift
 
-theorem sum_le_supᵢ {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * supᵢ f :=
-  by
+theorem sum_le_supᵢ {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ (#ι) * supᵢ f := by
   rw [← lift_id (#ι)]
   exact sum_le_supᵢ_lift f
 #align cardinal.sum_le_supr Cardinal.sum_le_supᵢ
 
 theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
-    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) :=
-  by
+    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by
   refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _
   simp only [mk_sum, mk_out, lift_id, mk_sigma]
 #align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ
@@ -1072,8 +1058,7 @@ theorem prod_le_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i ≤ g i) : pro
 #align cardinal.prod_le_prod Cardinal.prod_le_prod
 
 @[simp]
-theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 :=
-  by
+theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i = 0 := by
   lift f to ι → Type u using fun _ => trivial
   simp only [mk_eq_zero_iff, ← mk_pi, isEmpty_pi]
 #align cardinal.prod_eq_zero Cardinal.prod_eq_zero
@@ -1083,8 +1068,7 @@ theorem prod_ne_zero {ι} (f : ι → Cardinal) : prod f ≠ 0 ↔ ∀ i, f i 
 
 @[simp]
 theorem lift_prod {ι : Type u} (c : ι → Cardinal.{v}) :
-    lift.{w} (prod c) = prod fun i => lift.{w} (c i) :=
-  by
+    lift.{w} (prod c) = prod fun i => lift.{w} (c i) := by
   lift c to ι → Type v using fun _ => trivial
   simp only [← mk_pi, ← mk_uLift]
   exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)
@@ -1108,8 +1092,7 @@ theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{
 
 -- Porting note: Inserted .{u,v} below
 @[simp]
-theorem lift_infₛ (s : Set Cardinal) : lift.{u,v} (infₛ s) = infₛ (lift.{u,v} '' s) :=
-  by
+theorem lift_infₛ (s : Set Cardinal) : lift.{u,v} (infₛ s) = infₛ (lift.{u,v} '' s) := by
   rcases eq_empty_or_nonempty s with (rfl | hs)
   · simp
   · exact  lift_monotone.map_cinfₛ hs
@@ -1117,8 +1100,7 @@ theorem lift_infₛ (s : Set Cardinal) : lift.{u,v} (infₛ s) = infₛ (lift.{u
 
 -- Porting note: Inserted .{u,v} below
 @[simp]
-theorem lift_infᵢ {ι} (f : ι → Cardinal) : lift.{u,v} (infᵢ f) = ⨅ i, lift.{u,v} (f i) :=
-  by
+theorem lift_infᵢ {ι} (f : ι → Cardinal) : lift.{u,v} (infᵢ f) = ⨅ i, lift.{u,v} (f i) := by
   unfold infᵢ
   convert lift_infₛ (range f)
   simp_rw [←comp_apply (f := lift), range_comp]
@@ -1133,7 +1115,6 @@ theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
         Eq.symm <| lift_mk_eq.{_, _, v}.2
           ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)
               fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩
-
 #align cardinal.lift_down Cardinal.lift_down
 
 -- Porting note: Inserted .{u,v} below
@@ -1186,8 +1167,8 @@ theorem lift_max {a b : Cardinal} : lift.{u,v} (max a b) = max (lift.{u,v} a) (l
 #align cardinal.lift_max Cardinal.lift_max
 
 /-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) : lift.{u} (supₛ s) = supₛ (lift.{u} '' s) :=
-  by
+theorem lift_supₛ {s : Set Cardinal} (hs : BddAbove s) :
+    lift.{u} (supₛ s) = supₛ (lift.{u} '' s) := by
   apply ((le_csupₛ_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csupₛ_le' _)
   · intro c hc
     by_contra h
@@ -1209,8 +1190,7 @@ theorem lift_supᵢ {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (rang
 /-- To prove that the lift of a supremum is bounded by some cardinal `t`,
 it suffices to show that the lift of each cardinal is bounded by `t`. -/
 theorem lift_supᵢ_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
-    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (supᵢ f) ≤ t :=
-  by
+    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (supᵢ f) ≤ t := by
   rw [lift_supᵢ hf]
   exact csupᵢ_le' w
 #align cardinal.lift_supr_le Cardinal.lift_supᵢ_le
@@ -1230,8 +1210,7 @@ if bounded by the lift of some cardinal from the larger supremum.
 -/
 theorem lift_supᵢ_le_lift_supᵢ {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
     {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
-    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (supᵢ f) ≤ lift.{w} (supᵢ f') :=
-  by
+    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (supᵢ f) ≤ lift.{w} (supᵢ f') := by
   rw [lift_supᵢ hf, lift_supᵢ hf']
   exact csupᵢ_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩
 #align cardinal.lift_supr_le_lift_supr Cardinal.lift_supᵢ_le_lift_supᵢ
@@ -1400,8 +1379,7 @@ theorem nat_succ (n : ℕ) : (n.succ : Cardinal) = succ ↑n :=
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 #align cardinal.succ_zero Cardinal.succ_zero
 
-theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n :=
-  by
+theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n) : (#α) ≤ n := by
   refine' le_of_lt_succ (lt_of_not_ge fun hn => _)
   rw [← Cardinal.nat_succ, ← lift_mk_fin n.succ] at hn
   cases' hn with f
@@ -1410,8 +1388,7 @@ theorem card_le_of {α : Type u} {n : ℕ} (H : ∀ s : Finset α, s.card ≤ n)
   exact n.lt_succ_self
 #align cardinal.card_le_of Cardinal.card_le_of
 
-theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) :=
-  by
+theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < (b^a) := by
   rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
   exact (cantor a).trans_le (power_le_power_right hb)
 #align cardinal.cantor' Cardinal.cantor'
@@ -1459,8 +1436,7 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
 theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
-  isSuccLimit_of_succ_lt fun a ha =>
-    by
+  isSuccLimit_of_succ_lt fun a ha => by
     rcases lt_aleph0.1 ha with ⟨n, rfl⟩
     rw [← nat_succ]
     apply nat_lt_aleph0
@@ -1470,8 +1446,7 @@ theorem isLimit_aleph0 : IsLimit ℵ₀ :=
   ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
 #align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
 
-theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
-  by
+theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by
   by_contra' h'
   rcases lt_aleph0.1 h' with ⟨_ | n, rfl⟩
   · exact h.ne_zero.irrefl
@@ -1559,8 +1534,7 @@ theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a 
 #align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff
 
 /-- See also `Cardinal.nsmul_lt_aleph0_iff_of_ne_zero` if you already have `n ≠ 0`. -/
-theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ :=
-  by
+theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n = 0 ∨ a < ℵ₀ := by
   cases n with
   | zero => simpa using nat_lt_aleph0 0
   | succ n =>
@@ -1580,8 +1554,7 @@ theorem mul_lt_aleph0 {a b : Cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a *
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by rw [← Nat.cast_mul]; apply nat_lt_aleph0
 #align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0
 
-theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ :=
-  by
+theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ := by
   refine' ⟨fun h => _, _⟩
   · by_cases ha : a = 0
     · exact Or.inl ha
@@ -1599,15 +1572,13 @@ theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0
 #align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff
 
 /-- See also `Cardinal.aleph0_le_mul_iff`. -/
-theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) :=
-  by
+theorem aleph0_le_mul_iff {a b : Cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) := by
   let h := (@mul_lt_aleph0_iff a b).not
   rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h
 #align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff
 
 /-- See also `Cardinal.aleph0_le_mul_iff'`. -/
-theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 :=
-  by
+theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 := by
   have : ∀ {a : Cardinal.{u}}, ℵ₀ ≤ a → a ≠ 0 := fun a => ne_bot_of_le_ne_bot aleph0_ne_zero a
   simp only [aleph0_le_mul_iff, and_or_left, and_iff_right_of_imp this, @and_left_comm (a ≠ 0)]
   simp only [and_comm, or_comm]
@@ -1627,7 +1598,6 @@ theorem eq_one_iff_unique {α : Type _} : (#α) = 1 ↔ Subsingleton α ∧ None
     (#α) = 1 ↔ (#α) ≤ 1 ∧ 1 ≤ (#α) := le_antisymm_iff
     _ ↔ Subsingleton α ∧ Nonempty α :=
       le_one_iff_subsingleton.and (one_le_iff_ne_zero.trans mk_ne_zero_iff)
-
 #align cardinal.eq_one_iff_unique Cardinal.eq_one_iff_unique
 
 theorem infinite_iff {α : Type u} : Infinite α ↔ ℵ₀ ≤ (#α) := by
@@ -1645,8 +1615,7 @@ theorem mk_eq_aleph0 (α : Type _) [Countable α] [Infinite α] : (#α) = ℵ₀
 #align cardinal.mk_eq_aleph_0 Cardinal.mk_eq_aleph0
 
 theorem denumerable_iff {α : Type u} : Nonempty (Denumerable α) ↔ (#α) = ℵ₀ :=
-  ⟨fun ⟨h⟩ => mk_congr ((@Denumerable.eqv α h).trans Equiv.ulift.symm), fun h =>
-    by
+  ⟨fun ⟨h⟩ => mk_congr ((@Denumerable.eqv α h).trans Equiv.ulift.symm), fun h => by
     cases' Quotient.exact h with f
     exact ⟨Denumerable.mk' <| f.trans Equiv.ulift⟩⟩
 #align cardinal.denumerable_iff Cardinal.denumerable_iff
@@ -1695,8 +1664,7 @@ theorem nat_add_aleph0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ := by rw [add_comm, al
 /-- This function sends finite cardinals to the corresponding natural, and infinite cardinals
   to 0. -/
 def toNat : ZeroHom Cardinal ℕ :=
-  ⟨fun c => if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0,
-    by
+  ⟨fun c => if h : c < aleph0.{v} then Classical.choose (lt_aleph0.1 h) else 0, by
     have h : 0 < ℵ₀ := nat_lt_aleph0 0
     dsimp only
     rw [dif_pos h, ← Cardinal.natCast_inj, ← Classical.choose_spec (lt_aleph0.1 h),
@@ -1741,8 +1709,7 @@ theorem toNat_lt_of_lt_of_lt_aleph0 {c d : Cardinal} (hd : d < ℵ₀) (hcd : c
 #align cardinal.to_nat_lt_of_lt_of_lt_aleph_0 Cardinal.toNat_lt_of_lt_of_lt_aleph0
 
 @[simp]
-theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n :=
-  by
+theorem toNat_cast (n : ℕ) : Cardinal.toNat n = n := by
   rw [toNat_apply_of_lt_aleph0 (nat_lt_aleph0 n), ← natCast_inj]
   exact (Classical.choose_spec (lt_aleph0.1 (nat_lt_aleph0 n))).symm
 #align cardinal.to_nat_cast Cardinal.toNat_cast
@@ -1801,8 +1768,7 @@ theorem toNat_eq_one_iff_unique {α : Type _} : toNat (#α) = 1 ↔ Subsingleton
 #align cardinal.to_nat_eq_one_iff_unique Cardinal.toNat_eq_one_iff_unique
 
 @[simp]
-theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c :=
-  by
+theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by
   apply natCast_injective
   cases' lt_or_ge c ℵ₀ with hc hc
   · rw [cast_toNat_of_lt_aleph0, ← lift_natCast.{u,v}, cast_toNat_of_lt_aleph0 hc]
@@ -1817,8 +1783,7 @@ theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat (#α) = toNat (#β) :=
 #align cardinal.to_nat_congr Cardinal.toNat_congr
 
 @[simp]
-theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y :=
-  by
+theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y := by
   rcases eq_or_ne x 0 with (rfl | hx1)
   · rw [zero_mul, zero_toNat, zero_mul]
   rcases eq_or_ne y 0 with (rfl | hy1)
@@ -1850,8 +1815,7 @@ theorem toNat_finset_prod (s : Finset α) (f : α → Cardinal) :
 
 @[simp]
 theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
-    toNat (lift.{v, u} a + lift.{u, v} b) = toNat a + toNat b :=
-  by
+    toNat (lift.{v, u} a + lift.{u, v} b) = toNat a + toNat b := by
   apply Cardinal.natCast_injective
   replace ha : lift.{v, u} a < ℵ₀ := by rwa [lift_lt_aleph0]
   replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph0]
@@ -1924,10 +1888,8 @@ theorem mk_pNat : (#ℕ+) = ℵ₀ :=
 
 /-- **König's theorem** -/
 theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
-  lt_of_not_ge fun ⟨F⟩ =>
-    by
-    have : Inhabited (∀ i : ι, (g i).out) :=
-      by
+  lt_of_not_ge fun ⟨F⟩ => by
+    have : Inhabited (∀ i : ι, (g i).out) := by
       refine' ⟨fun i => Classical.choice <| mk_ne_zero_iff.1 _⟩
       rw [mk_out]
       exact (H i).ne_bot
@@ -1994,7 +1956,6 @@ theorem mk_list_eq_sum_pow (α : Type u) : (#List α) = sum fun n : ℕ => (#α)
   calc
     (#List α) = (#Σn, Vector α n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm
     _ = sum fun n : ℕ => (#α) ^ℕ n := by simp
-
 #align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow
 
 theorem mk_quot_le {α : Type u} {r : α → α → Prop} : (#Quot r) ≤ (#α) :=
@@ -2016,8 +1977,7 @@ theorem mk_emptyCollection (α : Type u) : (#(∅ : Set α)) = 0 :=
   mk_eq_zero _
 #align cardinal.mk_emptyc Cardinal.mk_emptyCollection
 
-theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅ :=
-  by
+theorem mk_emptyCollection_iff {α : Type u} {s : Set α} : (#s) = 0 ↔ s = ∅ := by
   constructor
   · intro h
     rw [mk_eq_zero_iff] at h
@@ -2071,7 +2031,6 @@ theorem mk_unionᵢ_le_sum_mk {α ι : Type u} {f : ι → Set α} : (#⋃ i, f
   calc
     (#⋃ i, f i) ≤ (#Σi, f i) := mk_le_of_surjective (Set.sigmaToUnionᵢ_surjective f)
     _ = sum fun i => #f i := mk_sigma _
-
 #align cardinal.mk_Union_le_sum_mk Cardinal.mk_unionᵢ_le_sum_mk
 
 theorem mk_unionᵢ_eq_sum_mk {α ι : Type u} {f : ι → Set α}
@@ -2079,22 +2038,19 @@ theorem mk_unionᵢ_eq_sum_mk {α ι : Type u} {f : ι → Set α}
   calc
     (#⋃ i, f i) = (#Σi, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)
     _ = sum fun i => #f i := mk_sigma _
-
 #align cardinal.mk_Union_eq_sum_mk Cardinal.mk_unionᵢ_eq_sum_mk
 
 theorem mk_unionᵢ_le {α ι : Type u} (f : ι → Set α) : (#⋃ i, f i) ≤ (#ι) * ⨆ i, #f i :=
   mk_unionᵢ_le_sum_mk.trans (sum_le_supᵢ _)
 #align cardinal.mk_Union_le Cardinal.mk_unionᵢ_le
 
-theorem mk_unionₛ_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s :=
-  by
+theorem mk_unionₛ_le {α : Type u} (A : Set (Set α)) : (#⋃₀ A) ≤ (#A) * ⨆ s : A, #s := by
   rw [unionₛ_eq_unionᵢ]
   apply mk_unionᵢ_le
 #align cardinal.mk_sUnion_le Cardinal.mk_unionₛ_le
 
 theorem mk_bunionᵢ_le {ι α : Type u} (A : ι → Set α) (s : Set ι) :
-    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 :=
-  by
+    (#⋃ x ∈ s, A x) ≤ (#s) * ⨆ x : s, #A x.1 := by
   rw [bunionᵢ_eq_unionᵢ]
   apply mk_unionᵢ_le
 #align cardinal.mk_bUnion_le Cardinal.mk_bunionᵢ_le
@@ -2104,8 +2060,7 @@ theorem finset_card_lt_aleph0 (s : Finset α) : (#(↑s : Set α)) < ℵ₀ :=
 #align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0
 
 theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
-    (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n :=
-  by
+    (#s) = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n := by
   constructor
   · intro h
     lift s to Finset α using lt_aleph0_iff_set_finite.1 (h.symm ▸ nat_lt_aleph0 n)
@@ -2119,8 +2074,7 @@ theorem mk_eq_nat_iff_finset {n : ℕ} :
   by rw [← mk_univ, mk_set_eq_nat_iff_finset]
 #align cardinal.mk_eq_nat_iff_finset Cardinal.mk_eq_nat_iff_finset
 
-theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fintype.card α h = n :=
-  by
+theorem mk_eq_nat_iff_fintype {n : ℕ} : (#α) = n ↔ ∃ h : Fintype α, @Fintype.card α h = n := by
   rw [mk_eq_nat_iff_finset]
   constructor
   · rintro ⟨t, ht, hn⟩
@@ -2146,8 +2100,7 @@ theorem mk_union_of_disjoint {α : Type u} {S T : Set α} (H : Disjoint S T) :
 #align cardinal.mk_union_of_disjoint Cardinal.mk_union_of_disjoint
 
 theorem mk_insert {α : Type u} {s : Set α} {a : α} (h : a ∉ s) :
-    (#(insert a s : Set α)) = (#s) + 1 :=
-  by
+    (#(insert a s : Set α)) = (#s) + 1 := by
   rw [← union_singleton, mk_union_of_disjoint, mk_singleton]
   simpa
 #align cardinal.mk_insert Cardinal.mk_insert
@@ -2213,8 +2166,7 @@ theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β
 #align cardinal.mk_preimage_of_injective_lift Cardinal.mk_preimage_of_injective_lift
 
 theorem mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
-    (h : s ⊆ range f) : lift.{u} (#s) ≤ lift.{v} (#f ⁻¹' s) :=
-  by
+    (h : s ⊆ range f) : lift.{u} (#s) ≤ lift.{v} (#f ⁻¹' s) := by
   rw [lift_mk_le.{v, u, 0}]
   refine' ⟨⟨_, _⟩⟩
   · rintro ⟨y, hy⟩
@@ -2237,7 +2189,6 @@ theorem mk_preimage_of_injective (f : α → β) (s : Set β) (h : Injective f)
     (#f ⁻¹' s) ≤ (#s) := by
   rw [← lift_id (#↑(f ⁻¹' s)), ← lift_id (#↑(s))]
   exact mk_preimage_of_injective_lift f s h
-
 #align cardinal.mk_preimage_of_injective Cardinal.mk_preimage_of_injective
 
 theorem mk_preimage_of_subset_range (f : α → β) (s : Set β) (h : s ⊆ range f) :
@@ -2254,8 +2205,7 @@ theorem mk_preimage_of_injective_of_subset_range (f : α → β) (s : Set β) (h
   Cardinal.mk_preimage_of_injective_of_subset_range
 
 theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : Set α}
-    {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) :=
-  by
+    {t : Set β} (h : t ⊆ f '' s) : lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : Set α)) := by
   rw [image_eq_range] at h
   convert mk_preimage_of_subset_range_lift _ _ h using 1
   rw [mk_sep]
@@ -2263,8 +2213,7 @@ theorem mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α 
 #align cardinal.mk_subset_ge_of_subset_image_lift Cardinal.mk_subset_ge_of_subset_image_lift
 
 theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (h : t ⊆ f '' s) :
-    (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) :=
-  by
+    (#t) ≤ (#({ x ∈ s | f x ∈ t } : Set α)) := by
   rw [image_eq_range] at h
   convert mk_preimage_of_subset_range _ _ h using 1
   rw [mk_sep]
@@ -2272,8 +2221,7 @@ theorem mk_subset_ge_of_subset_image (f : α → β) {s : Set α} {t : Set β} (
 #align cardinal.mk_subset_ge_of_subset_image Cardinal.mk_subset_ge_of_subset_image
 
 theorem le_mk_iff_exists_subset {c : Cardinal} {α : Type u} {s : Set α} :
-    c ≤ (#s) ↔ ∃ p : Set α, p ⊆ s ∧ (#p) = c :=
-  by
+    c ≤ (#s) ↔ ∃ p : Set α, p ⊆ s ∧ (#p) = c := by
   rw [le_mk_iff_exists_set, ← Subtype.exists_set_subtype]
   apply exists_congr; intro t; rw [mk_image_eq]; apply Subtype.val_injective
 #align cardinal.le_mk_iff_exists_subset Cardinal.le_mk_iff_exists_subset
@@ -2286,8 +2234,7 @@ theorem two_le_iff' (x : α) : (2 : Cardinal) ≤ (#α) ↔ ∃ y : α, y ≠ x
   rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x]
 #align cardinal.two_le_iff' Cardinal.two_le_iff'
 
-theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ :=
-  by
+theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α) = univ := by
   simp only [← @Nat.cast_two Cardinal, mk_eq_nat_iff_finset, Finset.card_eq_two]
   constructor
   · rintro ⟨t, ht, x, y, hne, rfl⟩
@@ -2296,8 +2243,7 @@ theorem mk_eq_two_iff : (#α) = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : Set α
     exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩
 #align cardinal.mk_eq_two_iff Cardinal.mk_eq_two_iff
 
-theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x :=
-  by
+theorem mk_eq_two_iff' (x : α) : (#α) = 2 ↔ ∃! y, y ≠ x := by
   rw [mk_eq_two_iff]; constructor
   · rintro ⟨a, b, hne, h⟩
     simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h
@@ -2315,11 +2261,9 @@ theorem exists_not_mem_of_length_lt {α : Type _} (l : List α) (h : ↑l.length
     _ ≤ (#l.toFinset) := mk_le_mk_of_subset fun x _ => List.mem_toFinset.mpr (h x)
     _ = l.toFinset.card := Cardinal.mk_coe_finset
     _ ≤ l.length := Cardinal.natCast_le.mpr (List.toFinset_card_le l)
-
 #align cardinal.exists_not_mem_of_length_lt Cardinal.exists_not_mem_of_length_lt
 
-theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y :=
-  by
+theorem three_le {α : Type _} (h : 3 ≤ (#α)) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by
   have : ↑(3 : ℕ) ≤ (#α); simpa using h
   have : ↑(2 : ℕ) < (#α); rwa [← succ_le_iff, ← Cardinal.nat_succ]
   have := exists_not_mem_of_length_lt [x, y] this
@@ -2335,15 +2279,13 @@ def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
 @[inherit_doc]
 infixl:80 " ^< " => powerlt
 
-theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b :=
-  by
+theorem le_powerlt {b c : Cardinal.{u}} (a) (h : c < b) : (a^c) ≤ a ^< b := by
   apply @le_csupᵢ _ _ _ (fun y : Iio b => a^y) _ ⟨c, h⟩
   rw [← image_eq_range]
   exact bddAbove_image.{u, u} _ bddAbove_Iio
 #align cardinal.le_powerlt Cardinal.le_powerlt
 
-theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c :=
-  by
+theorem powerlt_le {a b c : Cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, (a^x) ≤ c := by
   rw [powerlt, csupᵢ_le_iff']
   · simp
   · rw [← image_eq_range]
@@ -2370,8 +2312,7 @@ theorem powerlt_max {a b c : Cardinal} : a ^< max b c = max (a ^< b) (a ^< c) :=
   (powerlt_mono_left a).map_max
 #align cardinal.powerlt_max Cardinal.powerlt_max
 
-theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 :=
-  by
+theorem zero_powerlt {a : Cardinal} (h : a ≠ 0) : 0 ^< a = 1 := by
   apply (powerlt_le.2 fun c _ => zero_power_le _).antisymm
   rw [← power_zero]
   exact le_powerlt 0 (pos_iff_ne_zero.2 h)

feat: port changes to Cardinal files (#3343)

My bad for letting these pile up.

Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

f13648cc

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit ea050b44c0f9aba9d16a948c7cc7d2e7c8493567
+! leanprover-community/mathlib commit 9dba31df156d9d65b9d78db449542ca73d147c68
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,7 +14,7 @@ import Mathlib.Data.Nat.PartENat
 import Mathlib.Data.Set.Countable
 import Mathlib.Logic.Small.Basic
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Limit
 import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.Tactic.Positivity
 
@@ -32,6 +32,7 @@ We define cardinal numbers as a quotient of types under the equivalence relation
 * Multiplication `c₁ * c₂` is defined by `Cardinal.mul_def : #α * #β = #(α × β)`.
 * The order `c₁ ≤ c₂` is defined by `Cardinal.le_def α β : #α ≤ #β ↔ Nonempty (α ↪ β)`.
 * Exponentiation `c₁ ^ c₂` is defined by `Cardinal.power_def α β : #α ^ #β = #(β → α)`.
+* `Cardinal.isLimit c` means that `c` is a (weak) limit cardinal: `c ≠ 0 ∧ ∀ x < c, succ x < c`.
 * `Cardinal.aleph0` or `ℵ₀` is the cardinality of `ℕ`. This definition is universe polymorphic:
   `Cardinal.aleph0.{u} : Cardinal.{u}` (contrast with `ℕ : Type`, which lives in a specific
   universe). In some cases the universe level has to be given explicitly.
@@ -793,7 +794,7 @@ theorem infₛ_empty : infₛ (∅ : Set Cardinal.{u}) = 0 :=
   dif_neg Set.not_nonempty_empty
 #align cardinal.Inf_empty Cardinal.infₛ_empty
 
-/-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`.-/
+/-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal :=
   SuccOrder.ofSuccLeIff (fun c => infₛ { c' | c < c' })
     -- Porting note: Needed to insert `by apply` in the next line
@@ -805,6 +806,14 @@ theorem succ_def (c : Cardinal) : succ c = infₛ { c' | c < c' } :=
   rfl
 #align cardinal.succ_def Cardinal.succ_def
 
+theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
+  bot_lt_succ
+#align cardinal.succ_pos Cardinal.succ_pos
+
+theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
+  (succ_pos _).ne'
+#align cardinal.succ_ne_zero Cardinal.succ_ne_zero
+
 theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
   -- Porting note: rewrote the next three lines to avoid defeq abuse.
   have : Set.Nonempty { c' | c < c' } := exists_gt c
@@ -821,13 +830,29 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
 
 #align cardinal.add_one_le_succ Cardinal.add_one_le_succ
 
-theorem succ_pos : ∀ c : Cardinal, 0 < succ c :=
-  bot_lt_succ
-#align cardinal.succ_pos Cardinal.succ_pos
+/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
+  cardinal by this definition, but `0` isn't.
 
-theorem succ_ne_zero (c : Cardinal) : succ c ≠ 0 :=
-  (succ_pos _).ne'
-#align cardinal.succ_ne_zero Cardinal.succ_ne_zero
+  Use `is_succ_limit` if you want to include the `c = 0` case. -/
+def IsLimit (c : Cardinal) : Prop :=
+  c ≠ 0 ∧ IsSuccLimit c
+#align cardinal.is_limit Cardinal.IsLimit
+
+protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
+  h.1
+#align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero
+
+protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
+  h.2
+#align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit
+
+theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
+  h.isSuccLimit.succ_lt
+#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt
+
+theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
+  isSuccLimit_bot
+#align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero
 
 /-- The indexed sum of cardinals is the cardinality of the
   indexed disjoint union, i.e. sigma type. -/
@@ -1255,6 +1280,16 @@ theorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c ≤ ℵ₀ ↔ c ≤ ℵ
   rw [← lift_aleph0.{u,v}, lift_le]
 #align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0
 
+@[simp]
+theorem aleph0_lt_lift {c : Cardinal.{u}} : ℵ₀ < lift.{v} c ↔ ℵ₀ < c := by
+  rw [← lift_aleph0.{u,v}, lift_lt]
+#align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift
+
+@[simp]
+theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ := by
+  rw [← lift_aleph0.{u,v}, lift_lt]
+#align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0
+
 /-! ### Properties about the cast from `ℕ` -/
 
 -- Porting note : simp can prove this
@@ -1277,6 +1312,26 @@ theorem nat_eq_lift_iff {n : ℕ} {a : Cardinal.{u}} :
   rw [← lift_natCast.{v,u} n, lift_inj]
 #align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff
 
+@[simp]
+theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by
+  rw [← lift_natCast.{v,u}, lift_le]
+#align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff
+
+@[simp]
+theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} : n ≤ lift.{v} a ↔ n ≤ a := by
+  rw [← lift_natCast.{v,u}, lift_le]
+#align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff
+
+@[simp]
+theorem lift_lt_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a < n ↔ a < n := by
+  rw [← lift_natCast.{v,u}, lift_lt]
+#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff
+
+@[simp]
+theorem nat_lt_lift_iff {n : ℕ} {a : Cardinal.{u}} : n < lift.{v} a ↔ n < a := by
+  rw [← lift_natCast.{v,u}, lift_lt]
+#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff
+
 theorem lift_mk_fin (n : ℕ) : lift (#Fin n) = n := by simp
 #align cardinal.lift_mk_fin Cardinal.lift_mk_fin
 
@@ -1403,6 +1458,27 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))⟩
 #align cardinal.aleph_0_le Cardinal.aleph0_le
 
+theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
+  isSuccLimit_of_succ_lt fun a ha =>
+    by
+    rcases lt_aleph0.1 ha with ⟨n, rfl⟩
+    rw [← nat_succ]
+    apply nat_lt_aleph0
+#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0
+
+theorem isLimit_aleph0 : IsLimit ℵ₀ :=
+  ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
+#align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0
+
+theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c :=
+  by
+  by_contra' h'
+  rcases lt_aleph0.1 h' with ⟨_ | n, rfl⟩
+  · exact h.ne_zero.irrefl
+  · rw [nat_succ] at h
+    exact not_isSuccLimit_succ _ h.isSuccLimit
+#align cardinal.is_limit.aleph_0_le Cardinal.IsLimit.aleph0_le
+
 @[simp]
 theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ :=
   ext fun x => by simp only [mem_Iio, mem_range, eq_comm, lt_aleph0]
@@ -1730,9 +1806,9 @@ theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c :=
   apply natCast_injective
   cases' lt_or_ge c ℵ₀ with hc hc
   · rw [cast_toNat_of_lt_aleph0, ← lift_natCast.{u,v}, cast_toNat_of_lt_aleph0 hc]
-    rwa [← lift_aleph0.{v,u}, lift_lt]
+    rwa [lift_lt_aleph0]
   · rw [cast_toNat_of_aleph0_le, ← lift_natCast.{u,v}, cast_toNat_of_aleph0_le hc, lift_zero]
-    rwa [← lift_aleph0.{v,u}, lift_le]
+    rwa [aleph0_le_lift]
 #align cardinal.to_nat_lift Cardinal.toNat_lift
 
 theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat (#α) = toNat (#β) := by
@@ -1777,12 +1853,8 @@ theorem toNat_add_of_lt_aleph0 {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < 
     toNat (lift.{v, u} a + lift.{u, v} b) = toNat a + toNat b :=
   by
   apply Cardinal.natCast_injective
-  replace ha : lift.{v, u} a < ℵ₀ := by
-    rw [← lift_aleph0.{u,v}]
-    exact lift_lt.2 ha
-  replace hb : lift.{u, v} b < ℵ₀ := by
-    rw [← lift_aleph0.{v,u}]
-    exact lift_lt.2 hb
+  replace ha : lift.{v, u} a < ℵ₀ := by rwa [lift_lt_aleph0]
+  replace hb : lift.{u, v} b < ℵ₀ := by rwa [lift_lt_aleph0]
   rw [Nat.cast_add, ← toNat_lift.{v, u} a, ← toNat_lift.{u, v} b, cast_toNat_of_lt_aleph0 ha,
     cast_toNat_of_lt_aleph0 hb, cast_toNat_of_lt_aleph0 (add_lt_aleph0 ha hb)]
 #align cardinal.to_nat_add_of_lt_aleph_0 Cardinal.toNat_add_of_lt_aleph0

ea050b44

feat: tweak cardinal instances (#3247)

Mathlib 3: https://github.com/leanprover-community/mathlib/pull/18714

Co-authored-by: Scott Morrison <scott@tqft.net>

72b511e5

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 
 ! This file was ported from Lean 3 source module set_theory.cardinal.basic
-! leanprover-community/mathlib commit 4c19a16e4b705bf135cf9a80ac18fcc99c438514
+! leanprover-community/mathlib commit ea050b44c0f9aba9d16a948c7cc7d2e7c8493567
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -260,6 +260,13 @@ instance partialOrder : PartialOrder Cardinal.{u} where
     rintro ⟨α⟩ ⟨β⟩ ⟨e₁⟩ ⟨e₂⟩
     exact Quotient.sound (e₁.antisymm e₂)
 
+instance linearOrder : LinearOrder Cardinal.{u} :=
+  { Cardinal.partialOrder with
+    le_total := by
+      rintro ⟨α⟩ ⟨β⟩
+      apply Embedding.total
+    decidable_le := Classical.decRel _ }
+
 theorem le_def (α β : Type u) : (#α) ≤ (#β) ↔ Nonempty (α ↪ β) :=
   Iff.rfl
 #align cardinal.le_def Cardinal.le_def
@@ -535,6 +542,9 @@ instance commSemiring : CommSemiring Cardinal.{u} where
   npow_zero := @power_zero
   npow_succ n c := show (c ^ (n + 1)) = c * (c ^ n) by rw [power_add, power_one, mul_comm']
 
+-- Porting note: this ensures a computable instance.
+instance commMonoid : CommMonoid Cardinal.{u} := CommSemiring.toCommMonoid
+
 /-! Porting note: Deprecated section. Remove. -/
 section deprecated
 set_option linter.deprecated false
@@ -665,7 +675,7 @@ instance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (· +
   ⟨fun _ _ _ h => add_le_add' h le_rfl⟩
 #align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClass
 
-instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
+instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardinal.{u} :=
   { Cardinal.commSemiring,
     Cardinal.partialOrder with
     bot := 0
@@ -684,6 +694,15 @@ instance : CanonicallyOrderedCommSemiring Cardinal.{u} :=
       exact inductionOn₂ a b fun α β => by
         simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }
 
+instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
+  { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }
+
+instance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=
+  { Cardinal.commSemiring,
+    Cardinal.linearOrder with
+    mul_le_mul_left := @mul_le_mul_left' _ _ _ _
+    zero_le_one := zero_le _ }
+
 theorem zero_power_le (c : Cardinal.{u}) : ((0 : Cardinal.{u})^c) ≤ 1 := by
   by_cases h : c = 0
   · rw [h, power_zero]
@@ -715,17 +734,7 @@ theorem cantor (a : Cardinal.{u}) : a < (2^a) :=
   exact cantor_injective f hf
 #align cardinal.cantor Cardinal.cantor
 
-instance : NoMaxOrder Cardinal.{u} :=
-  { Cardinal.partialOrder with exists_gt := fun a => ⟨_, cantor a⟩ }
-
-instance : CanonicallyLinearOrderedAddMonoid Cardinal.{u} :=
-  { (inferInstance : CanonicallyOrderedAddMonoid Cardinal),
-    -- Porting note: Needed to add .{u} below
-    Cardinal.partialOrder.{u} with
-    le_total := by
-      rintro ⟨α⟩ ⟨β⟩
-      apply Embedding.total
-    decidable_le := Classical.decRel _}
+instance : NoMaxOrder Cardinal.{u} where exists_gt a := ⟨_, cantor a⟩
 
 -- short-circuit type class inference
 instance : DistribLattice Cardinal.{u} := inferInstance
@@ -769,6 +778,7 @@ protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
 instance : WellFoundedRelation Cardinal.{u} :=
   ⟨(· < ·), Cardinal.lt_wf⟩
 
+-- Porting note: this no longer is automatically inferred.
 instance : WellFoundedLT Cardinal.{u} :=
   ⟨Cardinal.lt_wf⟩

chore: tidy various files (#2742)

8c7b65ad

Diff

@@ -335,7 +335,7 @@ theorem lift_le {a b : Cardinal.{u}} : lift.{v, u} a ≤ lift.{v, u} b ↔ a ≤
 /-- `Cardinal.lift` as an `OrderEmbedding`. -/
 @[simps! (config := { fullyApplied := false })]
 def liftOrderEmbedding : Cardinal.{v} ↪o Cardinal.{max v u} :=
-  OrderEmbedding.ofMapLeIff lift.{u, v} fun _ _ => lift_le
+  OrderEmbedding.ofMapLEIff lift.{u, v} fun _ _ => lift_le
 #align cardinal.lift_order_embedding Cardinal.liftOrderEmbedding
 
 theorem lift_injective : Injective lift.{u, v} :=

feat: port SetTheory.Cardinal.Ordinal (#2463)

1231af35

Diff

@@ -482,10 +482,9 @@ instance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=
 -- with `HPow`, but somebody should figure out
 -- if this is still relevant in Lean4.
 -- mathport name: cardinal.pow
-local infixr:0 "^'" => @Pow.pow Cardinal Cardinal Cardinal.instPowCardinal
-
+local infixr:0 "^'" => @HPow.hPow Cardinal Cardinal Cardinal.instPowCardinal
 -- -- mathport name: cardinal.pow.nat
-local infixr:80 " ^ℕ " => @Pow.pow Cardinal ℕ Monoid.Pow
+local infixr:80 " ^ℕ " => @HPow.hPow Cardinal ℕ Cardinal instHPow
 
 theorem power_def (α β) : ((#α) ^ (#β)) = (#β → α) :=
   rfl