order.compactly_generated ⟷ Mathlib.Order.CompactlyGenerated.Basic

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

@@ -9,7 +9,7 @@ import Order.RelIso.Set
 import Order.SupIndep
 import Order.Zorn
 import Data.Finset.Order
-import Data.Set.Intervals.OrderIso
+import Order.Interval.Set.OrderIso
 import Data.Finite.Set
 import Tactic.Tfae
 

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

@@ -67,7 +67,7 @@ namespace CompleteLattice
 
 variable (α)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (a b «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (a b «expr ∈ » s) -/
 #print CompleteLattice.IsSupClosedCompact /-
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/

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

@@ -193,7 +193,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
     (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
     (hbelow : ∀ x ∈ s, x < k) : sSup s < k :=
   by
-  rw [is_compact_element_iff_le_of_directed_Sup_le] at hk 
+  rw [is_compact_element_iff_le_of_directed_Sup_le] at hk
   by_contra
   have sSup : Sup s ≤ k := sSup_le fun s hs => (hbelow s hs).le
   replace sSup : Sup s = k := eq_iff_le_not_lt.mpr ⟨sSup, h⟩
@@ -215,7 +215,7 @@ theorem isCompactElement_finsetSup {α β : Type _} [CompleteLattice α] {f : β
   rintro x hx
   obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
   specialize h p hps
-  rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
+  rw [is_compact_element_iff_le_of_directed_Sup_le] at h
   specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
   simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.isCompactElement_finsetSup
@@ -244,7 +244,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
   by
   intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
   cases' t.eq_empty_or_nonempty with h h
-  · subst h; rw [Finset.sup_empty] at ht₂ ; rw [ht₂]
+  · subst h; rw [Finset.sup_empty] at ht₂; rw [ht₂]
     simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
   · rw [ht₂]; exact t.sup_closed_of_sup_closed h ht₁ hsc
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
@@ -409,11 +409,11 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
       rw [le_iff_compact_le_imp]
       by_cases hs : s.nonempty
       · intro c hc hcinf
-        rw [le_inf_iff] at hcinf 
-        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc 
+        rw [le_inf_iff] at hcinf
+        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc
         rcases hc s hs h hcinf.2 with ⟨d, ds, cd⟩
         exact (le_inf hcinf.1 cd).trans (le_iSup₂ d ds)
-      · rw [Set.not_nonempty_iff_eq_empty] at hs 
+      · rw [Set.not_nonempty_iff_eq_empty] at hs
         simp [hs])
     iSup_inf_le_inf_sSup
 #align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
@@ -474,7 +474,7 @@ theorem inf_sSup_eq_iSup_inf_sup_finset :
     (by
       rw [le_iff_compact_le_imp]
       intro c hc hcinf
-      rw [le_inf_iff] at hcinf 
+      rw [le_inf_iff] at hcinf
       rcases hc s hcinf.2 with ⟨t, ht1, ht2⟩
       exact (le_inf hcinf.1 ht2).trans (le_iSup₂ t ht1))
     (iSup_le fun t =>
@@ -494,7 +494,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     intro ht
     classical
     have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
-    · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
+    · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h'
       exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
     · rw [Finset.coe_insert, Set.insert_subset_iff]
       exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
@@ -536,7 +536,7 @@ namespace CompleteLattice
 theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsCompactlyGenerated α :=
   by
-  rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h 
+  rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h
   -- x is the join of the set of compact elements {x}
   exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
@@ -586,14 +586,14 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
     · rcases Set.not_subset.1 h with ⟨c, ⟨hc, hcb⟩, hcbot⟩
       right
       have hc' := CompleteLattice.Iic_coatomic_of_compact_element hc
-      rw [← isAtomic_iff_isCoatomic] at hc' 
+      rw [← isAtomic_iff_isCoatomic] at hc'
       haveI := hc'
       obtain con | ⟨a, ha, hac⟩ := eq_bot_or_exists_atom_le (⟨c, le_refl c⟩ : Set.Iic c)
       · exfalso
         apply hcbot
-        simp only [Subtype.ext_iff, Set.Iic.coe_bot, Subtype.coe_mk] at con 
+        simp only [Subtype.ext_iff, Set.Iic.coe_bot, Subtype.coe_mk] at con
         exact Con
-      rw [← Subtype.coe_le_coe, Subtype.coe_mk] at hac 
+      rw [← Subtype.coe_le_coe, Subtype.coe_mk] at hac
       exact ⟨a, ha.of_is_atom_coe_Iic, hac.trans hcb⟩⟩
 #align is_atomic_of_complemented_lattice isAtomic_of_complementedLattice
 -/
@@ -651,11 +651,11 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup {a : α | IsAtom a} =
   refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
   rw [← Con, eq_comm, inf_eq_left]
   refine' (le_sSup _).trans le_sup_right
-  rw [← disjoint_iff] at con 
+  rw [← disjoint_iff] at con
   have a_dis_Sup_s : Disjoint a (Sup s) := con.mono_right le_sup_right
   rw [← s_max (s ∪ {a}) ⟨fun x hx => _, ⟨_, fun x hx => _⟩⟩ (Set.subset_union_left _ _)]
   · exact Set.mem_union_right _ (Set.mem_singleton _)
-  · rw [Set.mem_union, Set.mem_singleton_iff] at hx 
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain rfl | xa := eq_or_ne x a
     · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
       exact con.mono_right ((sSup_le_sSup <| Set.diff_subset _ _).trans le_sup_right)
@@ -672,7 +672,7 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup {a : α | IsAtom a} =
       rw [← sSup_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
   · rw [sSup_union, sSup_singleton]
     exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left Con.symm
-  · rw [Set.mem_union, Set.mem_singleton_iff] at hx 
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain hx | rfl := hx
     · exact s_atoms x hx
     · exact ha

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

@@ -97,7 +97,23 @@ def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
       ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
-  by classical
+  by
+  classical
+  constructor
+  · intro H ι s hs
+    obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
+    have : ∀ x : t, ∃ i, s i = x := fun x => ht x.Prop
+    choose f hf using this
+    refine' ⟨finset.univ.image f, ht'.trans _⟩
+    · rw [Finset.sup_le_iff]
+      intro b hb
+      rw [← show s (f ⟨b, hb⟩) = id b from hf _]
+      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ ⟨b, hb⟩)
+  · intro H s hs
+    obtain ⟨t, ht⟩ := H s coe (by delta iSup; rwa [Subtype.range_coe])
+    refine' ⟨t.image coe, by simp, ht.trans _⟩
+    rw [Finset.sup_le_iff]
+    exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 -/
 
@@ -107,17 +123,66 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
 theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
     IsCompactElement k ↔
       ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x :=
-  by classical
+  by
+  classical
+  constructor
+  · intro hk s hne hdir hsup
+    obtain ⟨t, ht⟩ := hk s hsup
+    -- certainly every element of t is below something in s, since ↑t ⊆ s.
+    have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
+    obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s
+    exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩
+  · intro hk s hsup
+    -- Consider the set of finite joins of elements of the (plain) set s.
+    let S : Set α := {x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id}
+    -- S is directed, nonempty, and still has sup above k.
+    have dir_US : DirectedOn (· ≤ ·) S :=
+      by
+      rintro x ⟨c, hc⟩ y ⟨d, hd⟩
+      use x ⊔ y
+      constructor
+      · use c ∪ d
+        constructor
+        · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]
+        · simp only [hc.right, hd.right, Finset.sup_union]
+      simp only [and_self_iff, le_sup_left, le_sup_right]
+    have sup_S : Sup s ≤ Sup S := by
+      apply sSup_le_sSup
+      intro x hx; use{x}
+      simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
+        Finset.sup_singleton, Set.singleton_subset_iff]
+    have Sne : S.nonempty := by
+      suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
+      use∅
+      simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
+        and_self_iff]
+    -- Now apply the defn of compact and finish.
+    obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)
+    obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
+    use t; exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 -/
 
 #print CompleteLattice.IsCompactElement.exists_finset_of_le_iSup /-
--- certainly every element of t is below something in s, since ↑t ⊆ s.
--- Consider the set of finite joins of elements of the (plain) set s.
--- S is directed, nonempty, and still has sup above k.
--- Now apply the defn of compact and finish.
 theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
-    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by classical
+    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
+  classical
+  let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
+  have h1 : DirectedOn (· ≤ ·) (Set.range g) :=
+    by
+    rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
+    exact
+      ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
+        iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
+  have h2 : k ≤ Sup (Set.range g) :=
+    h.trans
+      (iSup_le fun i =>
+        le_sSup_of_le ⟨{i}, rfl⟩
+          (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
+  obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
+    (is_compact_element_iff_le_of_directed_Sup_le α k).mp hk (Set.range g) (Set.range_nonempty g) h1
+      h2
+  exact ⟨s, hs⟩
 #align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 -/
 
@@ -140,7 +205,19 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
 
 #print CompleteLattice.isCompactElement_finsetSup /-
 theorem isCompactElement_finsetSup {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
-    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by classical
+    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
+  classical
+  rw [is_compact_element_iff_le_of_directed_Sup_le]
+  intro d hemp hdir hsup
+  change f with id ∘ f
+  rw [← Finset.sup_image]
+  apply Finset.sup_le_of_le_directed d hemp hdir
+  rintro x hx
+  obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
+  specialize h p hps
+  rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
+  specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
+  simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.isCompactElement_finsetSup
 -/
 
@@ -152,7 +229,11 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
     well_founded.well_founded_iff_has_min.mp h {x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x}
       ⟨⊥, ∅, by simp⟩
   refine' ⟨t, ht₁, (sSup_le fun y hy => _).antisymm _⟩
-  · classical
+  ·
+    classical
+    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
+        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]
+    simp
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
@@ -252,7 +333,21 @@ variable {α}
 
 #print CompleteLattice.WellFounded.finite_of_setIndependent /-
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
-    (hs : SetIndependent s) : s.Finite := by classical
+    (hs : SetIndependent s) : s.Finite := by
+  classical
+  refine' set.not_infinite.mp fun contra => _
+  obtain ⟨t, ht₁, ht₂⟩ := well_founded.is_Sup_finite_compact α h s
+  replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t
+  · have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
+    obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
+    exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
+  obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
+  replace hs : x ⊓ Sup s = ⊥
+  · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
+    simpa [Disjoint, hx₂, ← t.sup_id_eq_Sup, ← ht₂] using this.eq_bot
+  apply hx₁
+  rw [← hs, eq_comm, inf_eq_left]
+  exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 -/
 
@@ -397,7 +492,12 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     intro t
     rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
     intro ht
-    classical⟩
+    classical
+    have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
+    · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
+      exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
+    · rw [Finset.coe_insert, Set.insert_subset_iff]
+      exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 -/
 

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

@@ -97,23 +97,7 @@ def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
       ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
-  by
-  classical
-  constructor
-  · intro H ι s hs
-    obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
-    have : ∀ x : t, ∃ i, s i = x := fun x => ht x.Prop
-    choose f hf using this
-    refine' ⟨finset.univ.image f, ht'.trans _⟩
-    · rw [Finset.sup_le_iff]
-      intro b hb
-      rw [← show s (f ⟨b, hb⟩) = id b from hf _]
-      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ ⟨b, hb⟩)
-  · intro H s hs
-    obtain ⟨t, ht⟩ := H s coe (by delta iSup; rwa [Subtype.range_coe])
-    refine' ⟨t.image coe, by simp, ht.trans _⟩
-    rw [Finset.sup_le_iff]
-    exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
+  by classical
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 -/
 
@@ -123,66 +107,17 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
 theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
     IsCompactElement k ↔
       ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x :=
-  by
-  classical
-  constructor
-  · intro hk s hne hdir hsup
-    obtain ⟨t, ht⟩ := hk s hsup
-    -- certainly every element of t is below something in s, since ↑t ⊆ s.
-    have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
-    obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s
-    exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩
-  · intro hk s hsup
-    -- Consider the set of finite joins of elements of the (plain) set s.
-    let S : Set α := {x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id}
-    -- S is directed, nonempty, and still has sup above k.
-    have dir_US : DirectedOn (· ≤ ·) S :=
-      by
-      rintro x ⟨c, hc⟩ y ⟨d, hd⟩
-      use x ⊔ y
-      constructor
-      · use c ∪ d
-        constructor
-        · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]
-        · simp only [hc.right, hd.right, Finset.sup_union]
-      simp only [and_self_iff, le_sup_left, le_sup_right]
-    have sup_S : Sup s ≤ Sup S := by
-      apply sSup_le_sSup
-      intro x hx; use{x}
-      simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
-        Finset.sup_singleton, Set.singleton_subset_iff]
-    have Sne : S.nonempty := by
-      suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
-      use∅
-      simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
-        and_self_iff]
-    -- Now apply the defn of compact and finish.
-    obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)
-    obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
-    use t; exact ⟨htS, by rwa [← htsup]⟩
+  by classical
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 -/
 
 #print CompleteLattice.IsCompactElement.exists_finset_of_le_iSup /-
+-- certainly every element of t is below something in s, since ↑t ⊆ s.
+-- Consider the set of finite joins of elements of the (plain) set s.
+-- S is directed, nonempty, and still has sup above k.
+-- Now apply the defn of compact and finish.
 theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
-    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
-  classical
-  let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
-  have h1 : DirectedOn (· ≤ ·) (Set.range g) :=
-    by
-    rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
-    exact
-      ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
-        iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
-  have h2 : k ≤ Sup (Set.range g) :=
-    h.trans
-      (iSup_le fun i =>
-        le_sSup_of_le ⟨{i}, rfl⟩
-          (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
-  obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
-    (is_compact_element_iff_le_of_directed_Sup_le α k).mp hk (Set.range g) (Set.range_nonempty g) h1
-      h2
-  exact ⟨s, hs⟩
+    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by classical
 #align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 -/
 
@@ -205,19 +140,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
 
 #print CompleteLattice.isCompactElement_finsetSup /-
 theorem isCompactElement_finsetSup {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
-    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
-  classical
-  rw [is_compact_element_iff_le_of_directed_Sup_le]
-  intro d hemp hdir hsup
-  change f with id ∘ f
-  rw [← Finset.sup_image]
-  apply Finset.sup_le_of_le_directed d hemp hdir
-  rintro x hx
-  obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
-  specialize h p hps
-  rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
-  specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
-  simpa only [exists_prop]
+    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by classical
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.isCompactElement_finsetSup
 -/
 
@@ -229,11 +152,7 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
     well_founded.well_founded_iff_has_min.mp h {x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x}
       ⟨⊥, ∅, by simp⟩
   refine' ⟨t, ht₁, (sSup_le fun y hy => _).antisymm _⟩
-  ·
-    classical
-    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
-        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]
-    simp
+  · classical
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
@@ -333,21 +252,7 @@ variable {α}
 
 #print CompleteLattice.WellFounded.finite_of_setIndependent /-
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
-    (hs : SetIndependent s) : s.Finite := by
-  classical
-  refine' set.not_infinite.mp fun contra => _
-  obtain ⟨t, ht₁, ht₂⟩ := well_founded.is_Sup_finite_compact α h s
-  replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t
-  · have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
-    obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
-    exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
-  obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
-  replace hs : x ⊓ Sup s = ⊥
-  · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
-    simpa [Disjoint, hx₂, ← t.sup_id_eq_Sup, ← ht₂] using this.eq_bot
-  apply hx₁
-  rw [← hs, eq_comm, inf_eq_left]
-  exact le_sSup hx₀
+    (hs : SetIndependent s) : s.Finite := by classical
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 -/
 
@@ -492,12 +397,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     intro t
     rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
     intro ht
-    classical
-    have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
-    · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
-      exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
-    · rw [Finset.coe_insert, Set.insert_subset_iff]
-      exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
+    classical⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 -/
 

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

@@ -566,7 +566,7 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCo
 
 #print CompleteLattice.coatomic_of_top_compact /-
 theorem coatomic_of_top_compact (h : IsCompactElement (⊤ : α)) : IsCoatomic α :=
-  (@OrderIso.IicTop α _ _).IsCoatomic_iff.mp (Iic_coatomic_of_compact_element h)
+  (@OrderIso.IicTop α _ _).isCoatomic_iff.mp (Iic_coatomic_of_compact_element h)
 #align complete_lattice.coatomic_of_top_compact CompleteLattice.coatomic_of_top_compact
 -/
 

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

@@ -203,8 +203,8 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
 #align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 -/
 
-#print CompleteLattice.finset_sup_compact_of_compact /-
-theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
+#print CompleteLattice.isCompactElement_finsetSup /-
+theorem isCompactElement_finsetSup {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
   rw [is_compact_element_iff_le_of_directed_Sup_le]
@@ -218,7 +218,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
   rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
   specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
   simpa only [exists_prop]
-#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
+#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.isCompactElement_finsetSup
 -/
 
 #print CompleteLattice.WellFounded.isSupFiniteCompact /-

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

@@ -3,15 +3,15 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
-import Mathbin.Order.Atoms
-import Mathbin.Order.OrderIsoNat
-import Mathbin.Order.RelIso.Set
-import Mathbin.Order.SupIndep
-import Mathbin.Order.Zorn
-import Mathbin.Data.Finset.Order
-import Mathbin.Data.Set.Intervals.OrderIso
-import Mathbin.Data.Finite.Set
-import Mathbin.Tactic.Tfae
+import Order.Atoms
+import Order.OrderIsoNat
+import Order.RelIso.Set
+import Order.SupIndep
+import Order.Zorn
+import Data.Finset.Order
+import Data.Set.Intervals.OrderIso
+import Data.Finite.Set
+import Tactic.Tfae
 
 #align_import order.compactly_generated from "leanprover-community/mathlib"@"c813ed7de0f5115f956239124e9b30f3a621966f"
 
@@ -67,7 +67,7 @@ namespace CompleteLattice
 
 variable (α)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a b «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (a b «expr ∈ » s) -/
 #print CompleteLattice.IsSupClosedCompact /-
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/

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

@@ -56,7 +56,7 @@ complete lattice, well-founded, compact
 -/
 
 
-alias directedOn_range ↔ Directed.directedOn_range _
+alias ⟨Directed.directedOn_range, _⟩ := directedOn_range
 #align directed.directed_on_range Directed.directedOn_range
 
 attribute [protected] Directed.directedOn_range
@@ -319,14 +319,14 @@ theorem isSupClosedCompact_iff_wellFounded :
 #align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFounded
 -/
 
-alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
+alias ⟨_, is_Sup_finite_compact.well_founded⟩ := well_founded_iff_is_Sup_finite_compact
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
 
-alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔ _
-  is_sup_closed_compact.is_Sup_finite_compact
+alias ⟨_, is_sup_closed_compact.is_Sup_finite_compact⟩ :=
+  is_Sup_finite_compact_iff_is_sup_closed_compact
 #align complete_lattice.is_sup_closed_compact.is_Sup_finite_compact CompleteLattice.IsSupClosedCompact.isSupFiniteCompact
 
-alias is_sup_closed_compact_iff_well_founded ↔ _ _root_.well_founded.is_sup_closed_compact
+alias ⟨_, _root_.well_founded.is_sup_closed_compact⟩ := is_sup_closed_compact_iff_well_founded
 #align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompact
 
 variable {α}

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -148,12 +148,12 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
       simp only [and_self_iff, le_sup_left, le_sup_right]
     have sup_S : Sup s ≤ Sup S := by
       apply sSup_le_sSup
-      intro x hx; use {x}
+      intro x hx; use{x}
       simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
         Finset.sup_singleton, Set.singleton_subset_iff]
     have Sne : S.nonempty := by
       suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
-      use ∅
+      use∅
       simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
         and_self_iff]
     -- Now apply the defn of compact and finish.

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit c813ed7de0f5115f956239124e9b30f3a621966f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Order.Atoms
 import Mathbin.Order.OrderIsoNat
@@ -18,6 +13,8 @@ import Mathbin.Data.Set.Intervals.OrderIso
 import Mathbin.Data.Finite.Set
 import Mathbin.Tactic.Tfae
 
+#align_import order.compactly_generated from "leanprover-community/mathlib"@"c813ed7de0f5115f956239124e9b30f3a621966f"
+
 /-!
 # Compactness properties for complete lattices
 
@@ -70,7 +67,7 @@ namespace CompleteLattice
 
 variable (α)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (a b «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a b «expr ∈ » s) -/
 #print CompleteLattice.IsSupClosedCompact /-
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/

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

@@ -235,7 +235,7 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
   ·
     classical
     rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
-        (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
+        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]
     simp
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
@@ -499,7 +499,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
     · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
       exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
-    · rw [Finset.coe_insert, Set.insert_subset]
+    · rw [Finset.coe_insert, Set.insert_subset_iff]
       exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 -/

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

@@ -96,6 +96,7 @@ def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
 #align complete_lattice.is_compact_element CompleteLattice.IsCompactElement
 -/
 
+#print CompleteLattice.isCompactElement_iff /-
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
       ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
@@ -117,7 +118,9 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     rw [Finset.sup_le_iff]
     exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
+-/
 
+#print CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le /-
 /-- An element `k` is compact if and only if any directed set with `Sup` above
 `k` already got above `k` at some point in the set. -/
 theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
@@ -161,7 +164,9 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
     obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
     use t; exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
+-/
 
+#print CompleteLattice.IsCompactElement.exists_finset_of_le_iSup /-
 theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
@@ -182,7 +187,9 @@ theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElemen
       h2
   exact ⟨s, hs⟩
 #align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
+-/
 
+#print CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt /-
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
 its Sup strictly below `k`. -/
 theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
@@ -197,7 +204,9 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
   obtain hxk := hbelow x hxs
   exact hxk.ne (hxk.le.antisymm hkx)
 #align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
+-/
 
+#print CompleteLattice.finset_sup_compact_of_compact /-
 theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
@@ -213,6 +222,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
   specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
   simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
+-/
 
 #print CompleteLattice.WellFounded.isSupFiniteCompact /-
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
@@ -344,11 +354,13 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 -/
 
+#print CompleteLattice.WellFounded.finite_of_independent /-
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
   haveI := (well_founded.finite_of_set_independent hwf ht.set_independent_range).to_subtype
   Finite.of_injective_finite_range (ht.injective h_ne_bot)
 #align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independent
+-/
 
 end CompleteLattice
 
@@ -364,19 +376,23 @@ section
 
 variable {α} [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
+#print sSup_compact_le_eq /-
 @[simp]
 theorem sSup_compact_le_eq (b) : sSup {c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b :=
   by
   rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩
   exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)
 #align Sup_compact_le_eq sSup_compact_le_eq
+-/
 
+#print sSup_compact_eq_top /-
 @[simp]
 theorem sSup_compact_eq_top : sSup {a : α | CompleteLattice.IsCompactElement a} = ⊤ :=
   by
   refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq ⊤)
   exact (and_iff_left le_top).symm
 #align Sup_compact_eq_top sSup_compact_eq_top
+-/
 
 #print le_iff_compact_le_imp /-
 theorem le_iff_compact_le_imp {a b : α} :
@@ -388,6 +404,7 @@ theorem le_iff_compact_le_imp {a b : α} :
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 -/
 
+#print DirectedOn.inf_sSup_eq /-
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
@@ -403,40 +420,56 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
         simp [hs])
     iSup_inf_le_inf_sSup
 #align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
+-/
 
+#print DirectedOn.sSup_inf_eq /-
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
   by simp_rw [@inf_comm _ _ _ a, h.inf_Sup_eq]
 #align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
+-/
 
+#print Directed.inf_iSup_eq /-
 protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
   by rw [iSup, h.directed_on_range.inf_Sup_eq, iSup_range]
 #align directed.inf_supr_eq Directed.inf_iSup_eq
+-/
 
+#print Directed.iSup_inf_eq /-
 protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
   by rw [iSup, h.directed_on_range.Sup_inf_eq, iSup_range]
 #align directed.supr_inf_eq Directed.iSup_inf_eq
+-/
 
+#print DirectedOn.disjoint_sSup_right /-
 protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :
     Disjoint a (sSup s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
   simp_rw [disjoint_iff, h.inf_Sup_eq, iSup_eq_bot]
 #align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_right
+-/
 
+#print DirectedOn.disjoint_sSup_left /-
 protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
     Disjoint (sSup s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
   simp_rw [disjoint_iff, h.Sup_inf_eq, iSup_eq_bot]
 #align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_left
+-/
 
+#print Directed.disjoint_iSup_right /-
 protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
   simp_rw [disjoint_iff, h.inf_supr_eq, iSup_eq_bot]
 #align directed.disjoint_supr_right Directed.disjoint_iSup_right
+-/
 
+#print Directed.disjoint_iSup_left /-
 protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
   simp_rw [disjoint_iff, h.supr_inf_eq, iSup_eq_bot]
 #align directed.disjoint_supr_left Directed.disjoint_iSup_left
+-/
 
+#print inf_sSup_eq_iSup_inf_sup_finset /-
 /-- This property is equivalent to `α` being upper continuous. -/
 theorem inf_sSup_eq_iSup_inf_sup_finset :
     a ⊓ sSup s = ⨆ (t : Finset α) (H : ↑t ⊆ s), a ⊓ t.sup id :=
@@ -450,6 +483,7 @@ theorem inf_sSup_eq_iSup_inf_sup_finset :
     (iSup_le fun t =>
       iSup_le fun h => inf_le_inf_left _ ((Finset.sup_id_eq_sSup t).symm ▸ sSup_le_sSup h))
 #align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finset
+-/
 
 #print CompleteLattice.setIndependent_iff_finite /-
 theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
@@ -533,9 +567,11 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCo
 #align complete_lattice.Iic_coatomic_of_compact_element CompleteLattice.Iic_coatomic_of_compact_element
 -/
 
+#print CompleteLattice.coatomic_of_top_compact /-
 theorem coatomic_of_top_compact (h : IsCompactElement (⊤ : α)) : IsCoatomic α :=
   (@OrderIso.IicTop α _ _).IsCoatomic_iff.mp (Iic_coatomic_of_compact_element h)
 #align complete_lattice.coatomic_of_top_compact CompleteLattice.coatomic_of_top_compact
+-/
 
 end CompleteLattice
 
@@ -591,6 +627,7 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 -/
 
 
+#print exists_setIndependent_isCompl_sSup_atoms /-
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup {a : α | IsAtom a} = ⊤) (b : α) :
@@ -643,13 +680,17 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup {a : α | IsAtom a} =
     · exact s_atoms x hx
     · exact ha
 #align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atoms
+-/
 
+#print exists_setIndependent_of_sSup_atoms_eq_top /-
 theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup {a : α | IsAtom a} = ⊤) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h ⊥
   ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
 #align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_top
+-/
 
+#print complementedLattice_of_sSup_atoms_eq_top /-
 /-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup {a : α | IsAtom a} = ⊤) :
     ComplementedLattice α :=
@@ -657,6 +698,7 @@ theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup {a : α | IsAtom a} =
     let ⟨s, _, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h b
     ⟨sSup s, s_top⟩⟩
 #align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_top
+-/
 
 #print complementedLattice_of_isAtomistic /-
 /-- See [Theorem 6.6][calugareanu]. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -70,7 +70,7 @@ namespace CompleteLattice
 
 variable (α)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a b «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (a b «expr ∈ » s) -/
 #print CompleteLattice.IsSupClosedCompact /-
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/

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

@@ -101,21 +101,21 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
       ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
   by
   classical
-    constructor
-    · intro H ι s hs
-      obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
-      have : ∀ x : t, ∃ i, s i = x := fun x => ht x.Prop
-      choose f hf using this
-      refine' ⟨finset.univ.image f, ht'.trans _⟩
-      · rw [Finset.sup_le_iff]
-        intro b hb
-        rw [← show s (f ⟨b, hb⟩) = id b from hf _]
-        exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ ⟨b, hb⟩)
-    · intro H s hs
-      obtain ⟨t, ht⟩ := H s coe (by delta iSup; rwa [Subtype.range_coe])
-      refine' ⟨t.image coe, by simp, ht.trans _⟩
-      rw [Finset.sup_le_iff]
-      exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
+  constructor
+  · intro H ι s hs
+    obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
+    have : ∀ x : t, ∃ i, s i = x := fun x => ht x.Prop
+    choose f hf using this
+    refine' ⟨finset.univ.image f, ht'.trans _⟩
+    · rw [Finset.sup_le_iff]
+      intro b hb
+      rw [← show s (f ⟨b, hb⟩) = id b from hf _]
+      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ ⟨b, hb⟩)
+  · intro H s hs
+    obtain ⟨t, ht⟩ := H s coe (by delta iSup; rwa [Subtype.range_coe])
+    refine' ⟨t.image coe, by simp, ht.trans _⟩
+    rw [Finset.sup_le_iff]
+    exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 
 /-- An element `k` is compact if and only if any directed set with `Sup` above
@@ -125,62 +125,62 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
       ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x :=
   by
   classical
-    constructor
-    · intro hk s hne hdir hsup
-      obtain ⟨t, ht⟩ := hk s hsup
-      -- certainly every element of t is below something in s, since ↑t ⊆ s.
-      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
-      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s
-      exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩
-    · intro hk s hsup
-      -- Consider the set of finite joins of elements of the (plain) set s.
-      let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }
-      -- S is directed, nonempty, and still has sup above k.
-      have dir_US : DirectedOn (· ≤ ·) S :=
-        by
-        rintro x ⟨c, hc⟩ y ⟨d, hd⟩
-        use x ⊔ y
+  constructor
+  · intro hk s hne hdir hsup
+    obtain ⟨t, ht⟩ := hk s hsup
+    -- certainly every element of t is below something in s, since ↑t ⊆ s.
+    have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩
+    obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s
+    exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩
+  · intro hk s hsup
+    -- Consider the set of finite joins of elements of the (plain) set s.
+    let S : Set α := {x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id}
+    -- S is directed, nonempty, and still has sup above k.
+    have dir_US : DirectedOn (· ≤ ·) S :=
+      by
+      rintro x ⟨c, hc⟩ y ⟨d, hd⟩
+      use x ⊔ y
+      constructor
+      · use c ∪ d
         constructor
-        · use c ∪ d
-          constructor
-          · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]
-          · simp only [hc.right, hd.right, Finset.sup_union]
-        simp only [and_self_iff, le_sup_left, le_sup_right]
-      have sup_S : Sup s ≤ Sup S := by
-        apply sSup_le_sSup
-        intro x hx; use {x}
-        simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
-          Finset.sup_singleton, Set.singleton_subset_iff]
-      have Sne : S.nonempty := by
-        suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
-        use ∅
-        simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
-          and_self_iff]
-      -- Now apply the defn of compact and finish.
-      obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)
-      obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
-      use t; exact ⟨htS, by rwa [← htsup]⟩
+        · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]
+        · simp only [hc.right, hd.right, Finset.sup_union]
+      simp only [and_self_iff, le_sup_left, le_sup_right]
+    have sup_S : Sup s ≤ Sup S := by
+      apply sSup_le_sSup
+      intro x hx; use {x}
+      simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
+        Finset.sup_singleton, Set.singleton_subset_iff]
+    have Sne : S.nonempty := by
+      suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
+      use ∅
+      simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
+        and_self_iff]
+    -- Now apply the defn of compact and finish.
+    obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)
+    obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
+    use t; exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
 theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
-    let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
-    have h1 : DirectedOn (· ≤ ·) (Set.range g) :=
-      by
-      rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
-      exact
-        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
-          iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
-    have h2 : k ≤ Sup (Set.range g) :=
-      h.trans
-        (iSup_le fun i =>
-          le_sSup_of_le ⟨{i}, rfl⟩
-            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
-    obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
-      (is_compact_element_iff_le_of_directed_Sup_le α k).mp hk (Set.range g) (Set.range_nonempty g)
-        h1 h2
-    exact ⟨s, hs⟩
+  let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
+  have h1 : DirectedOn (· ≤ ·) (Set.range g) :=
+    by
+    rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
+    exact
+      ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
+        iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
+  have h2 : k ≤ Sup (Set.range g) :=
+    h.trans
+      (iSup_le fun i =>
+        le_sSup_of_le ⟨{i}, rfl⟩
+          (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
+  obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
+    (is_compact_element_iff_le_of_directed_Sup_le α k).mp hk (Set.range g) (Set.range_nonempty g) h1
+      h2
+  exact ⟨s, hs⟩
 #align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
@@ -201,17 +201,17 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
 theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
-    rw [is_compact_element_iff_le_of_directed_Sup_le]
-    intro d hemp hdir hsup
-    change f with id ∘ f
-    rw [← Finset.sup_image]
-    apply Finset.sup_le_of_le_directed d hemp hdir
-    rintro x hx
-    obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
-    specialize h p hps
-    rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
-    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
-    simpa only [exists_prop]
+  rw [is_compact_element_iff_le_of_directed_Sup_le]
+  intro d hemp hdir hsup
+  change f with id ∘ f
+  rw [← Finset.sup_image]
+  apply Finset.sup_le_of_le_directed d hemp hdir
+  rintro x hx
+  obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
+  specialize h p hps
+  rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
+  specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
+  simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
 #print CompleteLattice.WellFounded.isSupFiniteCompact /-
@@ -219,14 +219,14 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
     IsSupFiniteCompact α := fun s =>
   by
   obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ :=
-    well_founded.well_founded_iff_has_min.mp h { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
+    well_founded.well_founded_iff_has_min.mp h {x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x}
       ⟨⊥, ∅, by simp⟩
   refine' ⟨t, ht₁, (sSup_le fun y hy => _).antisymm _⟩
   ·
     classical
-      rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
-          (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
-      simp
+    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
+        (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
+    simp
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
@@ -315,8 +315,8 @@ theorem isSupClosedCompact_iff_wellFounded :
 alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
 
-alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔
-  _ is_sup_closed_compact.is_Sup_finite_compact
+alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔ _
+  is_sup_closed_compact.is_Sup_finite_compact
 #align complete_lattice.is_sup_closed_compact.is_Sup_finite_compact CompleteLattice.IsSupClosedCompact.isSupFiniteCompact
 
 alias is_sup_closed_compact_iff_well_founded ↔ _ _root_.well_founded.is_sup_closed_compact
@@ -328,19 +328,19 @@ variable {α}
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
     (hs : SetIndependent s) : s.Finite := by
   classical
-    refine' set.not_infinite.mp fun contra => _
-    obtain ⟨t, ht₁, ht₂⟩ := well_founded.is_Sup_finite_compact α h s
-    replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t
-    · have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
-      obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
-      exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
-    obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
-    replace hs : x ⊓ Sup s = ⊥
-    · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
-      simpa [Disjoint, hx₂, ← t.sup_id_eq_Sup, ← ht₂] using this.eq_bot
-    apply hx₁
-    rw [← hs, eq_comm, inf_eq_left]
-    exact le_sSup hx₀
+  refine' set.not_infinite.mp fun contra => _
+  obtain ⟨t, ht₁, ht₂⟩ := well_founded.is_Sup_finite_compact α h s
+  replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t
+  · have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
+    obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
+    exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
+  obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
+  replace hs : x ⊓ Sup s = ⊥
+  · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
+    simpa [Disjoint, hx₂, ← t.sup_id_eq_Sup, ← ht₂] using this.eq_bot
+  apply hx₁
+  rw [← hs, eq_comm, inf_eq_left]
+  exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 -/
 
@@ -365,14 +365,14 @@ section
 variable {α} [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
 @[simp]
-theorem sSup_compact_le_eq (b) : sSup { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b :=
+theorem sSup_compact_le_eq (b) : sSup {c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b :=
   by
   rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩
   exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)
 #align Sup_compact_le_eq sSup_compact_le_eq
 
 @[simp]
-theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ :=
+theorem sSup_compact_eq_top : sSup {a : α | CompleteLattice.IsCompactElement a} = ⊤ :=
   by
   refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq ⊤)
   exact (and_iff_left le_top).symm
@@ -462,11 +462,11 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
     intro ht
     classical
-      have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
-      · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
-        exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
-      · rw [Finset.coe_insert, Set.insert_subset]
-        exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
+    have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
+    · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
+      exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
+    · rw [Finset.coe_insert, Set.insert_subset]
+      exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 -/
 
@@ -546,7 +546,7 @@ variable [IsModularLattice α] [IsCompactlyGenerated α]
 #print isAtomic_of_complementedLattice /-
 instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice α] : IsAtomic α :=
   ⟨fun b => by
-    by_cases h : { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } ⊆ {⊥}
+    by_cases h : {c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b} ⊆ {⊥}
     · left
       rw [← sSup_compact_le_eq b, sSup_eq_bot]
       exact h
@@ -570,11 +570,11 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
 instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLattice α] :
     IsAtomistic α :=
   ⟨fun b =>
-    ⟨{ a | IsAtom a ∧ a ≤ b }, by
+    ⟨{a | IsAtom a ∧ a ≤ b}, by
       symm
-      have hle : Sup { a : α | IsAtom a ∧ a ≤ b } ≤ b := sSup_le fun _ => And.right
+      have hle : Sup {a : α | IsAtom a ∧ a ≤ b} ≤ b := sSup_le fun _ => And.right
       apply (lt_or_eq_of_le hle).resolve_left fun con => _
-      obtain ⟨c, hc⟩ := exists_is_compl (⟨Sup { a : α | IsAtom a ∧ a ≤ b }, hle⟩ : Set.Iic b)
+      obtain ⟨c, hc⟩ := exists_is_compl (⟨Sup {a : α | IsAtom a ∧ a ≤ b}, hle⟩ : Set.Iic b)
       obtain rfl | ⟨a, ha, hac⟩ := eq_bot_or_exists_atom_le c
       · exact ne_of_lt Con (Subtype.ext_iff.1 (eq_top_of_isCompl_bot hc))
       · apply ha.1
@@ -593,12 +593,12 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
-theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
+theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup {a : α | IsAtom a} = ⊤) (b : α) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   by
   obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ :=
     zorn_subset
-      { s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (Sup s) ∧ ∀ a ∈ s, IsAtom a }
+      {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (Sup s) ∧ ∀ a ∈ s, IsAtom a}
       fun c hc1 hc2 =>
       ⟨⋃₀ c,
         ⟨CompleteLattice.independent_sUnion_of_directed hc2.DirectedOn fun s hs => (hc1 hs).1, _,
@@ -644,14 +644,14 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
     · exact ha
 #align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atoms
 
-theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
+theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup {a : α | IsAtom a} = ⊤) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h ⊥
   ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
 #align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_top
 
 /-- See [Theorem 6.6][calugareanu]. -/
-theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
+theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup {a : α | IsAtom a} = ⊤) :
     ComplementedLattice α :=
   ⟨fun b =>
     let ⟨s, _, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h b

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

@@ -189,7 +189,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
     (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
     (hbelow : ∀ x ∈ s, x < k) : sSup s < k :=
   by
-  rw [is_compact_element_iff_le_of_directed_Sup_le] at hk
+  rw [is_compact_element_iff_le_of_directed_Sup_le] at hk 
   by_contra
   have sSup : Sup s ≤ k := sSup_le fun s hs => (hbelow s hs).le
   replace sSup : Sup s = k := eq_iff_le_not_lt.mpr ⟨sSup, h⟩
@@ -209,7 +209,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     rintro x hx
     obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx
     specialize h p hps
-    rw [is_compact_element_iff_le_of_directed_Sup_le] at h
+    rw [is_compact_element_iff_le_of_directed_Sup_le] at h 
     specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
     simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
@@ -237,7 +237,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
   by
   intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
   cases' t.eq_empty_or_nonempty with h h
-  · subst h; rw [Finset.sup_empty] at ht₂; rw [ht₂]
+  · subst h; rw [Finset.sup_empty] at ht₂ ; rw [ht₂]
     simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
   · rw [ht₂]; exact t.sup_closed_of_sup_closed h ht₁ hsc
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
@@ -395,11 +395,11 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
       rw [le_iff_compact_le_imp]
       by_cases hs : s.nonempty
       · intro c hc hcinf
-        rw [le_inf_iff] at hcinf
-        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc
+        rw [le_inf_iff] at hcinf 
+        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc 
         rcases hc s hs h hcinf.2 with ⟨d, ds, cd⟩
         exact (le_inf hcinf.1 cd).trans (le_iSup₂ d ds)
-      · rw [Set.not_nonempty_iff_eq_empty] at hs
+      · rw [Set.not_nonempty_iff_eq_empty] at hs 
         simp [hs])
     iSup_inf_le_inf_sSup
 #align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
@@ -444,7 +444,7 @@ theorem inf_sSup_eq_iSup_inf_sup_finset :
     (by
       rw [le_iff_compact_le_imp]
       intro c hc hcinf
-      rw [le_inf_iff] at hcinf
+      rw [le_inf_iff] at hcinf 
       rcases hc s hcinf.2 with ⟨t, ht1, ht2⟩
       exact (le_inf hcinf.1 ht2).trans (le_iSup₂ t ht1))
     (iSup_le fun t =>
@@ -463,7 +463,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     intro ht
     classical
       have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
-      · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h'
+      · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h' 
         exact fun con => ((Set.mem_diff a).1 (ht Con)).2 (Set.mem_singleton a)
       · rw [Finset.coe_insert, Set.insert_subset]
         exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
@@ -505,7 +505,7 @@ namespace CompleteLattice
 theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsCompactlyGenerated α :=
   by
-  rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h
+  rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h 
   -- x is the join of the set of compact elements {x}
   exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
@@ -553,14 +553,14 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
     · rcases Set.not_subset.1 h with ⟨c, ⟨hc, hcb⟩, hcbot⟩
       right
       have hc' := CompleteLattice.Iic_coatomic_of_compact_element hc
-      rw [← isAtomic_iff_isCoatomic] at hc'
+      rw [← isAtomic_iff_isCoatomic] at hc' 
       haveI := hc'
       obtain con | ⟨a, ha, hac⟩ := eq_bot_or_exists_atom_le (⟨c, le_refl c⟩ : Set.Iic c)
       · exfalso
         apply hcbot
-        simp only [Subtype.ext_iff, Set.Iic.coe_bot, Subtype.coe_mk] at con
+        simp only [Subtype.ext_iff, Set.Iic.coe_bot, Subtype.coe_mk] at con 
         exact Con
-      rw [← Subtype.coe_le_coe, Subtype.coe_mk] at hac
+      rw [← Subtype.coe_le_coe, Subtype.coe_mk] at hac 
       exact ⟨a, ha.of_is_atom_coe_Iic, hac.trans hcb⟩⟩
 #align is_atomic_of_complemented_lattice isAtomic_of_complementedLattice
 -/
@@ -617,11 +617,11 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
   refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
   rw [← Con, eq_comm, inf_eq_left]
   refine' (le_sSup _).trans le_sup_right
-  rw [← disjoint_iff] at con
+  rw [← disjoint_iff] at con 
   have a_dis_Sup_s : Disjoint a (Sup s) := con.mono_right le_sup_right
   rw [← s_max (s ∪ {a}) ⟨fun x hx => _, ⟨_, fun x hx => _⟩⟩ (Set.subset_union_left _ _)]
   · exact Set.mem_union_right _ (Set.mem_singleton _)
-  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx 
     obtain rfl | xa := eq_or_ne x a
     · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
       exact con.mono_right ((sSup_le_sSup <| Set.diff_subset _ _).trans le_sup_right)
@@ -638,7 +638,7 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
       rw [← sSup_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
   · rw [sSup_union, sSup_singleton]
     exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left Con.symm
-  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx 
     obtain hx | rfl := hx
     · exact s_atoms x hx
     · exact ha

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

@@ -214,6 +214,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
+#print CompleteLattice.WellFounded.isSupFiniteCompact /-
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsSupFiniteCompact α := fun s =>
   by
@@ -229,6 +230,7 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
+-/
 
 #print CompleteLattice.IsSupFiniteCompact.isSupClosedCompact /-
 theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSupClosedCompact α :=
@@ -241,6 +243,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
 -/
 
+#print CompleteLattice.IsSupClosedCompact.wellFounded /-
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     WellFounded ((· > ·) : α → α → Prop) :=
   by
@@ -254,6 +257,7 @@ theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
   · use a 37; apply Set.mem_range_self
   · rintro x ⟨m, hm⟩ y ⟨n, hn⟩; use m ⊔ n; rw [← hm, ← hn]; apply RelHomClass.map_sup a
 #align complete_lattice.is_sup_closed_compact.well_founded CompleteLattice.IsSupClosedCompact.wellFounded
+-/
 
 #print CompleteLattice.isSupFiniteCompact_iff_all_elements_compact /-
 theorem isSupFiniteCompact_iff_all_elements_compact :
@@ -274,6 +278,7 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
 #align complete_lattice.is_Sup_finite_compact_iff_all_elements_compact CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
 -/
 
+#print CompleteLattice.wellFounded_characterisations /-
 theorem wellFounded_characterisations :
     TFAE
       [WellFounded ((· > ·) : α → α → Prop), IsSupFiniteCompact α, IsSupClosedCompact α,
@@ -285,11 +290,14 @@ theorem wellFounded_characterisations :
   tfae_have 2 ↔ 4; · exact is_Sup_finite_compact_iff_all_elements_compact α
   tfae_finish
 #align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisations
+-/
 
+#print CompleteLattice.wellFounded_iff_isSupFiniteCompact /-
 theorem wellFounded_iff_isSupFiniteCompact :
     WellFounded ((· > ·) : α → α → Prop) ↔ IsSupFiniteCompact α :=
   (wellFounded_characterisations α).out 0 1
 #align complete_lattice.well_founded_iff_is_Sup_finite_compact CompleteLattice.wellFounded_iff_isSupFiniteCompact
+-/
 
 #print CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact /-
 theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ IsSupClosedCompact α :=
@@ -297,10 +305,12 @@ theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ Is
 #align complete_lattice.is_Sup_finite_compact_iff_is_sup_closed_compact CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact
 -/
 
+#print CompleteLattice.isSupClosedCompact_iff_wellFounded /-
 theorem isSupClosedCompact_iff_wellFounded :
     IsSupClosedCompact α ↔ WellFounded ((· > ·) : α → α → Prop) :=
   (wellFounded_characterisations α).out 2 0
 #align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFounded
+-/
 
 alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
@@ -314,6 +324,7 @@ alias is_sup_closed_compact_iff_well_founded ↔ _ _root_.well_founded.is_sup_cl
 
 variable {α}
 
+#print CompleteLattice.WellFounded.finite_of_setIndependent /-
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
     (hs : SetIndependent s) : s.Finite := by
   classical
@@ -331,6 +342,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     rw [← hs, eq_comm, inf_eq_left]
     exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
+-/
 
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
@@ -366,6 +378,7 @@ theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a
   exact (and_iff_left le_top).symm
 #align Sup_compact_eq_top sSup_compact_eq_top
 
+#print le_iff_compact_le_imp /-
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
   ⟨fun ab c hc ca => le_trans ca ab, fun h =>
@@ -373,6 +386,7 @@ theorem le_iff_compact_le_imp {a b : α} :
     rw [← sSup_compact_le_eq a, ← sSup_compact_le_eq b]
     exact sSup_le_sSup fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
+-/
 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
@@ -487,6 +501,7 @@ end
 
 namespace CompleteLattice
 
+#print CompleteLattice.isCompactlyGenerated_of_wellFounded /-
 theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsCompactlyGenerated α :=
   by
@@ -494,6 +509,7 @@ theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α →
   -- x is the join of the set of compact elements {x}
   exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
+-/
 
 #print CompleteLattice.Iic_coatomic_of_compact_element /-
 /-- A compact element `k` has the property that any `b < k` lies below a "maximal element below

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

@@ -96,12 +96,6 @@ def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
 #align complete_lattice.is_compact_element CompleteLattice.IsCompactElement
 -/
 
-/- warning: complete_lattice.is_compact_element_iff -> CompleteLattice.isCompactElement_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iffₓ'. -/
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
       ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
@@ -124,12 +118,6 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
       exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 
-/- warning: complete_lattice.is_compact_element_iff_le_of_directed_Sup_le -> CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
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-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_leₓ'. -/
 /-- An element `k` is compact if and only if any directed set with `Sup` above
 `k` already got above `k` at some point in the set. -/
 theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
@@ -174,12 +162,6 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
       use t; exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
-/- warning: complete_lattice.is_compact_element.exists_finset_of_le_supr -> CompleteLattice.IsCompactElement.exists_finset_of_le_iSup is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) -> (forall {ι : Type.{u2}} (f : ι -> α), (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u2} (Finset.{u2} ι) (fun (s : Finset.{u2} ι) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i))))))
-but is expected to have type
-  forall (α : Type.{u2}) [_inst_1 : CompleteLattice.{u2} α] {k : α}, (CompleteLattice.IsCompactElement.{u2} α _inst_1 k) -> (forall {ι : Type.{u1}} (f : ι -> α), (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (s : Finset.{u1} ι) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => iSup.{u2, 0} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i))))))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSupₓ'. -/
 theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
@@ -201,12 +183,6 @@ theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElemen
     exact ⟨s, hs⟩
 #align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 
-/- warning: complete_lattice.is_compact_element.directed_Sup_lt_of_lt -> CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1397 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 x._@.Mathlib.Order.CompactlyGenerated._hyg.1397) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_ltₓ'. -/
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
 its Sup strictly below `k`. -/
 theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
@@ -222,12 +198,6 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
   exact hxk.ne (hxk.le.antisymm hkx)
 #align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 
-/- warning: complete_lattice.finset_sup_compact_of_compact -> CompleteLattice.finset_sup_compact_of_compact is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : CompleteLattice.{u1} α] {f : β -> α} (s : Finset.{u2} β), (forall (x : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) x s) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (f x))) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (Finset.sup.{u1, u2} α β (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) s f))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : CompleteLattice.{u2} α] {f : β -> α} (s : Finset.{u1} β), (forall (x : β), (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) x s) -> (CompleteLattice.IsCompactElement.{u2} α _inst_2 (f x))) -> (CompleteLattice.IsCompactElement.{u2} α _inst_2 (Finset.sup.{u2, u1} α β (Lattice.toSemilatticeSup.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_2))) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeSup.toPartialOrder.{u2} α (Lattice.toSemilatticeSup.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u2} α _inst_2)) s f))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compactₓ'. -/
 theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
@@ -244,12 +214,6 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
-/- warning: complete_lattice.well_founded.is_Sup_finite_compact -> CompleteLattice.WellFounded.isSupFiniteCompact is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1797 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1799 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1797 x._@.Mathlib.Order.CompactlyGenerated._hyg.1799)) -> (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
-Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompactₓ'. -/
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsSupFiniteCompact α := fun s =>
   by
@@ -277,12 +241,6 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
 -/
 
-/- warning: complete_lattice.is_sup_closed_compact.well_founded -> CompleteLattice.IsSupClosedCompact.wellFounded is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.2160 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.2162 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.2160 x._@.Mathlib.Order.CompactlyGenerated._hyg.2162))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_sup_closed_compact.well_founded CompleteLattice.IsSupClosedCompact.wellFoundedₓ'. -/
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     WellFounded ((· > ·) : α → α → Prop) :=
   by
@@ -316,12 +274,6 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
 #align complete_lattice.is_Sup_finite_compact_iff_all_elements_compact CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
 -/
 
-/- warning: complete_lattice.well_founded_characterisations -> CompleteLattice.wellFounded_characterisations is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], List.TFAE (List.cons.{0} Prop (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (List.cons.{0} Prop (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) (List.cons.{0} Prop (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (List.cons.{0} Prop (forall (k : α), CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (List.nil.{0} Prop)))))
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], List.TFAE (List.cons.{0} Prop (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.2664 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.2666 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.2664 x._@.Mathlib.Order.CompactlyGenerated._hyg.2666)) (List.cons.{0} Prop (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) (List.cons.{0} Prop (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (List.cons.{0} Prop (forall (k : α), CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (List.nil.{0} Prop)))))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisationsₓ'. -/
 theorem wellFounded_characterisations :
     TFAE
       [WellFounded ((· > ·) : α → α → Prop), IsSupFiniteCompact α, IsSupClosedCompact α,
@@ -334,12 +286,6 @@ theorem wellFounded_characterisations :
   tfae_finish
 #align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisations
 
-/- warning: complete_lattice.well_founded_iff_is_Sup_finite_compact -> CompleteLattice.wellFounded_iff_isSupFiniteCompact is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5612 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 x._@.Mathlib.Order.CompactlyGenerated._hyg.5612)) (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
-Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded_iff_is_Sup_finite_compact CompleteLattice.wellFounded_iff_isSupFiniteCompactₓ'. -/
 theorem wellFounded_iff_isSupFiniteCompact :
     WellFounded ((· > ·) : α → α → Prop) ↔ IsSupFiniteCompact α :=
   (wellFounded_characterisations α).out 0 1
@@ -351,23 +297,11 @@ theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ Is
 #align complete_lattice.is_Sup_finite_compact_iff_is_sup_closed_compact CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact
 -/
 
-/- warning: complete_lattice.is_sup_closed_compact_iff_well_founded -> CompleteLattice.isSupClosedCompact_iff_wellFounded is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5719 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 x._@.Mathlib.Order.CompactlyGenerated._hyg.5719))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFoundedₓ'. -/
 theorem isSupClosedCompact_iff_wellFounded :
     IsSupClosedCompact α ↔ WellFounded ((· > ·) : α → α → Prop) :=
   (wellFounded_characterisations α).out 2 0
 #align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFounded
 
-/- warning: complete_lattice.is_Sup_finite_compact.well_founded -> CompleteLattice.IsSupFiniteCompact.wellFounded is a dubious translation:
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-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFoundedₓ'. -/
 alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
 
@@ -375,23 +309,11 @@ alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔
   _ is_sup_closed_compact.is_Sup_finite_compact
 #align complete_lattice.is_sup_closed_compact.is_Sup_finite_compact CompleteLattice.IsSupClosedCompact.isSupFiniteCompact
 
-/- warning: well_founded.is_sup_closed_compact -> WellFounded.isSupClosedCompact is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompactₓ'. -/
 alias is_sup_closed_compact_iff_well_founded ↔ _ _root_.well_founded.is_sup_closed_compact
 #align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompact
 
 variable {α}
 
-/- warning: complete_lattice.well_founded.finite_of_set_independent -> CompleteLattice.WellFounded.finite_of_setIndependent is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependentₓ'. -/
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
     (hs : SetIndependent s) : s.Finite := by
   classical
@@ -410,12 +332,6 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 
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-Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
   haveI := (well_founded.finite_of_set_independent hwf ht.set_independent_range).to_subtype
@@ -436,12 +352,6 @@ section
 
 variable {α} [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
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-Case conversion may be inaccurate. Consider using '#align Sup_compact_le_eq sSup_compact_le_eqₓ'. -/
 @[simp]
 theorem sSup_compact_le_eq (b) : sSup { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b :=
   by
@@ -449,12 +359,6 @@ theorem sSup_compact_le_eq (b) : sSup { c : α | CompleteLattice.IsCompactElemen
   exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)
 #align Sup_compact_le_eq sSup_compact_le_eq
 
-/- warning: Sup_compact_eq_top -> sSup_compact_eq_top is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align Sup_compact_eq_top sSup_compact_eq_topₓ'. -/
 @[simp]
 theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ :=
   by
@@ -462,12 +366,6 @@ theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a
   exact (and_iff_left le_top).symm
 #align Sup_compact_eq_top sSup_compact_eq_top
 
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-Case conversion may be inaccurate. Consider using '#align le_iff_compact_le_imp le_iff_compact_le_impₓ'. -/
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
   ⟨fun ab c hc ca => le_trans ca ab, fun h =>
@@ -476,12 +374,6 @@ theorem le_iff_compact_le_imp {a b : α} :
     exact sSup_le_sSup fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 
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-Case conversion may be inaccurate. Consider using '#align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
@@ -498,87 +390,39 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
     iSup_inf_le_inf_sSup
 #align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
 
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-Case conversion may be inaccurate. Consider using '#align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
   by simp_rw [@inf_comm _ _ _ a, h.inf_Sup_eq]
 #align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
 
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-Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_iSup_eqₓ'. -/
 protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
   by rw [iSup, h.directed_on_range.inf_Sup_eq, iSup_range]
 #align directed.inf_supr_eq Directed.inf_iSup_eq
 
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-Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.iSup_inf_eqₓ'. -/
 protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
   by rw [iSup, h.directed_on_range.Sup_inf_eq, iSup_range]
 #align directed.supr_inf_eq Directed.iSup_inf_eq
 
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-Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_rightₓ'. -/
 protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :
     Disjoint a (sSup s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
   simp_rw [disjoint_iff, h.inf_Sup_eq, iSup_eq_bot]
 #align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_right
 
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-Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_leftₓ'. -/
 protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
     Disjoint (sSup s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
   simp_rw [disjoint_iff, h.Sup_inf_eq, iSup_eq_bot]
 #align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_left
 
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-Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_iSup_rightₓ'. -/
 protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
   simp_rw [disjoint_iff, h.inf_supr_eq, iSup_eq_bot]
 #align directed.disjoint_supr_right Directed.disjoint_iSup_right
 
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-Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_iSup_leftₓ'. -/
 protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
   simp_rw [disjoint_iff, h.supr_inf_eq, iSup_eq_bot]
 #align directed.disjoint_supr_left Directed.disjoint_iSup_left
 
-/- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_sSup_eq_iSup_inf_sup_finset is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finsetₓ'. -/
 /-- This property is equivalent to `α` being upper continuous. -/
 theorem inf_sSup_eq_iSup_inf_sup_finset :
     a ⊓ sSup s = ⨆ (t : Finset α) (H : ↑t ⊆ s), a ⊓ t.sup id :=
@@ -643,12 +487,6 @@ end
 
 namespace CompleteLattice
 
-/- warning: complete_lattice.compactly_generated_of_well_founded -> CompleteLattice.isCompactlyGenerated_of_wellFounded is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (IsCompactlyGenerated.{u1} α _inst_1)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFoundedₓ'. -/
 theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsCompactlyGenerated α :=
   by
@@ -679,12 +517,6 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCo
 #align complete_lattice.Iic_coatomic_of_compact_element CompleteLattice.Iic_coatomic_of_compact_element
 -/
 
-/- warning: complete_lattice.coatomic_of_top_compact -> CompleteLattice.coatomic_of_top_compact is a dubious translation:
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-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsCompactElement.{u1} α _inst_1 (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (IsCoatomic.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderTop.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsCompactElement.{u1} α _inst_1 (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (IsCoatomic.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.coatomic_of_top_compact CompleteLattice.coatomic_of_top_compactₓ'. -/
 theorem coatomic_of_top_compact (h : IsCompactElement (⊤ : α)) : IsCoatomic α :=
   (@OrderIso.IicTop α _ _).IsCoatomic_iff.mp (Iic_coatomic_of_compact_element h)
 #align complete_lattice.coatomic_of_top_compact CompleteLattice.coatomic_of_top_compact
@@ -743,12 +575,6 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 -/
 
 
-/- warning: exists_set_independent_is_compl_Sup_atoms -> exists_setIndependent_isCompl_sSup_atoms 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 exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atomsₓ'. -/
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
@@ -802,24 +628,12 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
     · exact ha
 #align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atoms
 
-/- warning: exists_set_independent_of_Sup_atoms_eq_top -> exists_setIndependent_of_sSup_atoms_eq_top is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_topₓ'. -/
 theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h ⊥
   ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
 #align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_top
 
-/- warning: complemented_lattice_of_Sup_atoms_eq_top -> complementedLattice_of_sSup_atoms_eq_top is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_topₓ'. -/
 /-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
     ComplementedLattice α :=

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

@@ -118,11 +118,7 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
         rw [← show s (f ⟨b, hb⟩) = id b from hf _]
         exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ ⟨b, hb⟩)
     · intro H s hs
-      obtain ⟨t, ht⟩ :=
-        H s coe
-          (by
-            delta iSup
-            rwa [Subtype.range_coe])
+      obtain ⟨t, ht⟩ := H s coe (by delta iSup; rwa [Subtype.range_coe])
       refine' ⟨t.image coe, by simp, ht.trans _⟩
       rw [Finset.sup_le_iff]
       exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
@@ -164,21 +160,18 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
         simp only [and_self_iff, le_sup_left, le_sup_right]
       have sup_S : Sup s ≤ Sup S := by
         apply sSup_le_sSup
-        intro x hx
-        use {x}
+        intro x hx; use {x}
         simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
           Finset.sup_singleton, Set.singleton_subset_iff]
       have Sne : S.nonempty := by
-        suffices : ⊥ ∈ S
-        exact Set.nonempty_of_mem this
+        suffices : ⊥ ∈ S; exact Set.nonempty_of_mem this
         use ∅
         simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
           and_self_iff]
       -- Now apply the defn of compact and finish.
       obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)
       obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
-      use t
-      exact ⟨htS, by rwa [← htsup]⟩
+      use t; exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
 /- warning: complete_lattice.is_compact_element.exists_finset_of_le_supr -> CompleteLattice.IsCompactElement.exists_finset_of_le_iSup is a dubious translation:
@@ -278,12 +271,9 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
   by
   intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
   cases' t.eq_empty_or_nonempty with h h
-  · subst h
-    rw [Finset.sup_empty] at ht₂
-    rw [ht₂]
+  · subst h; rw [Finset.sup_empty] at ht₂; rw [ht₂]
     simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
-  · rw [ht₂]
-    exact t.sup_closed_of_sup_closed h ht₁ hsc
+  · rw [ht₂]; exact t.sup_closed_of_sup_closed h ht₁ hsc
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
 -/
 
@@ -300,21 +290,11 @@ theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
   suffices Sup (Set.range a) ∈ Set.range a
     by
     obtain ⟨n, hn⟩ := set.mem_range.mp this
-    have h' : Sup (Set.range a) < a (n + 1) :=
-      by
-      change _ > _
-      simp [← hn, a.map_rel_iff]
-    apply lt_irrefl (a (n + 1))
-    apply lt_of_le_of_lt _ h'
-    apply le_sSup
-    apply Set.mem_range_self
+    have h' : Sup (Set.range a) < a (n + 1) := by change _ > _; simp [← hn, a.map_rel_iff]
+    apply lt_irrefl (a (n + 1)); apply lt_of_le_of_lt _ h'; apply le_sSup; apply Set.mem_range_self
   apply h (Set.range a)
-  · use a 37
-    apply Set.mem_range_self
-  · rintro x ⟨m, hm⟩ y ⟨n, hn⟩
-    use m ⊔ n
-    rw [← hm, ← hn]
-    apply RelHomClass.map_sup a
+  · use a 37; apply Set.mem_range_self
+  · rintro x ⟨m, hm⟩ y ⟨n, hn⟩; use m ⊔ n; rw [← hm, ← hn]; apply RelHomClass.map_sup a
 #align complete_lattice.is_sup_closed_compact.well_founded CompleteLattice.IsSupClosedCompact.wellFounded
 
 #print CompleteLattice.isSupFiniteCompact_iff_all_elements_compact /-
@@ -647,8 +627,7 @@ theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type _} {s : η
       (h i).mono
         (Set.Subset.trans hI <| Set.iUnion₂_subset fun j hj => hi j (fi.mem_to_finset.2 hj))
   · rintro a ⟨_, ⟨i, _⟩, _⟩
-    exfalso
-    exact hη ⟨i⟩
+    exfalso; exact hη ⟨i⟩
 #align complete_lattice.set_independent_Union_of_directed CompleteLattice.setIndependent_iUnion_of_directed
 -/
 
@@ -684,9 +663,7 @@ theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α →
 theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCoatomic (Set.Iic k) :=
   ⟨fun ⟨b, hbk⟩ => by
     by_cases htriv : b = k
-    · left
-      ext
-      simp only [htriv, Set.Iic.coe_top, Subtype.coe_mk]
+    · left; ext; simp only [htriv, Set.Iic.coe_top, Subtype.coe_mk]
     right
     obtain ⟨a, a₀, ba, h⟩ := zorn_nonempty_partialOrder₀ (Set.Iio k) _ b (lt_of_le_of_ne hbk htriv)
     · refine' ⟨⟨a, le_of_lt a₀⟩, ⟨ne_of_lt a₀, fun c hck => by_contradiction fun c₀ => _⟩, ba⟩

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

@@ -98,7 +98,7 @@ def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
 
 /- warning: complete_lattice.is_compact_element_iff -> CompleteLattice.isCompactElement_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iffₓ'. -/
@@ -130,7 +130,7 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
 
 /- warning: complete_lattice.is_compact_element_iff_le_of_directed_Sup_le -> CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
 but is expected to have type
   forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.541 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.543 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.541 x._@.Mathlib.Order.CompactlyGenerated._hyg.543) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_leₓ'. -/
@@ -183,7 +183,7 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
 
 /- warning: complete_lattice.is_compact_element.exists_finset_of_le_supr -> CompleteLattice.IsCompactElement.exists_finset_of_le_iSup is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) -> (forall {ι : Type.{u2}} (f : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u2} (Finset.{u2} ι) (fun (s : Finset.{u2} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i))))))
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) -> (forall {ι : Type.{u2}} (f : ι -> α), (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u2} (Finset.{u2} ι) (fun (s : Finset.{u2} ι) => LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i))))))
 but is expected to have type
   forall (α : Type.{u2}) [_inst_1 : CompleteLattice.{u2} α] {k : α}, (CompleteLattice.IsCompactElement.{u2} α _inst_1 k) -> (forall {ι : Type.{u1}} (f : ι -> α), (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (s : Finset.{u1} ι) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => iSup.{u2, 0} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i))))))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSupₓ'. -/
@@ -210,7 +210,7 @@ theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElemen
 
 /- warning: complete_lattice.is_compact_element.directed_Sup_lt_of_lt -> CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1397 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 x._@.Mathlib.Order.CompactlyGenerated._hyg.1397) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_ltₓ'. -/
@@ -231,7 +231,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
 
 /- warning: complete_lattice.finset_sup_compact_of_compact -> CompleteLattice.finset_sup_compact_of_compact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : CompleteLattice.{u1} α] {f : β -> α} (s : Finset.{u2} β), (forall (x : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) x s) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (f x))) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (Finset.sup.{u1, u2} α β (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) s f))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : CompleteLattice.{u1} α] {f : β -> α} (s : Finset.{u2} β), (forall (x : β), (Membership.Mem.{u2, u2} β (Finset.{u2} β) (Finset.hasMem.{u2} β) x s) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (f x))) -> (CompleteLattice.IsCompactElement.{u1} α _inst_2 (Finset.sup.{u1, u2} α β (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) s f))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : CompleteLattice.{u2} α] {f : β -> α} (s : Finset.{u1} β), (forall (x : β), (Membership.mem.{u1, u1} β (Finset.{u1} β) (Finset.instMembershipFinset.{u1} β) x s) -> (CompleteLattice.IsCompactElement.{u2} α _inst_2 (f x))) -> (CompleteLattice.IsCompactElement.{u2} α _inst_2 (Finset.sup.{u2, u1} α β (Lattice.toSemilatticeSup.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_2))) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeSup.toPartialOrder.{u2} α (Lattice.toSemilatticeSup.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u2} α _inst_2)) s f))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compactₓ'. -/
@@ -251,7 +251,12 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
-#print CompleteLattice.WellFounded.isSupFiniteCompact /-
+/- warning: complete_lattice.well_founded.is_Sup_finite_compact -> CompleteLattice.WellFounded.isSupFiniteCompact is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1797 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1799 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1797 x._@.Mathlib.Order.CompactlyGenerated._hyg.1799)) -> (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompactₓ'. -/
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsSupFiniteCompact α := fun s =>
   by
@@ -267,7 +272,6 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
--/
 
 #print CompleteLattice.IsSupFiniteCompact.isSupClosedCompact /-
 theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSupClosedCompact α :=
@@ -283,7 +287,12 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
 -/
 
-#print CompleteLattice.IsSupClosedCompact.wellFounded /-
+/- warning: complete_lattice.is_sup_closed_compact.well_founded -> CompleteLattice.IsSupClosedCompact.wellFounded is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.2160 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.2162 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.2160 x._@.Mathlib.Order.CompactlyGenerated._hyg.2162))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_sup_closed_compact.well_founded CompleteLattice.IsSupClosedCompact.wellFoundedₓ'. -/
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     WellFounded ((· > ·) : α → α → Prop) :=
   by
@@ -307,7 +316,6 @@ theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     rw [← hm, ← hn]
     apply RelHomClass.map_sup a
 #align complete_lattice.is_sup_closed_compact.well_founded CompleteLattice.IsSupClosedCompact.wellFounded
--/
 
 #print CompleteLattice.isSupFiniteCompact_iff_all_elements_compact /-
 theorem isSupFiniteCompact_iff_all_elements_compact :
@@ -328,7 +336,12 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
 #align complete_lattice.is_Sup_finite_compact_iff_all_elements_compact CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
 -/
 
-#print CompleteLattice.wellFounded_characterisations /-
+/- warning: complete_lattice.well_founded_characterisations -> CompleteLattice.wellFounded_characterisations is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], List.TFAE (List.cons.{0} Prop (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (List.cons.{0} Prop (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) (List.cons.{0} Prop (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (List.cons.{0} Prop (forall (k : α), CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (List.nil.{0} Prop)))))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], List.TFAE (List.cons.{0} Prop (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.2664 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.2666 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.2664 x._@.Mathlib.Order.CompactlyGenerated._hyg.2666)) (List.cons.{0} Prop (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) (List.cons.{0} Prop (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (List.cons.{0} Prop (forall (k : α), CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (List.nil.{0} Prop)))))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisationsₓ'. -/
 theorem wellFounded_characterisations :
     TFAE
       [WellFounded ((· > ·) : α → α → Prop), IsSupFiniteCompact α, IsSupClosedCompact α,
@@ -340,14 +353,17 @@ theorem wellFounded_characterisations :
   tfae_have 2 ↔ 4; · exact is_Sup_finite_compact_iff_all_elements_compact α
   tfae_finish
 #align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisations
--/
 
-#print CompleteLattice.wellFounded_iff_isSupFiniteCompact /-
+/- warning: complete_lattice.well_founded_iff_is_Sup_finite_compact -> CompleteLattice.wellFounded_iff_isSupFiniteCompact is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5612 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 x._@.Mathlib.Order.CompactlyGenerated._hyg.5612)) (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded_iff_is_Sup_finite_compact CompleteLattice.wellFounded_iff_isSupFiniteCompactₓ'. -/
 theorem wellFounded_iff_isSupFiniteCompact :
     WellFounded ((· > ·) : α → α → Prop) ↔ IsSupFiniteCompact α :=
   (wellFounded_characterisations α).out 0 1
 #align complete_lattice.well_founded_iff_is_Sup_finite_compact CompleteLattice.wellFounded_iff_isSupFiniteCompact
--/
 
 #print CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact /-
 theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ IsSupClosedCompact α :=
@@ -355,13 +371,23 @@ theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ Is
 #align complete_lattice.is_Sup_finite_compact_iff_is_sup_closed_compact CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact
 -/
 
-#print CompleteLattice.isSupClosedCompact_iff_wellFounded /-
+/- warning: complete_lattice.is_sup_closed_compact_iff_well_founded -> CompleteLattice.isSupClosedCompact_iff_wellFounded is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], Iff (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1) (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5719 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 x._@.Mathlib.Order.CompactlyGenerated._hyg.5719))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFoundedₓ'. -/
 theorem isSupClosedCompact_iff_wellFounded :
     IsSupClosedCompact α ↔ WellFounded ((· > ·) : α → α → Prop) :=
   (wellFounded_characterisations α).out 2 0
 #align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFounded
--/
 
+/- warning: complete_lattice.is_Sup_finite_compact.well_founded -> CompleteLattice.IsSupFiniteCompact.wellFounded is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsSupFiniteCompact.{u1} α _inst_1) -> (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5612 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5610 x._@.Mathlib.Order.CompactlyGenerated._hyg.5612))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFoundedₓ'. -/
 alias well_founded_iff_is_Sup_finite_compact ↔ _ is_Sup_finite_compact.well_founded
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
 
@@ -369,12 +395,23 @@ alias is_Sup_finite_compact_iff_is_sup_closed_compact ↔
   _ is_sup_closed_compact.is_Sup_finite_compact
 #align complete_lattice.is_sup_closed_compact.is_Sup_finite_compact CompleteLattice.IsSupClosedCompact.isSupFiniteCompact
 
+/- warning: well_founded.is_sup_closed_compact -> WellFounded.isSupClosedCompact is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1)
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5719 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5717 x._@.Mathlib.Order.CompactlyGenerated._hyg.5719)) -> (CompleteLattice.IsSupClosedCompact.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompactₓ'. -/
 alias is_sup_closed_compact_iff_well_founded ↔ _ _root_.well_founded.is_sup_closed_compact
 #align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompact
 
 variable {α}
 
-#print CompleteLattice.WellFounded.finite_of_setIndependent /-
+/- warning: complete_lattice.well_founded.finite_of_set_independent -> CompleteLattice.WellFounded.finite_of_setIndependent is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {s : Set.{u1} α}, (CompleteLattice.SetIndependent.{u1} α _inst_1 s) -> (Set.Finite.{u1} α s))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.5794 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.5796 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.5794 x._@.Mathlib.Order.CompactlyGenerated._hyg.5796)) -> (forall {s : Set.{u1} α}, (CompleteLattice.SetIndependent.{u1} α _inst_1 s) -> (Set.Finite.{u1} α s))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependentₓ'. -/
 theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
     (hs : SetIndependent s) : s.Finite := by
   classical
@@ -392,11 +429,10 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     rw [← hs, eq_comm, inf_eq_left]
     exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
--/
 
 /- warning: complete_lattice.well_founded.finite_of_independent -> CompleteLattice.WellFounded.finite_of_independent is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
 but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6107 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6109 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6107 x._@.Mathlib.Order.CompactlyGenerated._hyg.6109)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
@@ -422,7 +458,7 @@ variable {α} [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
 /- warning: Sup_compact_le_eq -> sSup_compact_le_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
 Case conversion may be inaccurate. Consider using '#align Sup_compact_le_eq sSup_compact_le_eqₓ'. -/
@@ -446,7 +482,12 @@ theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a
   exact (and_iff_left le_top).symm
 #align Sup_compact_eq_top sSup_compact_eq_top
 
-#print le_iff_compact_le_imp /-
+/- warning: le_iff_compact_le_imp -> le_iff_compact_le_imp is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) a b) (forall (c : α), (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c a) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) a b) (forall (c : α), (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c a) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b))
+Case conversion may be inaccurate. Consider using '#align le_iff_compact_le_imp le_iff_compact_le_impₓ'. -/
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
   ⟨fun ab c hc ca => le_trans ca ab, fun h =>
@@ -454,11 +495,10 @@ theorem le_iff_compact_le_imp {a b : α} :
     rw [← sSup_compact_le_eq a, ← sSup_compact_le_eq b]
     exact sSup_le_sSup fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
--/
 
 /- warning: directed_on.inf_Sup_eq -> DirectedOn.inf_sSup_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6528 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 x._@.Mathlib.Order.CompactlyGenerated._hyg.6528) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
 Case conversion may be inaccurate. Consider using '#align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eqₓ'. -/
@@ -480,7 +520,7 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
 
 /- warning: directed_on.Sup_inf_eq -> DirectedOn.sSup_inf_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) b a))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) b a))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6796 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 x._@.Mathlib.Order.CompactlyGenerated._hyg.6796) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) b a))))
 Case conversion may be inaccurate. Consider using '#align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eqₓ'. -/
@@ -491,7 +531,7 @@ protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s
 
 /- warning: directed.inf_supr_eq -> Directed.inf_iSup_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
 but is expected to have type
   forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 x._@.Mathlib.Order.CompactlyGenerated._hyg.6887) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
 Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_iSup_eqₓ'. -/
@@ -501,7 +541,7 @@ protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆
 
 /- warning: directed.supr_inf_eq -> Directed.iSup_inf_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
 but is expected to have type
   forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 x._@.Mathlib.Order.CompactlyGenerated._hyg.7002) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
 Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.iSup_inf_eqₓ'. -/
@@ -511,7 +551,7 @@ protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f
 
 /- warning: directed_on.disjoint_Sup_right -> DirectedOn.disjoint_sSup_right is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 x._@.Mathlib.Order.CompactlyGenerated._hyg.7119) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
 Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_rightₓ'. -/
@@ -522,7 +562,7 @@ protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s)
 
 /- warning: directed_on.disjoint_Sup_left -> DirectedOn.disjoint_sSup_left is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7190 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 x._@.Mathlib.Order.CompactlyGenerated._hyg.7190) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
 Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_leftₓ'. -/
@@ -533,7 +573,7 @@ protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
 
 /- warning: directed.disjoint_supr_right -> Directed.disjoint_iSup_right is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
 but is expected to have type
   forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7261 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 x._@.Mathlib.Order.CompactlyGenerated._hyg.7261) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
 Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_iSup_rightₓ'. -/
@@ -544,7 +584,7 @@ protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
 
 /- warning: directed.disjoint_supr_left -> Directed.disjoint_iSup_left is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
 but is expected to have type
   forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7345 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 x._@.Mathlib.Order.CompactlyGenerated._hyg.7345) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
 Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_iSup_leftₓ'. -/
@@ -555,7 +595,7 @@ protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
 
 /- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_sSup_eq_iSup_inf_sup_finset is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finsetₓ'. -/
@@ -624,7 +664,12 @@ end
 
 namespace CompleteLattice
 
-#print CompleteLattice.isCompactlyGenerated_of_wellFounded /-
+/- warning: complete_lattice.compactly_generated_of_well_founded -> CompleteLattice.isCompactlyGenerated_of_wellFounded is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toHasLt.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (IsCompactlyGenerated.{u1} α _inst_1)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.8170 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.8172 : α) => GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.8170 x._@.Mathlib.Order.CompactlyGenerated._hyg.8172)) -> (IsCompactlyGenerated.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFoundedₓ'. -/
 theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsCompactlyGenerated α :=
   by
@@ -632,7 +677,6 @@ theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α →
   -- x is the join of the set of compact elements {x}
   exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
--/
 
 #print CompleteLattice.Iic_coatomic_of_compact_element /-
 /-- A compact element `k` has the property that any `b < k` lies below a "maximal element below
@@ -660,7 +704,7 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCo
 
 /- warning: complete_lattice.coatomic_of_top_compact -> CompleteLattice.coatomic_of_top_compact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsCompactElement.{u1} α _inst_1 (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (IsCoatomic.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsCompactElement.{u1} α _inst_1 (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (IsCoatomic.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderTop.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (CompleteLattice.IsCompactElement.{u1} α _inst_1 (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (IsCoatomic.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.coatomic_of_top_compact CompleteLattice.coatomic_of_top_compactₓ'. -/
@@ -724,7 +768,7 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 
 /- warning: exists_set_independent_is_compl_Sup_atoms -> exists_setIndependent_isCompl_sSup_atoms is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 Case conversion may be inaccurate. Consider using '#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atomsₓ'. -/
@@ -783,7 +827,7 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
 
 /- warning: exists_set_independent_of_Sup_atoms_eq_top -> exists_setIndependent_of_sSup_atoms_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 Case conversion may be inaccurate. Consider using '#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_topₓ'. -/
@@ -795,7 +839,7 @@ theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a
 
 /- warning: complemented_lattice_of_Sup_atoms_eq_top -> complementedLattice_of_sSup_atoms_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_topₓ'. -/

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

@@ -75,7 +75,7 @@ variable (α)
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/
 def IsSupClosedCompact : Prop :=
-  ∀ (s : Set α) (h : s.Nonempty), (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), a ⊔ b ∈ s) → supₛ s ∈ s
+  ∀ (s : Set α) (h : s.Nonempty), (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), a ⊔ b ∈ s) → sSup s ∈ s
 #align complete_lattice.is_sup_closed_compact CompleteLattice.IsSupClosedCompact
 -/
 
@@ -83,7 +83,7 @@ def IsSupClosedCompact : Prop :=
 /-- A compactness property for a complete lattice is that any subset has a finite subset with the
 same `Sup`. -/
 def IsSupFiniteCompact : Prop :=
-  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ supₛ s = t.sup id
+  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id
 #align complete_lattice.is_Sup_finite_compact CompleteLattice.IsSupFiniteCompact
 -/
 
@@ -92,19 +92,19 @@ def IsSupFiniteCompact : Prop :=
 above `k` has a finite subset with `Sup` above `k`.  Such an element is also called
 "finite" or "S-compact". -/
 def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
-  ∀ s : Set α, k ≤ supₛ s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id
+  ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id
 #align complete_lattice.is_compact_element CompleteLattice.IsCompactElement
 -/
 
 /- warning: complete_lattice.is_compact_element_iff -> CompleteLattice.isCompactElement_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) (forall (ι : Type.{u1}) (s : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) ι s)) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (t : Finset.{u1} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) k (Finset.sup.{u1, u1} α ι (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_2)) t s))))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iffₓ'. -/
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
-      ∀ (ι : Type u) (s : ι → α), k ≤ supᵢ s → ∃ t : Finset ι, k ≤ t.sup s :=
+      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s :=
   by
   classical
     constructor
@@ -121,24 +121,24 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
       obtain ⟨t, ht⟩ :=
         H s coe
           (by
-            delta supᵢ
+            delta iSup
             rwa [Subtype.range_coe])
       refine' ⟨t.image coe, by simp, ht.trans _⟩
       rw [Finset.sup_le_iff]
       exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem coe hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 
-/- warning: complete_lattice.is_compact_element_iff_le_of_directed_Sup_le -> CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_le is a dubious translation:
+/- warning: complete_lattice.is_compact_element_iff_le_of_directed_Sup_le -> CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
 but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.541 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.543 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.541 x._@.Mathlib.Order.CompactlyGenerated._hyg.543) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_leₓ'. -/
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.541 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.543 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.541 x._@.Mathlib.Order.CompactlyGenerated._hyg.543) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_leₓ'. -/
 /-- An element `k` is compact if and only if any directed set with `Sup` above
 `k` already got above `k` at some point in the set. -/
-theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
+theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
     IsCompactElement k ↔
-      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ supₛ s → ∃ x : α, x ∈ s ∧ k ≤ x :=
+      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x :=
   by
   classical
     constructor
@@ -163,7 +163,7 @@ theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
           · simp only [hc.right, hd.right, Finset.sup_union]
         simp only [and_self_iff, le_sup_left, le_sup_right]
       have sup_S : Sup s ≤ Sup S := by
-        apply supₛ_le_supₛ
+        apply sSup_le_sSup
         intro x hx
         use {x}
         simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
@@ -179,15 +179,15 @@ theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
       obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
       use t
       exact ⟨htS, by rwa [← htsup]⟩
-#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_le
+#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
-/- warning: complete_lattice.is_compact_element.exists_finset_of_le_supr -> CompleteLattice.IsCompactElement.exists_finset_of_le_supᵢ is a dubious translation:
+/- warning: complete_lattice.is_compact_element.exists_finset_of_le_supr -> CompleteLattice.IsCompactElement.exists_finset_of_le_iSup is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) -> (forall {ι : Type.{u2}} (f : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (supᵢ.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u2} (Finset.{u2} ι) (fun (s : Finset.{u2} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (supᵢ.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i))))))
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) -> (forall {ι : Type.{u2}} (f : ι -> α), (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u2} (Finset.{u2} ι) (fun (s : Finset.{u2} ι) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (iSup.{u1, succ u2} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) ι (fun (i : ι) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i))))))
 but is expected to have type
-  forall (α : Type.{u2}) [_inst_1 : CompleteLattice.{u2} α] {k : α}, (CompleteLattice.IsCompactElement.{u2} α _inst_1 k) -> (forall {ι : Type.{u1}} (f : ι -> α), (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (supᵢ.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (s : Finset.{u1} ι) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (supᵢ.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => supᵢ.{u2, 0} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i))))))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_supᵢₓ'. -/
-theorem IsCompactElement.exists_finset_of_le_supᵢ {k : α} (hk : IsCompactElement k) {ι : Type _}
+  forall (α : Type.{u2}) [_inst_1 : CompleteLattice.{u2} α] {k : α}, (CompleteLattice.IsCompactElement.{u2} α _inst_1 k) -> (forall {ι : Type.{u1}} (f : ι -> α), (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) -> (Exists.{succ u1} (Finset.{u1} ι) (fun (s : Finset.{u1} ι) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) k (iSup.{u2, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => iSup.{u2, 0} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i))))))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSupₓ'. -/
+theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
     let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
@@ -195,39 +195,39 @@ theorem IsCompactElement.exists_finset_of_le_supᵢ {k : α} (hk : IsCompactElem
       by
       rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
       exact
-        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, supᵢ_le_supᵢ_of_subset (Finset.subset_union_left s t),
-          supᵢ_le_supᵢ_of_subset (Finset.subset_union_right s t)⟩
+        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
+          iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
     have h2 : k ≤ Sup (Set.range g) :=
       h.trans
-        (supᵢ_le fun i =>
-          le_supₛ_of_le ⟨{i}, rfl⟩
-            (le_supᵢ_of_le i (le_supᵢ_of_le (Finset.mem_singleton_self i) le_rfl)))
+        (iSup_le fun i =>
+          le_sSup_of_le ⟨{i}, rfl⟩
+            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
     obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
       (is_compact_element_iff_le_of_directed_Sup_le α k).mp hk (Set.range g) (Set.range_nonempty g)
         h1 h2
     exact ⟨s, hs⟩
-#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_supᵢ
+#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 
-/- warning: complete_lattice.is_compact_element.directed_Sup_lt_of_lt -> CompleteLattice.IsCompactElement.directed_supₛ_lt_of_lt is a dubious translation:
+/- warning: complete_lattice.is_compact_element.directed_Sup_lt_of_lt -> CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1397 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 x._@.Mathlib.Order.CompactlyGenerated._hyg.1397) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
-Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_supₛ_lt_of_ltₓ'. -/
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1397 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 x._@.Mathlib.Order.CompactlyGenerated._hyg.1397) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
+Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_ltₓ'. -/
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
 its Sup strictly below `k`. -/
-theorem IsCompactElement.directed_supₛ_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
+theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
     (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
-    (hbelow : ∀ x ∈ s, x < k) : supₛ s < k :=
+    (hbelow : ∀ x ∈ s, x < k) : sSup s < k :=
   by
   rw [is_compact_element_iff_le_of_directed_Sup_le] at hk
   by_contra
-  have sSup : Sup s ≤ k := supₛ_le fun s hs => (hbelow s hs).le
+  have sSup : Sup s ≤ k := sSup_le fun s hs => (hbelow s hs).le
   replace sSup : Sup s = k := eq_iff_le_not_lt.mpr ⟨sSup, h⟩
   obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le
   obtain hxk := hbelow x hxs
   exact hxk.ne (hxk.le.antisymm hkx)
-#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_supₛ_lt_of_lt
+#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 
 /- warning: complete_lattice.finset_sup_compact_of_compact -> CompleteLattice.finset_sup_compact_of_compact is a dubious translation:
 lean 3 declaration is
@@ -258,14 +258,14 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
   obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ :=
     well_founded.well_founded_iff_has_min.mp h { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
       ⟨⊥, ∅, by simp⟩
-  refine' ⟨t, ht₁, (supₛ_le fun y hy => _).antisymm _⟩
+  refine' ⟨t, ht₁, (sSup_le fun y hy => _).antisymm _⟩
   ·
     classical
       rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
           (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
       simp
-  · rw [Finset.sup_id_eq_supₛ]
-    exact supₛ_le_supₛ ht₁
+  · rw [Finset.sup_id_eq_sSup]
+    exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
 -/
 
@@ -277,7 +277,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
   · subst h
     rw [Finset.sup_empty] at ht₂
     rw [ht₂]
-    simp [eq_singleton_bot_of_supₛ_eq_bot_of_nonempty ht₂ hne]
+    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
   · rw [ht₂]
     exact t.sup_closed_of_sup_closed h ht₁ hsc
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
@@ -297,7 +297,7 @@ theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
       simp [← hn, a.map_rel_iff]
     apply lt_irrefl (a (n + 1))
     apply lt_of_le_of_lt _ h'
-    apply le_supₛ
+    apply le_sSup
     apply Set.mem_range_self
   apply h (Set.range a)
   · use a 37
@@ -323,7 +323,7 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
       suffices t.sup id ≤ Sup s by apply le_antisymm <;> assumption
       simp only [id.def, Finset.sup_le_iff]
       intro x hx
-      exact le_supₛ (hts hx)
+      exact le_sSup (hts hx)
     use t, hts, this
 #align complete_lattice.is_Sup_finite_compact_iff_all_elements_compact CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
 -/
@@ -390,7 +390,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
       simpa [Disjoint, hx₂, ← t.sup_id_eq_Sup, ← ht₂] using this.eq_bot
     apply hx₁
     rw [← hs, eq_comm, inf_eq_left]
-    exact le_supₛ hx₀
+    exact le_sSup hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 -/
 
@@ -412,7 +412,7 @@ end CompleteLattice
 /-- A complete lattice is said to be compactly generated if any
 element is the `Sup` of compact elements. -/
 class IsCompactlyGenerated (α : Type _) [CompleteLattice α] : Prop where
-  exists_supₛ_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ supₛ s = x
+  exists_sSup_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ sSup s = x
 #align is_compactly_generated IsCompactlyGenerated
 -/
 
@@ -420,158 +420,158 @@ section
 
 variable {α} [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
-/- warning: Sup_compact_le_eq -> supₛ_compact_le_eq is a dubious translation:
+/- warning: Sup_compact_le_eq -> sSup_compact_le_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
-Case conversion may be inaccurate. Consider using '#align Sup_compact_le_eq supₛ_compact_le_eqₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] (b : α), Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (c : α) => And (CompleteLattice.IsCompactElement.{u1} α _inst_1 c) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) c b)))) b
+Case conversion may be inaccurate. Consider using '#align Sup_compact_le_eq sSup_compact_le_eqₓ'. -/
 @[simp]
-theorem supₛ_compact_le_eq (b) : supₛ { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b :=
+theorem sSup_compact_le_eq (b) : sSup { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b :=
   by
-  rcases IsCompactlyGenerated.exists_supₛ_eq b with ⟨s, hs, rfl⟩
-  exact le_antisymm (supₛ_le fun c hc => hc.2) (supₛ_le_supₛ fun c cs => ⟨hs c cs, le_supₛ cs⟩)
-#align Sup_compact_le_eq supₛ_compact_le_eq
+  rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩
+  exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)
+#align Sup_compact_le_eq sSup_compact_le_eq
 
-/- warning: Sup_compact_eq_top -> supₛ_compact_eq_top is a dubious translation:
+/- warning: Sup_compact_eq_top -> sSup_compact_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1], Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => CompleteLattice.IsCompactElement.{u1} α _inst_1 a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1], Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => CompleteLattice.IsCompactElement.{u1} α _inst_1 a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1], Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => CompleteLattice.IsCompactElement.{u1} α _inst_1 a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))
-Case conversion may be inaccurate. Consider using '#align Sup_compact_eq_top supₛ_compact_eq_topₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1], Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => CompleteLattice.IsCompactElement.{u1} α _inst_1 a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))
+Case conversion may be inaccurate. Consider using '#align Sup_compact_eq_top sSup_compact_eq_topₓ'. -/
 @[simp]
-theorem supₛ_compact_eq_top : supₛ { a : α | CompleteLattice.IsCompactElement a } = ⊤ :=
+theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ :=
   by
-  refine' Eq.trans (congr rfl (Set.ext fun x => _)) (supₛ_compact_le_eq ⊤)
+  refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq ⊤)
   exact (and_iff_left le_top).symm
-#align Sup_compact_eq_top supₛ_compact_eq_top
+#align Sup_compact_eq_top sSup_compact_eq_top
 
 #print le_iff_compact_le_imp /-
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
   ⟨fun ab c hc ca => le_trans ca ab, fun h =>
     by
-    rw [← supₛ_compact_le_eq a, ← supₛ_compact_le_eq b]
-    exact supₛ_le_supₛ fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
+    rw [← sSup_compact_le_eq a, ← sSup_compact_le_eq b]
+    exact sSup_le_sSup fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 -/
 
-/- warning: directed_on.inf_Sup_eq -> DirectedOn.inf_supₛ_eq is a dubious translation:
+/- warning: directed_on.inf_Sup_eq -> DirectedOn.inf_sSup_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6528 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 x._@.Mathlib.Order.CompactlyGenerated._hyg.6528) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
-Case conversion may be inaccurate. Consider using '#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eqₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6528 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 x._@.Mathlib.Order.CompactlyGenerated._hyg.6528) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+Case conversion may be inaccurate. Consider using '#align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
+theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
       by_cases hs : s.nonempty
       · intro c hc hcinf
         rw [le_inf_iff] at hcinf
-        rw [CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_le] at hc
+        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc
         rcases hc s hs h hcinf.2 with ⟨d, ds, cd⟩
-        exact (le_inf hcinf.1 cd).trans (le_supᵢ₂ d ds)
+        exact (le_inf hcinf.1 cd).trans (le_iSup₂ d ds)
       · rw [Set.not_nonempty_iff_eq_empty] at hs
         simp [hs])
-    supᵢ_inf_le_inf_supₛ
-#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eq
+    iSup_inf_le_inf_sSup
+#align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
 
-/- warning: directed_on.Sup_inf_eq -> DirectedOn.supₛ_inf_eq is a dubious translation:
+/- warning: directed_on.Sup_inf_eq -> DirectedOn.sSup_inf_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) b a))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) b a))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6796 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 x._@.Mathlib.Order.CompactlyGenerated._hyg.6796) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) b a))))
-Case conversion may be inaccurate. Consider using '#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eqₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6796 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 x._@.Mathlib.Order.CompactlyGenerated._hyg.6796) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) b a))))
+Case conversion may be inaccurate. Consider using '#align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : supₛ s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
+protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
   by simp_rw [@inf_comm _ _ _ a, h.inf_Sup_eq]
-#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eq
+#align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
 
-/- warning: directed.inf_supr_eq -> Directed.inf_supᵢ_eq is a dubious translation:
+/- warning: directed.inf_supr_eq -> Directed.inf_iSup_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 x._@.Mathlib.Order.CompactlyGenerated._hyg.6887) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
-Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_supᵢ_eqₓ'. -/
-protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
-  by rw [supᵢ, h.directed_on_range.inf_Sup_eq, supᵢ_range]
-#align directed.inf_supr_eq Directed.inf_supᵢ_eq
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 x._@.Mathlib.Order.CompactlyGenerated._hyg.6887) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
+Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_iSup_eqₓ'. -/
+protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
+  by rw [iSup, h.directed_on_range.inf_Sup_eq, iSup_range]
+#align directed.inf_supr_eq Directed.inf_iSup_eq
 
-/- warning: directed.supr_inf_eq -> Directed.supᵢ_inf_eq is a dubious translation:
+/- warning: directed.supr_inf_eq -> Directed.iSup_inf_eq is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 x._@.Mathlib.Order.CompactlyGenerated._hyg.7002) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
-Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.supᵢ_inf_eqₓ'. -/
-protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
-  by rw [supᵢ, h.directed_on_range.Sup_inf_eq, supᵢ_range]
-#align directed.supr_inf_eq Directed.supᵢ_inf_eq
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 x._@.Mathlib.Order.CompactlyGenerated._hyg.7002) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
+Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.iSup_inf_eqₓ'. -/
+protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
+  by rw [iSup, h.directed_on_range.Sup_inf_eq, iSup_range]
+#align directed.supr_inf_eq Directed.iSup_inf_eq
 
-/- warning: directed_on.disjoint_Sup_right -> DirectedOn.disjoint_supₛ_right is a dubious translation:
+/- warning: directed_on.disjoint_Sup_right -> DirectedOn.disjoint_sSup_right is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 x._@.Mathlib.Order.CompactlyGenerated._hyg.7119) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
-Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_rightₓ'. -/
-protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
-    Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
-  simp_rw [disjoint_iff, h.inf_Sup_eq, supᵢ_eq_bot]
-#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_right
-
-/- warning: directed_on.disjoint_Sup_left -> DirectedOn.disjoint_supₛ_left is a dubious translation:
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 x._@.Mathlib.Order.CompactlyGenerated._hyg.7119) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_rightₓ'. -/
+protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :
+    Disjoint a (sSup s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
+  simp_rw [disjoint_iff, h.inf_Sup_eq, iSup_eq_bot]
+#align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_right
+
+/- warning: directed_on.disjoint_Sup_left -> DirectedOn.disjoint_sSup_left is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7190 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 x._@.Mathlib.Order.CompactlyGenerated._hyg.7190) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
-Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_leftₓ'. -/
-protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
-    Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
-  simp_rw [disjoint_iff, h.Sup_inf_eq, supᵢ_eq_bot]
-#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_left
-
-/- warning: directed.disjoint_supr_right -> Directed.disjoint_supᵢ_right is a dubious translation:
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7190 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 x._@.Mathlib.Order.CompactlyGenerated._hyg.7190) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_leftₓ'. -/
+protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
+    Disjoint (sSup s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
+  simp_rw [disjoint_iff, h.Sup_inf_eq, iSup_eq_bot]
+#align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_left
+
+/- warning: directed.disjoint_supr_right -> Directed.disjoint_iSup_right is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7261 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 x._@.Mathlib.Order.CompactlyGenerated._hyg.7261) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
-Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_supᵢ_rightₓ'. -/
-protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7261 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 x._@.Mathlib.Order.CompactlyGenerated._hyg.7261) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
+Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_iSup_rightₓ'. -/
+protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
-  simp_rw [disjoint_iff, h.inf_supr_eq, supᵢ_eq_bot]
-#align directed.disjoint_supr_right Directed.disjoint_supᵢ_right
+  simp_rw [disjoint_iff, h.inf_supr_eq, iSup_eq_bot]
+#align directed.disjoint_supr_right Directed.disjoint_iSup_right
 
-/- warning: directed.disjoint_supr_left -> Directed.disjoint_supᵢ_left is a dubious translation:
+/- warning: directed.disjoint_supr_left -> Directed.disjoint_iSup_left is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7345 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 x._@.Mathlib.Order.CompactlyGenerated._hyg.7345) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
-Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_supᵢ_leftₓ'. -/
-protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7345 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 x._@.Mathlib.Order.CompactlyGenerated._hyg.7345) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (iSup.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
+Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_iSup_leftₓ'. -/
+protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
-  simp_rw [disjoint_iff, h.supr_inf_eq, supᵢ_eq_bot]
-#align directed.disjoint_supr_left Directed.disjoint_supᵢ_left
+  simp_rw [disjoint_iff, h.supr_inf_eq, iSup_eq_bot]
+#align directed.disjoint_supr_left Directed.disjoint_iSup_left
 
-/- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_supₛ_eq_supᵢ_inf_sup_finset is a dubious translation:
+/- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_sSup_eq_iSup_inf_sup_finset is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
-Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_supr_inf_sup_finset inf_supₛ_eq_supᵢ_inf_sup_finsetₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (iSup.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
+Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finsetₓ'. -/
 /-- This property is equivalent to `α` being upper continuous. -/
-theorem inf_supₛ_eq_supᵢ_inf_sup_finset :
-    a ⊓ supₛ s = ⨆ (t : Finset α) (H : ↑t ⊆ s), a ⊓ t.sup id :=
+theorem inf_sSup_eq_iSup_inf_sup_finset :
+    a ⊓ sSup s = ⨆ (t : Finset α) (H : ↑t ⊆ s), a ⊓ t.sup id :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
       intro c hc hcinf
       rw [le_inf_iff] at hcinf
       rcases hc s hcinf.2 with ⟨t, ht1, ht2⟩
-      exact (le_inf hcinf.1 ht2).trans (le_supᵢ₂ t ht1))
-    (supᵢ_le fun t =>
-      supᵢ_le fun h => inf_le_inf_left _ ((Finset.sup_id_eq_supₛ t).symm ▸ supₛ_le_supₛ h))
-#align inf_Sup_eq_supr_inf_sup_finset inf_supₛ_eq_supᵢ_inf_sup_finset
+      exact (le_inf hcinf.1 ht2).trans (le_iSup₂ t ht1))
+    (iSup_le fun t =>
+      iSup_le fun h => inf_le_inf_left _ ((Finset.sup_id_eq_sSup t).symm ▸ sSup_le_sSup h))
+#align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finset
 
 #print CompleteLattice.setIndependent_iff_finite /-
 theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
@@ -579,9 +579,9 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
       ∀ t : Finset α, ↑t ⊆ s → CompleteLattice.SetIndependent (↑t : Set α) :=
   ⟨fun hs t ht => hs.mono ht, fun h a ha =>
     by
-    rw [disjoint_iff, inf_supₛ_eq_supᵢ_inf_sup_finset, supᵢ_eq_bot]
+    rw [disjoint_iff, inf_sSup_eq_iSup_inf_sup_finset, iSup_eq_bot]
     intro t
-    rw [supᵢ_eq_bot, Finset.sup_id_eq_supₛ]
+    rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
     intro ht
     classical
       have h' := (h (insert a t) _ (t.mem_insert_self a)).eq_bot
@@ -592,8 +592,8 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 -/
 
-#print CompleteLattice.setIndependent_unionᵢ_of_directed /-
-theorem CompleteLattice.setIndependent_unionᵢ_of_directed {η : Type _} {s : η → Set α}
+#print CompleteLattice.setIndependent_iUnion_of_directed /-
+theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type _} {s : η → Set α}
     (hs : Directed (· ⊆ ·) s) (h : ∀ i, CompleteLattice.SetIndependent (s i)) :
     CompleteLattice.SetIndependent (⋃ i, s i) :=
   by
@@ -601,23 +601,23 @@ theorem CompleteLattice.setIndependent_unionᵢ_of_directed {η : Type _} {s : 
   · skip
     rw [CompleteLattice.setIndependent_iff_finite]
     intro t ht
-    obtain ⟨I, fi, hI⟩ := Set.finite_subset_unionᵢ t.finite_to_set ht
+    obtain ⟨I, fi, hI⟩ := Set.finite_subset_iUnion t.finite_to_set ht
     obtain ⟨i, hi⟩ := hs.finset_le fi.to_finset
     exact
       (h i).mono
-        (Set.Subset.trans hI <| Set.unionᵢ₂_subset fun j hj => hi j (fi.mem_to_finset.2 hj))
+        (Set.Subset.trans hI <| Set.iUnion₂_subset fun j hj => hi j (fi.mem_to_finset.2 hj))
   · rintro a ⟨_, ⟨i, _⟩, _⟩
     exfalso
     exact hη ⟨i⟩
-#align complete_lattice.set_independent_Union_of_directed CompleteLattice.setIndependent_unionᵢ_of_directed
+#align complete_lattice.set_independent_Union_of_directed CompleteLattice.setIndependent_iUnion_of_directed
 -/
 
-#print CompleteLattice.independent_unionₛ_of_directed /-
-theorem CompleteLattice.independent_unionₛ_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
+#print CompleteLattice.independent_sUnion_of_directed /-
+theorem CompleteLattice.independent_sUnion_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
     (h : ∀ a ∈ s, CompleteLattice.SetIndependent a) : CompleteLattice.SetIndependent (⋃₀ s) := by
-  rw [Set.unionₛ_eq_unionᵢ] <;>
-    exact CompleteLattice.setIndependent_unionᵢ_of_directed hs.directed_coe (by simpa using h)
-#align complete_lattice.independent_sUnion_of_directed CompleteLattice.independent_unionₛ_of_directed
+  rw [Set.sUnion_eq_iUnion] <;>
+    exact CompleteLattice.setIndependent_iUnion_of_directed hs.directed_coe (by simpa using h)
+#align complete_lattice.independent_sUnion_of_directed CompleteLattice.independent_sUnion_of_directed
 -/
 
 end
@@ -630,7 +630,7 @@ theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α →
   by
   rw [well_founded_iff_is_Sup_finite_compact, is_Sup_finite_compact_iff_all_elements_compact] at h
   -- x is the join of the set of compact elements {x}
-  exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, supₛ_singleton⟩⟩⟩
+  exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
 -/
 
@@ -650,7 +650,7 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) : IsCo
       exact lt_irrefl _ hck
     · intro S SC cC I IS
       by_cases hS : S.nonempty
-      · exact ⟨Sup S, h.directed_Sup_lt_of_lt hS cC.directed_on SC, fun _ => le_supₛ⟩
+      · exact ⟨Sup S, h.directed_Sup_lt_of_lt hS cC.directed_on SC, fun _ => le_sSup⟩
       exact
         ⟨b, lt_of_le_of_ne hbk htriv, by
           simp only [set.not_nonempty_iff_eq_empty.mp hS, Set.mem_empty_iff_false, forall_const,
@@ -679,7 +679,7 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
   ⟨fun b => by
     by_cases h : { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } ⊆ {⊥}
     · left
-      rw [← supₛ_compact_le_eq b, supₛ_eq_bot]
+      rw [← sSup_compact_le_eq b, sSup_eq_bot]
       exact h
     · rcases Set.not_subset.1 h with ⟨c, ⟨hc, hcb⟩, hcbot⟩
       right
@@ -703,7 +703,7 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
   ⟨fun b =>
     ⟨{ a | IsAtom a ∧ a ≤ b }, by
       symm
-      have hle : Sup { a : α | IsAtom a ∧ a ≤ b } ≤ b := supₛ_le fun _ => And.right
+      have hle : Sup { a : α | IsAtom a ∧ a ≤ b } ≤ b := sSup_le fun _ => And.right
       apply (lt_or_eq_of_le hle).resolve_left fun con => _
       obtain ⟨c, hc⟩ := exists_is_compl (⟨Sup { a : α | IsAtom a ∧ a ≤ b }, hle⟩ : Set.Iic b)
       obtain rfl | ⟨a, ha, hac⟩ := eq_bot_or_exists_atom_le c
@@ -712,7 +712,7 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
         rw [eq_bot_iff]
         apply le_trans (le_inf _ hac) hc.disjoint.le_bot
         rw [← Subtype.coe_le_coe, Subtype.coe_mk]
-        exact le_supₛ ⟨ha.of_is_atom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
+        exact le_sSup ⟨ha.of_is_atom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
 #align is_atomistic_of_complemented_lattice isAtomistic_of_complementedLattice
 -/
 
@@ -722,38 +722,38 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 -/
 
 
-/- warning: exists_set_independent_is_compl_Sup_atoms -> exists_setIndependent_isCompl_supₛ_atoms is a dubious translation:
+/- warning: exists_set_independent_is_compl_Sup_atoms -> exists_setIndependent_isCompl_sSup_atoms is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
-Case conversion may be inaccurate. Consider using '#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atomsₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+Case conversion may be inaccurate. Consider using '#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atomsₓ'. -/
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
-theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
-    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   by
   obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ :=
     zorn_subset
       { s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (Sup s) ∧ ∀ a ∈ s, IsAtom a }
       fun c hc1 hc2 =>
       ⟨⋃₀ c,
-        ⟨CompleteLattice.independent_unionₛ_of_directed hc2.DirectedOn fun s hs => (hc1 hs).1, _,
+        ⟨CompleteLattice.independent_sUnion_of_directed hc2.DirectedOn fun s hs => (hc1 hs).1, _,
           fun a ⟨s, sc, as⟩ => (hc1 sc).2.2 a as⟩,
-        fun _ => Set.subset_unionₛ_of_mem⟩
+        fun _ => Set.subset_sUnion_of_mem⟩
   swap
-  · rw [supₛ_unionₛ, ← supₛ_image, DirectedOn.disjoint_supₛ_right]
+  · rw [sSup_sUnion, ← sSup_image, DirectedOn.disjoint_sSup_right]
     · rintro _ ⟨s, hs, rfl⟩
       exact (hc1 hs).2.1
     · rw [directedOn_image]
-      exact hc2.directed_on.mono fun s t => supₛ_le_supₛ
+      exact hc2.directed_on.mono fun s t => sSup_le_sSup
   refine' ⟨s, s_ind, ⟨b_inf_Sup_s, _⟩, s_atoms⟩
-  rw [codisjoint_iff_le_sup, ← h, supₛ_le_iff]
+  rw [codisjoint_iff_le_sup, ← h, sSup_le_iff]
   intro a ha
   rw [← inf_eq_left]
   refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
   rw [← Con, eq_comm, inf_eq_left]
-  refine' (le_supₛ _).trans le_sup_right
+  refine' (le_sSup _).trans le_sup_right
   rw [← disjoint_iff] at con
   have a_dis_Sup_s : Disjoint a (Sup s) := con.mono_right le_sup_right
   rw [← s_max (s ∪ {a}) ⟨fun x hx => _, ⟨_, fun x hx => _⟩⟩ (Set.subset_union_left _ _)]
@@ -761,56 +761,56 @@ theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom
   · rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain rfl | xa := eq_or_ne x a
     · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
-      exact con.mono_right ((supₛ_le_supₛ <| Set.diff_subset _ _).trans le_sup_right)
+      exact con.mono_right ((sSup_le_sSup <| Set.diff_subset _ _).trans le_sup_right)
     · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
         by
         simp only [Set.union_singleton]
         rw [Set.insert_diff_of_not_mem]
         rw [Set.mem_singleton_iff]
         exact Ne.symm xa
-      rw [h, supₛ_union, supₛ_singleton]
+      rw [h, sSup_union, sSup_singleton]
       apply
         (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
           (a_dis_Sup_s.mono_right _).symm
-      rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
-  · rw [supₛ_union, supₛ_singleton]
+      rw [← sSup_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
+  · rw [sSup_union, sSup_singleton]
     exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left Con.symm
   · rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain hx | rfl := hx
     · exact s_atoms x hx
     · exact ha
-#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atoms
+#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atoms
 
-/- warning: exists_set_independent_of_Sup_atoms_eq_top -> exists_setIndependent_of_supₛ_atoms_eq_top is a dubious translation:
+/- warning: exists_set_independent_of_Sup_atoms_eq_top -> exists_setIndependent_of_sSup_atoms_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
-Case conversion may be inaccurate. Consider using '#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_topₓ'. -/
-theorem exists_setIndependent_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
-    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ supₛ s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
-  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h ⊥
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+Case conversion may be inaccurate. Consider using '#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_topₓ'. -/
+theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h ⊥
   ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
-#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_top
+#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_top
 
-/- warning: complemented_lattice_of_Sup_atoms_eq_top -> complementedLattice_of_supₛ_atoms_eq_top is a dubious translation:
+/- warning: complemented_lattice_of_Sup_atoms_eq_top -> complementedLattice_of_sSup_atoms_eq_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
-Case conversion may be inaccurate. Consider using '#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_topₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.sSup.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
+Case conversion may be inaccurate. Consider using '#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_topₓ'. -/
 /-- See [Theorem 6.6][calugareanu]. -/
-theorem complementedLattice_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
+theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
     ComplementedLattice α :=
   ⟨fun b =>
-    let ⟨s, _, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h b
-    ⟨supₛ s, s_top⟩⟩
-#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_top
+    let ⟨s, _, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h b
+    ⟨sSup s, s_top⟩⟩
+#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_top
 
 #print complementedLattice_of_isAtomistic /-
 /-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_isAtomistic [IsAtomistic α] : ComplementedLattice α :=
-  complementedLattice_of_supₛ_atoms_eq_top supₛ_atoms_eq_top
+  complementedLattice_of_sSup_atoms_eq_top sSup_atoms_eq_top
 #align complemented_lattice_of_is_atomistic complementedLattice_of_isAtomistic
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -493,7 +493,7 @@ protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : sup
 lean 3 declaration is
   forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6889 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 x._@.Mathlib.Order.CompactlyGenerated._hyg.6889) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6885 x._@.Mathlib.Order.CompactlyGenerated._hyg.6887) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
 Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_supᵢ_eqₓ'. -/
 protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
   by rw [supᵢ, h.directed_on_range.inf_Sup_eq, supᵢ_range]
@@ -503,7 +503,7 @@ protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ 
 lean 3 declaration is
   forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7004 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 x._@.Mathlib.Order.CompactlyGenerated._hyg.7004) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7000 x._@.Mathlib.Order.CompactlyGenerated._hyg.7002) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
 Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.supᵢ_inf_eqₓ'. -/
 protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
   by rw [supᵢ, h.directed_on_range.Sup_inf_eq, supᵢ_range]
@@ -513,7 +513,7 @@ protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i,
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7121 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 x._@.Mathlib.Order.CompactlyGenerated._hyg.7121) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7117 x._@.Mathlib.Order.CompactlyGenerated._hyg.7119) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
 Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_rightₓ'. -/
 protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
     Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
@@ -524,7 +524,7 @@ protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7192 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7194 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7192 x._@.Mathlib.Order.CompactlyGenerated._hyg.7194) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7190 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7188 x._@.Mathlib.Order.CompactlyGenerated._hyg.7190) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
 Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_leftₓ'. -/
 protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
     Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
@@ -535,7 +535,7 @@ protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s)
 lean 3 declaration is
   forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7265 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7267 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7265 x._@.Mathlib.Order.CompactlyGenerated._hyg.7267) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7261 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7259 x._@.Mathlib.Order.CompactlyGenerated._hyg.7261) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
 Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_supᵢ_rightₓ'. -/
 protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
@@ -546,7 +546,7 @@ protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
 lean 3 declaration is
   forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
 but is expected to have type
-  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7351 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7353 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7351 x._@.Mathlib.Order.CompactlyGenerated._hyg.7353) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7345 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7343 x._@.Mathlib.Order.CompactlyGenerated._hyg.7345) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
 Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_supᵢ_leftₓ'. -/
 protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/a4f99eae998680d3a2c240da4a2b16354c85ee49

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit e8cf0cfec5fcab9baf46dc17d30c5e22048468be
+! leanprover-community/mathlib commit c813ed7de0f5115f956239124e9b30f3a621966f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -241,7 +241,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     rw [is_compact_element_iff_le_of_directed_Sup_le]
     intro d hemp hdir hsup
     change f with id ∘ f
-    rw [← Finset.sup_finset_image]
+    rw [← Finset.sup_image]
     apply Finset.sup_le_of_le_directed d hemp hdir
     rintro x hx
     obtain ⟨p, ⟨hps, rfl⟩⟩ := finset.mem_image.mp hx

mathlib commit https://github.com/leanprover-community/mathlib/commit/fa78268d4d77cb2b2fbc89f0527e2e7807763780

@@ -132,7 +132,7 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
 lean 3 declaration is
   forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
 but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.511 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.513 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.511 x._@.Mathlib.Order.CompactlyGenerated._hyg.513) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
+  forall (α : Type.{u1}) [_inst_1 : CompleteLattice.{u1} α] (k : α), Iff (CompleteLattice.IsCompactElement.{u1} α _inst_1 k) (forall (s : Set.{u1} α), (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.541 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.543 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.541 x._@.Mathlib.Order.CompactlyGenerated._hyg.543) s) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) k x))))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_leₓ'. -/
 /-- An element `k` is compact if and only if any directed set with `Sup` above
 `k` already got above `k` at some point in the set. -/
@@ -212,7 +212,7 @@ theorem IsCompactElement.exists_finset_of_le_supᵢ {k : α} (hk : IsCompactElem
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2))))) s) -> (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1357 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1359 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1357 x._@.Mathlib.Order.CompactlyGenerated._hyg.1359) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
+  forall {α : Type.{u1}} [_inst_2 : CompleteLattice.{u1} α] {k : α}, (CompleteLattice.IsCompactElement.{u1} α _inst_2 k) -> (forall {s : Set.{u1} α}, (Set.Nonempty.{u1} α s) -> (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.1397 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.1395 x._@.Mathlib.Order.CompactlyGenerated._hyg.1397) s) -> (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) x k)) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_2)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_2)) s) k))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_supₛ_lt_of_ltₓ'. -/
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
 its Sup strictly below `k`. -/
@@ -398,7 +398,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6021 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6023 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6021 x._@.Mathlib.Order.CompactlyGenerated._hyg.6023)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
+  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6107 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6109 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6107 x._@.Mathlib.Order.CompactlyGenerated._hyg.6109)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
@@ -456,6 +456,12 @@ theorem le_iff_compact_le_imp {a b : α} :
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 -/
 
+/- warning: directed_on.inf_Sup_eq -> DirectedOn.inf_supₛ_eq is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6528 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6526 x._@.Mathlib.Order.CompactlyGenerated._hyg.6528) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+Case conversion may be inaccurate. Consider using '#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
@@ -472,34 +478,76 @@ theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s
     supᵢ_inf_le_inf_supₛ
 #align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eq
 
+/- warning: directed_on.Sup_inf_eq -> DirectedOn.supₛ_inf_eq is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) b a))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6796 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6794 x._@.Mathlib.Order.CompactlyGenerated._hyg.6796) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) b a))))
+Case conversion may be inaccurate. Consider using '#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eqₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : supₛ s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
   by simp_rw [@inf_comm _ _ _ a, h.inf_Sup_eq]
 #align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eq
 
+/- warning: directed.inf_supr_eq -> Directed.inf_supᵢ_eq is a dubious translation:
+lean 3 declaration is
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) a (f i))))
+but is expected to have type
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6889 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6887 x._@.Mathlib.Order.CompactlyGenerated._hyg.6889) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) a (_inst_1 i))))
+Case conversion may be inaccurate. Consider using '#align directed.inf_supr_eq Directed.inf_supᵢ_eqₓ'. -/
 protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
   by rw [supᵢ, h.directed_on_range.inf_Sup_eq, supᵢ_range]
 #align directed.inf_supr_eq Directed.inf_supᵢ_eq
 
+/- warning: directed.supr_inf_eq -> Directed.supᵢ_inf_eq is a dubious translation:
+lean 3 declaration is
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Eq.{succ u2} α (Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => Inf.inf.{u2} α (SemilatticeInf.toHasInf.{u2} α (Lattice.toSemilatticeInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)))) (f i) a)))
+but is expected to have type
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7004 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7002 x._@.Mathlib.Order.CompactlyGenerated._hyg.7004) _inst_1) -> (Eq.{succ u2} α (Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => Inf.inf.{u2} α (Lattice.toInf.{u2} α (ConditionallyCompleteLattice.toLattice.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f))) (_inst_1 i) a)))
+Case conversion may be inaccurate. Consider using '#align directed.supr_inf_eq Directed.supᵢ_inf_eqₓ'. -/
 protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
   by rw [supᵢ, h.directed_on_range.Sup_inf_eq, supᵢ_range]
 #align directed.supr_inf_eq Directed.supᵢ_inf_eq
 
+/- warning: directed_on.disjoint_Sup_right -> DirectedOn.disjoint_supₛ_right is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7121 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7119 x._@.Mathlib.Order.CompactlyGenerated._hyg.7121) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a b)))
+Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_rightₓ'. -/
 protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
     Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
   simp_rw [disjoint_iff, h.inf_Sup_eq, supᵢ_eq_bot]
 #align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_right
 
+/- warning: directed_on.disjoint_Sup_left -> DirectedOn.disjoint_supₛ_left is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7192 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7194 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7192 x._@.Mathlib.Order.CompactlyGenerated._hyg.7194) s) -> (Iff (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) a) (forall {{b : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) -> (Disjoint.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) b a)))
+Case conversion may be inaccurate. Consider using '#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_leftₓ'. -/
 protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
     Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
   simp_rw [disjoint_iff, h.Sup_inf_eq, supᵢ_eq_bot]
 #align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_left
 
+/- warning: directed.disjoint_supr_right -> Directed.disjoint_supᵢ_right is a dubious translation:
+lean 3 declaration is
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) a (f i)))
+but is expected to have type
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7265 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7267 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7265 x._@.Mathlib.Order.CompactlyGenerated._hyg.7267) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i))) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) a (_inst_1 i)))
+Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_right Directed.disjoint_supᵢ_rightₓ'. -/
 protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
   simp_rw [disjoint_iff, h.inf_supr_eq, supᵢ_eq_bot]
 #align directed.disjoint_supr_right Directed.disjoint_supᵢ_right
 
+/- warning: directed.disjoint_supr_left -> Directed.disjoint_supᵢ_left is a dubious translation:
+lean 3 declaration is
+  forall {ι : Sort.{u1}} {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α] {f : ι -> α} [_inst_2 : IsCompactlyGenerated.{u2} α _inst_1] {a : α}, (Directed.{u2, u1} α ι (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1))))) f) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toHasSup.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α _inst_1)) ι (fun (i : ι) => f i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u2} α _inst_1)) (f i) a))
+but is expected to have type
+  forall {ι : Sort.{u1}} {α : Type.{u2}} {_inst_1 : ι -> α} [f : CompleteLattice.{u2} α] [_inst_2 : IsCompactlyGenerated.{u2} α f] {a : α}, (Directed.{u2, u1} α ι (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.7351 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.7353 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.7351 x._@.Mathlib.Order.CompactlyGenerated._hyg.7353) _inst_1) -> (Iff (Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (supᵢ.{u2, u1} α (ConditionallyCompleteLattice.toSupSet.{u2} α (CompleteLattice.toConditionallyCompleteLattice.{u2} α f)) ι (fun (i : ι) => _inst_1 i)) a) (forall (i : ι), Disjoint.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)) (BoundedOrder.toOrderBot.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α f)))) (CompleteLattice.toBoundedOrder.{u2} α f)) (_inst_1 i) a))
+Case conversion may be inaccurate. Consider using '#align directed.disjoint_supr_left Directed.disjoint_supᵢ_leftₓ'. -/
 protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
   simp_rw [disjoint_iff, h.supr_inf_eq, supᵢ_eq_bot]
@@ -674,6 +722,12 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 -/
 
 
+/- warning: exists_set_independent_is_compl_Sup_atoms -> exists_setIndependent_isCompl_supₛ_atoms is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (forall (b : α), Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (IsCompl.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1) b (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+Case conversion may be inaccurate. Consider using '#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atomsₓ'. -/
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
@@ -727,6 +781,12 @@ theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom
     · exact ha
 #align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atoms
 
+/- warning: exists_set_independent_of_Sup_atoms_eq_top -> exists_setIndependent_of_supₛ_atoms_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (CompleteLattice.SetIndependent.{u1} α _inst_1 s) (And (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) (forall {{a : α}}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a)))))
+Case conversion may be inaccurate. Consider using '#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_topₓ'. -/
 theorem exists_setIndependent_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧ supₛ s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
   let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h ⊥

mathlib commit https://github.com/leanprover-community/mathlib/commit/52932b3a083d4142e78a15dc928084a22fea9ba0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
+! leanprover-community/mathlib commit e8cf0cfec5fcab9baf46dc17d30c5e22048468be
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -59,7 +59,12 @@ complete lattice, well-founded, compact
 -/
 
 
-variable {α : Type _} [CompleteLattice α]
+alias directedOn_range ↔ Directed.directedOn_range _
+#align directed.directed_on_range Directed.directedOn_range
+
+attribute [protected] Directed.directedOn_range
+
+variable {ι : Sort _} {α : Type _} [CompleteLattice α] {f : ι → α}
 
 namespace CompleteLattice
 
@@ -451,14 +456,8 @@ theorem le_iff_compact_le_imp {a b : α} :
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 -/
 
-/- warning: inf_Sup_eq_of_directed_on -> inf_supₛ_eq_of_directedOn is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6409 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6411 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6409 x._@.Mathlib.Order.CompactlyGenerated._hyg.6411) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
-Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOnₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
+theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
@@ -471,7 +470,40 @@ theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ sup
       · rw [Set.not_nonempty_iff_eq_empty] at hs
         simp [hs])
     supᵢ_inf_le_inf_supₛ
-#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOn
+#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eq
+
+/-- This property is sometimes referred to as `α` being upper continuous. -/
+protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : supₛ s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
+  by simp_rw [@inf_comm _ _ _ a, h.inf_Sup_eq]
+#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eq
+
+protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
+  by rw [supᵢ, h.directed_on_range.inf_Sup_eq, supᵢ_range]
+#align directed.inf_supr_eq Directed.inf_supᵢ_eq
+
+protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
+  by rw [supᵢ, h.directed_on_range.Sup_inf_eq, supᵢ_range]
+#align directed.supr_inf_eq Directed.supᵢ_inf_eq
+
+protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
+    Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
+  simp_rw [disjoint_iff, h.inf_Sup_eq, supᵢ_eq_bot]
+#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_right
+
+protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
+    Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
+  simp_rw [disjoint_iff, h.Sup_inf_eq, supᵢ_eq_bot]
+#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_left
+
+protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
+    Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
+  simp_rw [disjoint_iff, h.inf_supr_eq, supᵢ_eq_bot]
+#align directed.disjoint_supr_right Directed.disjoint_supᵢ_right
+
+protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
+    Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
+  simp_rw [disjoint_iff, h.supr_inf_eq, supᵢ_eq_bot]
+#align directed.disjoint_supr_left Directed.disjoint_supᵢ_left
 
 /- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_supₛ_eq_supᵢ_inf_sup_finset is a dubious translation:
 lean 3 declaration is
@@ -617,7 +649,7 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
 -/
 
 #print isAtomistic_of_complementedLattice /-
-/-- See Lemma 5.1, Călugăreanu -/
+/-- See [Lemma 5.1][calugareanu]. -/
 instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLattice α] :
     IsAtomistic α :=
   ⟨fun b =>
@@ -636,73 +668,87 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
 #align is_atomistic_of_complemented_lattice isAtomistic_of_complementedLattice
 -/
 
+/-!
+Now we will prove that a compactly generated modular atomistic lattice is a complemented lattice.
+Most explicitly, every element is the complement of a supremum of indepedendent atoms.
+-/
+
+
+/-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
+of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
+theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  by
+  obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ :=
+    zorn_subset
+      { s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (Sup s) ∧ ∀ a ∈ s, IsAtom a }
+      fun c hc1 hc2 =>
+      ⟨⋃₀ c,
+        ⟨CompleteLattice.independent_unionₛ_of_directed hc2.DirectedOn fun s hs => (hc1 hs).1, _,
+          fun a ⟨s, sc, as⟩ => (hc1 sc).2.2 a as⟩,
+        fun _ => Set.subset_unionₛ_of_mem⟩
+  swap
+  · rw [supₛ_unionₛ, ← supₛ_image, DirectedOn.disjoint_supₛ_right]
+    · rintro _ ⟨s, hs, rfl⟩
+      exact (hc1 hs).2.1
+    · rw [directedOn_image]
+      exact hc2.directed_on.mono fun s t => supₛ_le_supₛ
+  refine' ⟨s, s_ind, ⟨b_inf_Sup_s, _⟩, s_atoms⟩
+  rw [codisjoint_iff_le_sup, ← h, supₛ_le_iff]
+  intro a ha
+  rw [← inf_eq_left]
+  refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
+  rw [← Con, eq_comm, inf_eq_left]
+  refine' (le_supₛ _).trans le_sup_right
+  rw [← disjoint_iff] at con
+  have a_dis_Sup_s : Disjoint a (Sup s) := con.mono_right le_sup_right
+  rw [← s_max (s ∪ {a}) ⟨fun x hx => _, ⟨_, fun x hx => _⟩⟩ (Set.subset_union_left _ _)]
+  · exact Set.mem_union_right _ (Set.mem_singleton _)
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
+    obtain rfl | xa := eq_or_ne x a
+    · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
+      exact con.mono_right ((supₛ_le_supₛ <| Set.diff_subset _ _).trans le_sup_right)
+    · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
+        by
+        simp only [Set.union_singleton]
+        rw [Set.insert_diff_of_not_mem]
+        rw [Set.mem_singleton_iff]
+        exact Ne.symm xa
+      rw [h, supₛ_union, supₛ_singleton]
+      apply
+        (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
+          (a_dis_Sup_s.mono_right _).symm
+      rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
+  · rw [supₛ_union, supₛ_singleton]
+    exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left Con.symm
+  · rw [Set.mem_union, Set.mem_singleton_iff] at hx
+    obtain hx | rfl := hx
+    · exact s_atoms x hx
+    · exact ha
+#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atoms
+
+theorem exists_setIndependent_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ supₛ s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h ⊥
+  ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
+#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_top
+
 /- warning: complemented_lattice_of_Sup_atoms_eq_top -> complementedLattice_of_supₛ_atoms_eq_top is a dubious translation:
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toHasTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsModularLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))] [_inst_3 : IsCompactlyGenerated.{u1} α _inst_1], (Eq.{succ u1} α (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (setOf.{u1} α (fun (a : α) => IsAtom.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) a))) (Top.top.{u1} α (CompleteLattice.toTop.{u1} α _inst_1))) -> (ComplementedLattice.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (CompleteLattice.toBoundedOrder.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_topₓ'. -/
-/-- See Theorem 6.6, Călugăreanu -/
+/-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
     ComplementedLattice α :=
   ⟨fun b =>
-    by
-    obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ :=
-      zorn_subset
-        { s : Set α | CompleteLattice.SetIndependent s ∧ b ⊓ Sup s = ⊥ ∧ ∀ a ∈ s, IsAtom a } _
-    · refine'
-        ⟨Sup s, disjoint_iff.mpr b_inf_Sup_s,
-          codisjoint_iff_le_sup.mpr <| h.symm.trans_le <| supₛ_le_iff.2 fun a ha => _⟩
-      rw [← inf_eq_left]
-      refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
-      rw [eq_bot_iff, ← Con]
-      refine' le_inf (le_refl a) ((le_supₛ _).trans le_sup_right)
-      rw [← disjoint_iff] at *
-      have a_dis_Sup_s : Disjoint a (Sup s) := con.mono_right le_sup_right
-      rw [← s_max (s ∪ {a}) ⟨fun x hx => _, ⟨_, fun x hx => _⟩⟩ (Set.subset_union_left _ _)]
-      · exact Set.mem_union_right _ (Set.mem_singleton _)
-      · rw [Set.mem_union, Set.mem_singleton_iff] at hx
-        by_cases xa : x = a
-        · simp only [xa, Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
-          exact con.mono_right (le_trans (supₛ_le_supₛ (Set.diff_subset s {a})) le_sup_right)
-        · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
-            by
-            simp only [Set.union_singleton]
-            rw [Set.insert_diff_of_not_mem]
-            rw [Set.mem_singleton_iff]
-            exact Ne.symm xa
-          rw [h, supₛ_union, supₛ_singleton]
-          apply
-            (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
-              (a_dis_Sup_s.mono_right _).symm
-          rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
-      · rw [supₛ_union, supₛ_singleton, ← disjoint_iff]
-        exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left Con.symm
-      · rw [Set.mem_union, Set.mem_singleton_iff] at hx
-        cases hx
-        · exact s_atoms x hx
-        · rw [hx]
-          exact ha
-    · intro c hc1 hc2
-      refine'
-        ⟨⋃₀ c,
-          ⟨CompleteLattice.independent_unionₛ_of_directed hc2.directed_on fun s hs => (hc1 hs).1, _,
-            fun a ha => _⟩,
-          fun _ => Set.subset_unionₛ_of_mem⟩
-      · rw [supₛ_unionₛ, ← supₛ_image, inf_supₛ_eq_of_directedOn, supᵢ_eq_bot]
-        · intro i
-          rw [supᵢ_eq_bot]
-          intro hi
-          obtain ⟨x, xc, rfl⟩ := (Set.mem_image _ _ _).1 hi
-          exact (hc1 xc).2.1
-        · rw [directedOn_image]
-          refine' hc2.directed_on.mono fun s t => supₛ_le_supₛ
-      · rcases Set.mem_unionₛ.1 ha with ⟨s, sc, as⟩
-        exact (hc1 sc).2.2 a as⟩
+    let ⟨s, _, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h b
+    ⟨supₛ s, s_top⟩⟩
 #align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_top
 
 #print complementedLattice_of_isAtomistic /-
-/-- See Theorem 6.6, Călugăreanu -/
+/-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_isAtomistic [IsAtomistic α] : ComplementedLattice α :=
   complementedLattice_of_supₛ_atoms_eq_top supₛ_atoms_eq_top
 #align complemented_lattice_of_is_atomistic complementedLattice_of_isAtomistic

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -393,7 +393,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6211 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6213 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6211 x._@.Mathlib.Order.CompactlyGenerated._hyg.6213)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
+  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6021 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6023 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6021 x._@.Mathlib.Order.CompactlyGenerated._hyg.6023)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
@@ -455,7 +455,7 @@ theorem le_iff_compact_le_imp {a b : α} :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6599 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6601 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6599 x._@.Mathlib.Order.CompactlyGenerated._hyg.6601) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6409 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6411 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6409 x._@.Mathlib.Order.CompactlyGenerated._hyg.6411) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOnₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/88fcb83fe7996142dfcfe7368d31304a9adc874a

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit f16e7a22e11fc09c71f25446ac1db23a24e8a0bd
+! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -248,30 +248,18 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
 
 #print CompleteLattice.WellFounded.isSupFiniteCompact /-
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
-    IsSupFiniteCompact α := by
-  intro s
-  let p : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
-  have hp : p.nonempty := by
-    use ⊥, ∅
-    simp
-  obtain ⟨m, ⟨t, ⟨ht₁, ht₂⟩⟩, hm⟩ := well_founded.well_founded_iff_has_max'.mp h p hp
-  use t
-  simp only [ht₁, ht₂, true_and_iff]
-  apply le_antisymm
-  · apply supₛ_le
-    intro y hy
+    IsSupFiniteCompact α := fun s =>
+  by
+  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ :=
+    well_founded.well_founded_iff_has_min.mp h { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
+      ⟨⊥, ∅, by simp⟩
+  refine' ⟨t, ht₁, (supₛ_le fun y hy => _).antisymm _⟩
+  ·
     classical
-      have hy' : (insert y t).sup id ∈ p := by
-        use insert y t
-        simp
-        rw [Set.insert_subset]
-        exact ⟨hy, ht₁⟩
-      have hm' : m ≤ (insert y t).sup id := by
-        rw [← ht₂]
-        exact Finset.sup_mono (t.subset_insert y)
-      rw [← hm _ hy' hm']
+      rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
+          (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
       simp
-  · rw [← ht₂, Finset.sup_id_eq_supₛ]
+  · rw [Finset.sup_id_eq_supₛ]
     exact supₛ_le_supₛ ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

@@ -405,7 +405,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3543 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3545 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3543 x._@.Mathlib.Order.CompactlyGenerated._hyg.3545)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
+  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6211 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6213 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6211 x._@.Mathlib.Order.CompactlyGenerated._hyg.6213)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
@@ -467,7 +467,7 @@ theorem le_iff_compact_le_imp {a b : α} :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3931 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3933 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3931 x._@.Mathlib.Order.CompactlyGenerated._hyg.3933) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.6599 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.6601 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.6599 x._@.Mathlib.Order.CompactlyGenerated._hyg.6601) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOnₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=

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

@@ -405,7 +405,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α], (WellFounded.{succ u1} α (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))))) -> (forall {ι : Type.{u2}} {t : ι -> α}, (CompleteLattice.Independent.{succ u2, u1} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u1} α (t i) (Bot.bot.{u1} α (CompleteLattice.toHasBot.{u1} α _inst_1))) -> (Finite.{succ u2} ι))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3536 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3538 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3536 x._@.Mathlib.Order.CompactlyGenerated._hyg.3538)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
+  forall {α : Type.{u2}} [_inst_1 : CompleteLattice.{u2} α], (WellFounded.{succ u2} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3543 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3545 : α) => GT.gt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (CompleteSemilatticeInf.toPartialOrder.{u2} α (CompleteLattice.toCompleteSemilatticeInf.{u2} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3543 x._@.Mathlib.Order.CompactlyGenerated._hyg.3545)) -> (forall {ι : Type.{u1}} {t : ι -> α}, (CompleteLattice.Independent.{succ u1, u2} ι α _inst_1 t) -> (forall (i : ι), Ne.{succ u2} α (t i) (Bot.bot.{u2} α (CompleteLattice.toBot.{u2} α _inst_1))) -> (Finite.{succ u1} ι))
 Case conversion may be inaccurate. Consider using '#align complete_lattice.well_founded.finite_of_independent CompleteLattice.WellFounded.finite_of_independentₓ'. -/
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
@@ -467,7 +467,7 @@ theorem le_iff_compact_le_imp {a b : α} :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3926 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 x._@.Mathlib.Order.CompactlyGenerated._hyg.3926) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3931 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3933 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3931 x._@.Mathlib.Order.CompactlyGenerated._hyg.3933) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOnₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=

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

@@ -65,7 +65,7 @@ namespace CompleteLattice
 
 variable (α)
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (a b «expr ∈ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a b «expr ∈ » s) -/
 #print CompleteLattice.IsSupClosedCompact /-
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `Sup`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -465,9 +465,9 @@ theorem le_iff_compact_le_imp {a b : α} :
 
 /- warning: inf_Sup_eq_of_directed_on -> inf_supₛ_eq_of_directedOn is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (HasInf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => HasInf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1))))) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) b s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a b))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3926 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 x._@.Mathlib.Order.CompactlyGenerated._hyg.3926) s) -> (Eq.{succ u1} α (HasInf.inf.{u1} α (Lattice.toHasInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => HasInf.inf.{u1} α (Lattice.toHasInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (DirectedOn.{u1} α (fun (x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 : α) (x._@.Mathlib.Order.CompactlyGenerated._hyg.3926 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) x._@.Mathlib.Order.CompactlyGenerated._hyg.3924 x._@.Mathlib.Order.CompactlyGenerated._hyg.3926) s) -> (Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) α (fun (b : α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) b s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a b))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOnₓ'. -/
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
@@ -487,9 +487,9 @@ theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ sup
 
 /- warning: inf_Sup_eq_supr_inf_sup_finset -> inf_supₛ_eq_supᵢ_inf_sup_finset is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (HasInf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => HasInf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{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))) t) s) => Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (HasInf.inf.{u1} α (Lattice.toHasInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) => HasInf.inf.{u1} α (Lattice.toHasInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] [_inst_2 : IsCompactlyGenerated.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Eq.{succ u1} α (Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (SupSet.supₛ.{u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) s)) (supᵢ.{u1, succ u1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (Finset.{u1} α) (fun (t : Finset.{u1} α) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (Finset.toSet.{u1} α t) s) => Inf.inf.{u1} α (Lattice.toInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) a (Finset.sup.{u1, u1} α α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) (BoundedOrder.toOrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} α _inst_1)) t (id.{succ u1} α)))))
 Case conversion may be inaccurate. Consider using '#align inf_Sup_eq_supr_inf_sup_finset inf_supₛ_eq_supᵢ_inf_sup_finsetₓ'. -/
 /-- This property is equivalent to `α` being upper continuous. -/
 theorem inf_supₛ_eq_supᵢ_inf_sup_finset :

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

A PR accompanying #12339.

Zulip discussion

@@ -593,13 +593,13 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
         ⟨CompleteLattice.independent_sUnion_of_directed hc2.directedOn fun s hs => (hc1 hs).1, ?_,
           fun a ⟨s, sc, as⟩ => (hc1 sc).2.2 a as⟩,
         fun _ => Set.subset_sUnion_of_mem⟩
-  obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ := this
   swap
   · rw [sSup_sUnion, ← sSup_image, DirectedOn.disjoint_sSup_right]
     · rintro _ ⟨s, hs, rfl⟩
       exact (hc1 hs).2.1
     · rw [directedOn_image]
       exact hc2.directedOn.mono @fun s t => sSup_le_sSup
+  obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ := this
   refine' ⟨s, s_ind, ⟨b_inf_Sup_s, _⟩, s_atoms⟩
   rw [codisjoint_iff_le_sup, ← h, sSup_le_iff]
   intro a ha

These · are scoping when there is a single active goal.

These were found using a modification of the linter at #12339.

@@ -90,10 +90,10 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
       have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop
       choose f hf using this
       refine' ⟨Finset.univ.image f, ht'.trans _⟩
-      · rw [Finset.sup_le_iff]
-        intro b hb
-        rw [← show s (f ⟨b, hb⟩) = id b from hf _]
-        exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))
+      rw [Finset.sup_le_iff]
+      intro b hb
+      rw [← show s (f ⟨b, hb⟩) = id b from hf _]
+      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))
     · intro H s hs
       obtain ⟨t, ht⟩ :=
         H s Subtype.val

Move Set.Ixx, Finset.Ixx, Multiset.Ixx together under two different folders:

• Order.Interval for their definition and basic properties
• Algebra.Order.Interval for their algebraic properties

Move the definitions of Multiset.Ixx to what is now Order.Interval.Multiset. I believe we could just delete this file in a later PR as nothing uses it (and I already had doubts when defining Multiset.Ixx three years ago).

Move the algebraic results out of what is now Order.Interval.Finset.Basic to a new file Algebra.Order.Interval.Finset.Basic.

@@ -10,7 +10,7 @@ import Mathlib.Order.SupClosed
 import Mathlib.Order.SupIndep
 import Mathlib.Order.Zorn
 import Mathlib.Data.Finset.Order
-import Mathlib.Data.Set.Intervals.OrderIso
+import Mathlib.Order.Interval.Set.OrderIso
 import Mathlib.Data.Finite.Set
 import Mathlib.Tactic.TFAE
 

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

@@ -135,7 +135,7 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
         apply sSup_le_sSup
         intro x hx
         use {x}
-        simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
+        simpa only [and_true_iff, id, Finset.coe_singleton, eq_self_iff_true,
           Finset.sup_singleton, Set.singleton_subset_iff]
       have Sne : S.Nonempty := by
         suffices ⊥ ∈ S from Set.nonempty_of_mem this
@@ -254,7 +254,7 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
   · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)
     have : sSup s = t.sup id := by
       suffices t.sup id ≤ sSup s by apply le_antisymm <;> assumption
-      simp only [id.def, Finset.sup_le_iff]
+      simp only [id, Finset.sup_le_iff]
       intro x hx
       exact le_sSup _ _ (hts hx)
     exact ⟨t, hts, this⟩

Those lemmas have historically been very annoying to use in rw since all their arguments were implicit. One too many people complained about it on Zulip, so I'm changing them.

Downstream code broken by this change can fix it by adding appropriately many _s.

Also marks CauSeq.ext @[ext].

Order.BoundedOrder

• top_sup_eq
• sup_top_eq
• bot_sup_eq
• sup_bot_eq
• top_inf_eq
• inf_top_eq
• bot_inf_eq
• inf_bot_eq

Order.Lattice

• sup_idem
• sup_comm
• sup_assoc
• sup_left_idem
• sup_right_idem
• inf_idem
• inf_comm
• inf_assoc
• inf_left_idem
• inf_right_idem
• sup_inf_left
• sup_inf_right
• inf_sup_left
• inf_sup_right

Order.MinMax

• max_min_distrib_left
• max_min_distrib_right
• min_max_distrib_left
• min_max_distrib_right

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

@@ -383,7 +383,7 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
 protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) :
     sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a := by
-  simp_rw [@inf_comm _ _ _ a, h.inf_sSup_eq]
+  simp_rw [inf_comm _ a, h.inf_sSup_eq]
 #align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
 
 protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) :

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -456,11 +456,12 @@ lemma CompleteLattice.independent_iff_supIndep_of_injOn {ι : Type*} {f : ι →
   classical
   rw [← Finset.sup_erase_bot]
   set t := s.erase ⊥
-  replace hf : InjOn f (f ⁻¹' t) := fun i hi j _ hij ↦ by refine hf ?_ ?_ hij <;> aesop
+  replace hf : InjOn f (f ⁻¹' t) := fun i hi j _ hij ↦ by
+    refine hf ?_ ?_ hij <;> aesop (add norm simp [t])
   have : (Finset.erase (insert i (t.preimage _ hf)) i).image f = t := by
     ext a
     simp only [Finset.mem_preimage, Finset.mem_erase, ne_eq, Finset.mem_insert, true_or, not_true,
-      Finset.erase_insert_eq_erase, not_and, Finset.mem_image]
+      Finset.erase_insert_eq_erase, not_and, Finset.mem_image, t]
     refine ⟨by aesop, fun ⟨ha, has⟩ ↦ ?_⟩
     obtain ⟨j, hj, rfl⟩ := hs has
     exact ⟨j, ⟨hj, ha, has⟩, rfl⟩

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".

@@ -608,7 +608,7 @@ theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a }
   refine' (le_sSup _).trans le_sup_right
   rw [← disjoint_iff] at con
   have a_dis_Sup_s : Disjoint a (sSup s) := con.mono_right le_sup_right
-  -- porting note: The two following `fun x hx => _` are no-op
+  -- Porting note: The two following `fun x hx => _` are no-op
   rw [← s_max (s ∪ {a}) ⟨fun x hx => _, _, fun x hx => _⟩ (Set.subset_union_left _ _)]
   · exact Set.mem_union_right _ (Set.mem_singleton _)
   · intro x hx

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>

@@ -138,8 +138,7 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
         simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
           Finset.sup_singleton, Set.singleton_subset_iff]
       have Sne : S.Nonempty := by
-        suffices : ⊥ ∈ S
-        exact Set.nonempty_of_mem this
+        suffices ⊥ ∈ S from Set.nonempty_of_mem this
         use ∅
         simp only [Set.empty_subset, Finset.coe_empty, Finset.sup_empty, eq_self_iff_true,
           and_self_iff]
@@ -306,13 +305,13 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
   classical
     refine' Set.not_infinite.mp fun contra => _
     obtain ⟨t, ht₁, ht₂⟩ := WellFounded.isSupFiniteCompact α h s
-    replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t
-    · have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
+    replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t := by
+      have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
       obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
       exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
     obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
-    replace hs : x ⊓ sSup s = ⊥
-    · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
+    replace hs : x ⊓ sSup s = ⊥ := by
+      have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
       simpa [Disjoint, hx₂, ← t.sup_id_eq_sSup, ← ht₂] using this.eq_bot
     apply hx₁
     rw [← hs, eq_comm, inf_eq_left]

• Merge Function.left_id and Function.comp.left_id into Function.id_comp.
• Merge Function.right_id and Function.comp.right_id into Function.comp_id.
• Merge Function.comp_const_right and Function.comp_const into Function.comp_const, use explicit arguments.
• Move Function.const_comp to Mathlib.Init.Function, use explicit arguments.

@@ -189,7 +189,7 @@ theorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β
   classical
     rw [isCompactElement_iff_le_of_directed_sSup_le]
     intro d hemp hdir hsup
-    rw [← Function.comp.left_id f]
+    rw [← Function.id_comp f]
     rw [← Finset.sup_image]
     apply Finset.sup_le_of_le_directed d hemp hdir
     rintro x hx

Change a few lemma names that have historically bothered me.

• Finset.card_le_of_subset → Finset.card_le_card
• Multiset.card_le_of_le → Multiset.card_le_card
• Multiset.card_lt_of_lt → Multiset.card_lt_card
• Set.ncard_le_of_subset → Set.ncard_le_ncard
• Finset.image_filter → Finset.filter_image
• CompleteLattice.finset_sup_compact_of_compact → CompleteLattice.isCompactElement_finset_sup

@@ -184,7 +184,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α
   exact hxk.ne (hxk.le.antisymm hkx)
 #align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 
-theorem finset_sup_compact_of_compact {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)
+theorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
     rw [isCompactElement_iff_le_of_directed_sSup_le]
@@ -198,7 +198,7 @@ theorem finset_sup_compact_of_compact {α β : Type*} [CompleteLattice α] {f :
     rw [isCompactElement_iff_le_of_directed_sSup_le] at h
     specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
     simpa only [exists_prop]
-#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
+#align complete_lattice.finset_sup_compact_of_compact CompleteLattice.isCompactElement_finsetSup
 
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
     IsSupFiniteCompact α := fun s => by

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -216,7 +216,7 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
 theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
     IsSupClosedCompact α := by
   intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
-  cases' t.eq_empty_or_nonempty with h h
+  rcases t.eq_empty_or_nonempty with h | h
   · subst h
     rw [Finset.sup_empty] at ht₂
     rw [ht₂]

I used the regex \(\(· (.) ·\) (.)\), replacing with ($2 $1 ·).

@@ -263,7 +263,7 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
 
 open List in
 theorem wellFounded_characterisations : List.TFAE
-    [WellFounded (( · > · ) : α → α → Prop),
+    [WellFounded ((· > ·) : α → α → Prop),
       IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] := by
   tfae_have 1 → 2
   · exact WellFounded.isSupFiniteCompact α

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

@@ -382,16 +382,19 @@ theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = 
 #align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
-  by simp_rw [@inf_comm _ _ _ a, h.inf_sSup_eq]
+protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) :
+    sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a := by
+  simp_rw [@inf_comm _ _ _ a, h.inf_sSup_eq]
 #align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
 
-protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
-  by rw [iSup, h.directedOn_range.inf_sSup_eq, iSup_range]
+protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) :
+    (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i := by
+  rw [iSup, h.directedOn_range.inf_sSup_eq, iSup_range]
 #align directed.inf_supr_eq Directed.inf_iSup_eq
 
-protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
-  by rw [iSup, h.directedOn_range.sSup_inf_eq, iSup_range]
+protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) :
+    (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a := by
+  rw [iSup, h.directedOn_range.sSup_inf_eq, iSup_range]
 #align directed.supr_inf_eq Directed.iSup_inf_eq
 
 protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :

@@ -319,6 +319,11 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     exact le_sSup _ _ hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 
+theorem WellFounded.finite_ne_bot_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop))
+    {ι : Type*} {t : ι → α} (ht : Independent t) : Set.Finite {i | t i ≠ ⊥} := by
+  refine Finite.of_finite_image (Finite.subset ?_ (image_subset_range t _)) ht.injOn
+  exact WellFounded.finite_of_setIndependent hwf ht.setIndependent_range
+
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type*}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
   haveI := (WellFounded.finite_of_setIndependent hwf ht.setIndependent_range).to_subtype

Also some loosely-related short lemmas

@@ -53,6 +53,8 @@ This is demonstrated by means of the following four lemmas:
 complete lattice, well-founded, compact
 -/
 
+open Set
+
 variable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}
 
 namespace CompleteLattice
@@ -438,6 +440,28 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
         exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 
+lemma CompleteLattice.independent_iff_supIndep_of_injOn {ι : Type*} {f : ι → α}
+    (hf : InjOn f {i | f i ≠ ⊥}) :
+    CompleteLattice.Independent f ↔ ∀ (s : Finset ι), s.SupIndep f := by
+  refine ⟨fun h ↦ h.supIndep', fun h ↦ CompleteLattice.independent_def'.mpr fun i ↦ ?_⟩
+  simp_rw [disjoint_iff, inf_sSup_eq_iSup_inf_sup_finset, iSup_eq_bot, ← disjoint_iff]
+  intro s hs
+  classical
+  rw [← Finset.sup_erase_bot]
+  set t := s.erase ⊥
+  replace hf : InjOn f (f ⁻¹' t) := fun i hi j _ hij ↦ by refine hf ?_ ?_ hij <;> aesop
+  have : (Finset.erase (insert i (t.preimage _ hf)) i).image f = t := by
+    ext a
+    simp only [Finset.mem_preimage, Finset.mem_erase, ne_eq, Finset.mem_insert, true_or, not_true,
+      Finset.erase_insert_eq_erase, not_and, Finset.mem_image]
+    refine ⟨by aesop, fun ⟨ha, has⟩ ↦ ?_⟩
+    obtain ⟨j, hj, rfl⟩ := hs has
+    exact ⟨j, ⟨hj, ha, has⟩, rfl⟩
+  rw [← this, Finset.sup_image]
+  specialize h (insert i (t.preimage _ hf))
+  rw [Finset.supIndep_iff_disjoint_erase] at h
+  exact h i (Finset.mem_insert_self i _)
+
 theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type*} {s : η → Set α}
     (hs : Directed (· ⊆ ·) s) (h : ∀ i, CompleteLattice.SetIndependent (s i)) :
     CompleteLattice.SetIndependent (⋃ i, s i) := by

This defines sets closed under supremum/infimum, shows that every set has a sup-closure/inf-closure and prove that if every sup-closed/inf-closed set in a sup-semilattice/inf-semilattice has a least upper bound/greatest lower, then the lattice is in fact complete.

As a bonus, we use our new predicate in Order.CompactlyGenerated.

Co-authored-by: Christopher Hoskin <christopher.hoskin@gmail.com>

@@ -6,6 +6,7 @@ Authors: Oliver Nash
 import Mathlib.Order.Atoms
 import Mathlib.Order.OrderIsoNat
 import Mathlib.Order.RelIso.Set
+import Mathlib.Order.SupClosed
 import Mathlib.Order.SupIndep
 import Mathlib.Order.Zorn
 import Mathlib.Data.Finset.Order
@@ -61,7 +62,7 @@ variable (α)
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
 contains its `sSup`. -/
 def IsSupClosedCompact : Prop :=
-  ∀ (s : Set α) (_ : s.Nonempty), (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), a ⊔ b ∈ s) → sSup s ∈ s
+  ∀ (s : Set α) (_ : s.Nonempty), SupClosed s → sSup s ∈ s
 #align complete_lattice.is_sup_closed_compact CompleteLattice.IsSupClosedCompact
 
 /-- A compactness property for a complete lattice is that any subset has a finite subset with the
@@ -219,7 +220,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
     rw [ht₂]
     simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
   · rw [ht₂]
-    exact t.sup_closed_of_sup_closed h ht₁ hsc
+    exact hsc.finsetSup_mem h ht₁
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
 
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :

@@ -287,13 +287,13 @@ theorem isSupClosedCompact_iff_wellFounded :
   (wellFounded_characterisations α).out 2 0
 #align complete_lattice.is_sup_closed_compact_iff_well_founded CompleteLattice.isSupClosedCompact_iff_wellFounded
 
-alias wellFounded_iff_isSupFiniteCompact ↔ _ IsSupFiniteCompact.wellFounded
+alias ⟨_, IsSupFiniteCompact.wellFounded⟩ := wellFounded_iff_isSupFiniteCompact
 #align complete_lattice.is_Sup_finite_compact.well_founded CompleteLattice.IsSupFiniteCompact.wellFounded
 
-alias isSupFiniteCompact_iff_isSupClosedCompact ↔ _ IsSupClosedCompact.isSupFiniteCompact
+alias ⟨_, IsSupClosedCompact.isSupFiniteCompact⟩ := isSupFiniteCompact_iff_isSupClosedCompact
 #align complete_lattice.is_sup_closed_compact.is_Sup_finite_compact CompleteLattice.IsSupClosedCompact.isSupFiniteCompact
 
-alias isSupClosedCompact_iff_wellFounded ↔ _ _root_.WellFounded.isSupClosedCompact
+alias ⟨_, _root_.WellFounded.isSupClosedCompact⟩ := isSupClosedCompact_iff_wellFounded
 #align well_founded.is_sup_closed_compact WellFounded.isSupClosedCompact
 
 variable {α}

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

This has nice performance benefits.

@@ -52,7 +52,7 @@ This is demonstrated by means of the following four lemmas:
 complete lattice, well-founded, compact
 -/
 
-variable {ι : Sort _} {α : Type _} [CompleteLattice α] {f : ι → α}
+variable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}
 
 namespace CompleteLattice
 
@@ -73,7 +73,7 @@ def IsSupFiniteCompact : Prop :=
 /-- An element `k` of a complete lattice is said to be compact if any set with `sSup`
 above `k` has a finite subset with `sSup` above `k`.  Such an element is also called
 "finite" or "S-compact". -/
-def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
+def IsCompactElement {α : Type*} [CompleteLattice α] (k : α) :=
   ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id
 #align complete_lattice.is_compact_element CompleteLattice.IsCompactElement
 
@@ -147,7 +147,7 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
       exact ⟨htS, by rwa [← htsup]⟩
 #align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
-theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
+theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
     let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
@@ -169,7 +169,7 @@ theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElemen
 
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
 its `sSup` strictly below `k`. -/
-theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
+theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}
     (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
     (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by
   rw [isCompactElement_iff_le_of_directed_sSup_le] at hk
@@ -181,7 +181,7 @@ theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice 
   exact hxk.ne (hxk.le.antisymm hkx)
 #align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 
-theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
+theorem finset_sup_compact_of_compact {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
     rw [isCompactElement_iff_le_of_directed_sSup_le]
@@ -316,7 +316,7 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     exact le_sSup _ _ hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 
-theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
+theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type*}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
   haveI := (WellFounded.finite_of_setIndependent hwf ht.setIndependent_range).to_subtype
   Finite.of_injective_finite_range (ht.injective h_ne_bot)
@@ -326,7 +326,7 @@ end CompleteLattice
 
 /-- A complete lattice is said to be compactly generated if any
 element is the `sSup` of compact elements. -/
-class IsCompactlyGenerated (α : Type _) [CompleteLattice α] : Prop where
+class IsCompactlyGenerated (α : Type*) [CompleteLattice α] : Prop where
   /-- In a compactly generated complete lattice,
     every element is the `sSup` of some set of compact elements. -/
   exists_sSup_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ sSup s = x
@@ -437,7 +437,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
         exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 
-theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type _} {s : η → Set α}
+theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type*} {s : η → Set α}
     (hs : Directed (· ⊆ ·) s) (h : ∀ i, CompleteLattice.SetIndependent (s i)) :
     CompleteLattice.SetIndependent (⋃ i, s i) := by
   by_cases hη : Nonempty η

@@ -546,12 +546,13 @@ Now we will prove that a compactly generated modular atomistic lattice is a comp
 Most explicitly, every element is the complement of a supremum of indepedendent atoms.
 -/
 
-/-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
-of `s` is an atom. See also `complementedLattice_of_sSup_atoms_eq_top`. -/
+/-- In an atomic lattice, every element `b` has a complement of the form `sSup s`, where each
+element of `s` is an atom. See also `complementedLattice_of_sSup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧
     IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a := by
-  -- porting note: `obtain` chokes on the placeholder.
+  -- porting note(https://github.com/leanprover-community/mathlib4/issues/5732):
+  -- `obtain` chokes on the placeholder.
   have := zorn_subset
     {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a}
     fun c hc1 hc2 =>

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit c813ed7de0f5115f956239124e9b30f3a621966f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Order.Atoms
 import Mathlib.Order.OrderIsoNat
@@ -18,6 +13,8 @@ import Mathlib.Data.Set.Intervals.OrderIso
 import Mathlib.Data.Finite.Set
 import Mathlib.Tactic.TFAE
 
+#align_import order.compactly_generated from "leanprover-community/mathlib"@"c813ed7de0f5115f956239124e9b30f3a621966f"
+
 /-!
 # Compactness properties for complete lattices
 

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -481,8 +481,7 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) :
   obtain rfl | H := eq_or_ne b k
   · left; ext; simp only [Set.Iic.coe_top, Subtype.coe_mk]
   right
-  have ih : ?_ := ?_ -- Porting note: this is an ugly hack, but `?ih` on the next line fails
-  obtain ⟨a, a₀, ba, h⟩ := zorn_nonempty_partialOrder₀ (Set.Iio k) ih b (lt_of_le_of_ne hbk H)
+  have ⟨a, a₀, ba, h⟩ := zorn_nonempty_partialOrder₀ (Set.Iio k) ?_ b (lt_of_le_of_ne hbk H)
   · refine' ⟨⟨a, le_of_lt a₀⟩, ⟨ne_of_lt a₀, fun c hck => by_contradiction fun c₀ => _⟩, ba⟩
     cases h c.1 (lt_of_le_of_ne c.2 fun con => c₀ (Subtype.ext con)) hck.le
     exact lt_irrefl _ hck

Currently, (for both Set and Finset) insert_subset is an iff lemma stating that insert a s ⊆ t if and only if a ∈ t and s ⊆ t. For both types, this PR renames this lemma to insert_subset_iff, and adds an insert_subset lemma that gives the implication just in the reverse direction : namely theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t .

This both aligns the naming with union_subset and union_subset_iff, and removes the need for the awkward insert_subset.mpr ⟨_,_⟩ idiom. It touches a lot of files (too many to list), but in a trivial way.

@@ -207,7 +207,7 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
   refine' ⟨t, ht₁, (sSup_le _ _ fun y hy => _).antisymm _⟩
   · classical
     rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
-        (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
+        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]
     simp
   · rw [Finset.sup_id_eq_sSup]
     exact sSup_le_sSup ht₁
@@ -436,7 +436,7 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
       have h' := (h (insert a t) ?_ (t.mem_insert_self a)).eq_bot
       · rwa [Finset.coe_insert, Set.insert_diff_self_of_not_mem] at h'
         exact fun con => ((Set.mem_diff a).1 (ht con)).2 (Set.mem_singleton a)
-      · rw [Finset.coe_insert, Set.insert_subset]
+      · rw [Finset.coe_insert, Set.insert_subset_iff]
         exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 

@@ -411,7 +411,7 @@ protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
 
 /-- This property is equivalent to `α` being upper continuous. -/
 theorem inf_sSup_eq_iSup_inf_sup_finset :
-    a ⊓ sSup s = ⨆ (t : Finset α) (_H : ↑t ⊆ s), a ⊓ t.sup id :=
+    a ⊓ sSup s = ⨆ (t : Finset α) (_ : ↑t ⊆ s), a ⊓ t.sup id :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -551,7 +551,7 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 -/
 
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
-of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
+of `s` is an atom. See also `complementedLattice_of_sSup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧
     IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a := by

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>

@@ -62,27 +62,27 @@ namespace CompleteLattice
 variable (α)
 
 /-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset
-contains its `supₛ`. -/
+contains its `sSup`. -/
 def IsSupClosedCompact : Prop :=
-  ∀ (s : Set α) (_ : s.Nonempty), (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), a ⊔ b ∈ s) → supₛ s ∈ s
+  ∀ (s : Set α) (_ : s.Nonempty), (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), a ⊔ b ∈ s) → sSup s ∈ s
 #align complete_lattice.is_sup_closed_compact CompleteLattice.IsSupClosedCompact
 
 /-- A compactness property for a complete lattice is that any subset has a finite subset with the
-same `supₛ`. -/
+same `sSup`. -/
 def IsSupFiniteCompact : Prop :=
-  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ supₛ s = t.sup id
+  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id
 #align complete_lattice.is_Sup_finite_compact CompleteLattice.IsSupFiniteCompact
 
-/-- An element `k` of a complete lattice is said to be compact if any set with `supₛ`
-above `k` has a finite subset with `supₛ` above `k`.  Such an element is also called
+/-- An element `k` of a complete lattice is said to be compact if any set with `sSup`
+above `k` has a finite subset with `sSup` above `k`.  Such an element is also called
 "finite" or "S-compact". -/
 def IsCompactElement {α : Type _} [CompleteLattice α] (k : α) :=
-  ∀ s : Set α, k ≤ supₛ s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id
+  ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id
 #align complete_lattice.is_compact_element CompleteLattice.IsCompactElement
 
 theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
     CompleteLattice.IsCompactElement k ↔
-      ∀ (ι : Type u) (s : ι → α), k ≤ supᵢ s → ∃ t : Finset ι, k ≤ t.sup s := by
+      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by
   classical
     constructor
     · intro H ι s hs
@@ -98,18 +98,18 @@ theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
       obtain ⟨t, ht⟩ :=
         H s Subtype.val
           (by
-            delta supᵢ
+            delta iSup
             rwa [Subtype.range_coe])
       refine' ⟨t.image Subtype.val, by simp, ht.trans _⟩
       rw [Finset.sup_le_iff]
       exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)
 #align complete_lattice.is_compact_element_iff CompleteLattice.isCompactElement_iff
 
-/-- An element `k` is compact if and only if any directed set with `supₛ` above
+/-- An element `k` is compact if and only if any directed set with `sSup` above
 `k` already got above `k` at some point in the set. -/
-theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
+theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
     IsCompactElement k ↔
-      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ supₛ s → ∃ x : α, x ∈ s ∧ k ≤ x := by
+      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by
   classical
     constructor
     · intro hk s hne hdir hsup
@@ -131,8 +131,8 @@ theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
           · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]
           · simp only [hc.right, hd.right, Finset.sup_union]
         simp only [and_self_iff, le_sup_left, le_sup_right]
-      have sup_S : supₛ s ≤ supₛ S := by
-        apply supₛ_le_supₛ
+      have sup_S : sSup s ≤ sSup S := by
+        apply sSup_le_sSup
         intro x hx
         use {x}
         simpa only [and_true_iff, id.def, Finset.coe_singleton, eq_self_iff_true,
@@ -148,46 +148,46 @@ theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
       obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
       use t
       exact ⟨htS, by rwa [← htsup]⟩
-#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_le
+#align complete_lattice.is_compact_element_iff_le_of_directed_Sup_le CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
 
-theorem IsCompactElement.exists_finset_of_le_supᵢ {k : α} (hk : IsCompactElement k) {ι : Type _}
+theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type _}
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
     let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
     have h1 : DirectedOn (· ≤ ·) (Set.range g) := by
       rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
       exact
-        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, supᵢ_le_supᵢ_of_subset (Finset.subset_union_left s t),
-          supᵢ_le_supᵢ_of_subset (Finset.subset_union_right s t)⟩
-    have h2 : k ≤ supₛ (Set.range g) :=
+        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset (Finset.subset_union_left s t),
+          iSup_le_iSup_of_subset (Finset.subset_union_right s t)⟩
+    have h2 : k ≤ sSup (Set.range g) :=
       h.trans
-        (supᵢ_le fun i =>
-          le_supₛ_of_le ⟨{i}, rfl⟩
-            (le_supᵢ_of_le i (le_supᵢ_of_le (Finset.mem_singleton_self i) le_rfl)))
+        (iSup_le fun i =>
+          le_sSup_of_le ⟨{i}, rfl⟩
+            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))
     obtain ⟨-, ⟨s, rfl⟩, hs⟩ :=
-      (isCompactElement_iff_le_of_directed_supₛ_le α k).mp hk (Set.range g) (Set.range_nonempty g)
+      (isCompactElement_iff_le_of_directed_sSup_le α k).mp hk (Set.range g) (Set.range_nonempty g)
         h1 h2
     exact ⟨s, hs⟩
-#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_supᵢ
+#align complete_lattice.is_compact_element.exists_finset_of_le_supr CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
 
 /-- A compact element `k` has the property that any directed set lying strictly below `k` has
-its `supₛ` strictly below `k`. -/
-theorem IsCompactElement.directed_supₛ_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
+its `sSup` strictly below `k`. -/
+theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type _} [CompleteLattice α] {k : α}
     (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
-    (hbelow : ∀ x ∈ s, x < k) : supₛ s < k := by
-  rw [isCompactElement_iff_le_of_directed_supₛ_le] at hk
+    (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by
+  rw [isCompactElement_iff_le_of_directed_sSup_le] at hk
   by_contra h
-  have sSup : supₛ s ≤ k := supₛ_le s k fun s hs => (hbelow s hs).le
-  replace sSup : supₛ s = k := eq_iff_le_not_lt.mpr ⟨sSup, h⟩
+  have sSup' : sSup s ≤ k := sSup_le s k fun s hs => (hbelow s hs).le
+  replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩
   obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le
   obtain hxk := hbelow x hxs
   exact hxk.ne (hxk.le.antisymm hkx)
-#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_supₛ_lt_of_lt
+#align complete_lattice.is_compact_element.directed_Sup_lt_of_lt CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
 
 theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f : β → α} (s : Finset β)
     (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by
   classical
-    rw [isCompactElement_iff_le_of_directed_supₛ_le]
+    rw [isCompactElement_iff_le_of_directed_sSup_le]
     intro d hemp hdir hsup
     rw [← Function.comp.left_id f]
     rw [← Finset.sup_image]
@@ -195,7 +195,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     rintro x hx
     obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx
     specialize h p hps
-    rw [isCompactElement_iff_le_of_directed_supₛ_le] at h
+    rw [isCompactElement_iff_le_of_directed_sSup_le] at h
     specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
     simpa only [exists_prop]
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
@@ -204,13 +204,13 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
     IsSupFiniteCompact α := fun s => by
   let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
   obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := h.has_min S ⟨⊥, ∅, by simp⟩
-  refine' ⟨t, ht₁, (supₛ_le _ _ fun y hy => _).antisymm _⟩
+  refine' ⟨t, ht₁, (sSup_le _ _ fun y hy => _).antisymm _⟩
   · classical
     rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
         (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
     simp
-  · rw [Finset.sup_id_eq_supₛ]
-    exact supₛ_le_supₛ ht₁
+  · rw [Finset.sup_id_eq_sSup]
+    exact sSup_le_sSup ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
 
 theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
@@ -220,7 +220,7 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
   · subst h
     rw [Finset.sup_empty] at ht₂
     rw [ht₂]
-    simp [eq_singleton_bot_of_supₛ_eq_bot_of_nonempty ht₂ hne]
+    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]
   · rw [ht₂]
     exact t.sup_closed_of_sup_closed h ht₁ hsc
 #align complete_lattice.is_Sup_finite_compact.is_sup_closed_compact CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
@@ -228,14 +228,14 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     WellFounded ((· > ·) : α → α → Prop) := by
   refine' RelEmbedding.wellFounded_iff_no_descending_seq.mpr ⟨fun a => _⟩
-  suffices supₛ (Set.range a) ∈ Set.range a by
+  suffices sSup (Set.range a) ∈ Set.range a by
     obtain ⟨n, hn⟩ := Set.mem_range.mp this
-    have h' : supₛ (Set.range a) < a (n + 1) := by
+    have h' : sSup (Set.range a) < a (n + 1) := by
       change _ > _
       simp [← hn, a.map_rel_iff]
     apply lt_irrefl (a (n + 1))
     apply lt_of_le_of_lt _ h'
-    apply le_supₛ
+    apply le_sSup
     apply Set.mem_range_self
   apply h (Set.range a)
   · use a 37
@@ -252,12 +252,12 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
   · obtain ⟨t, ⟨hts, htsup⟩⟩ := h s
     use t, hts
     rwa [← htsup]
-  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (supₛ s) s (by rfl)
-    have : supₛ s = t.sup id := by
-      suffices t.sup id ≤ supₛ s by apply le_antisymm <;> assumption
+  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)
+    have : sSup s = t.sup id := by
+      suffices t.sup id ≤ sSup s by apply le_antisymm <;> assumption
       simp only [id.def, Finset.sup_le_iff]
       intro x hx
-      exact le_supₛ _ _ (hts hx)
+      exact le_sSup _ _ (hts hx)
     exact ⟨t, hts, this⟩
 #align complete_lattice.is_Sup_finite_compact_iff_all_elements_compact CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
 
@@ -311,12 +311,12 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
       obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
       exact ⟨x, hx₁, by simpa [not_or] using hx₂⟩
     obtain ⟨x, hx₀, hx₁, hx₂⟩ := contra
-    replace hs : x ⊓ supₛ s = ⊥
+    replace hs : x ⊓ sSup s = ⊥
     · have := hs.mono (by simp [ht₁, hx₀, -Set.union_singleton] : ↑t ∪ {x} ≤ s) (by simp : x ∈ _)
-      simpa [Disjoint, hx₂, ← t.sup_id_eq_supₛ, ← ht₂] using this.eq_bot
+      simpa [Disjoint, hx₂, ← t.sup_id_eq_sSup, ← ht₂] using this.eq_bot
     apply hx₁
     rw [← hs, eq_comm, inf_eq_left]
-    exact le_supₛ _ _ hx₀
+    exact le_sSup _ _ hx₀
 #align complete_lattice.well_founded.finite_of_set_independent CompleteLattice.WellFounded.finite_of_setIndependent
 
 theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type _}
@@ -328,11 +328,11 @@ theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α →
 end CompleteLattice
 
 /-- A complete lattice is said to be compactly generated if any
-element is the `supₛ` of compact elements. -/
+element is the `sSup` of compact elements. -/
 class IsCompactlyGenerated (α : Type _) [CompleteLattice α] : Prop where
   /-- In a compactly generated complete lattice,
-    every element is the `supₛ` of some set of compact elements. -/
-  exists_supₛ_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ supₛ s = x
+    every element is the `sSup` of some set of compact elements. -/
+  exists_sSup_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ sSup s = x
 #align is_compactly_generated IsCompactlyGenerated
 
 section
@@ -340,97 +340,97 @@ section
 variable [CompleteLattice α] [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
 @[simp]
-theorem supₛ_compact_le_eq (b) :
-    supₛ { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b := by
-  rcases IsCompactlyGenerated.exists_supₛ_eq b with ⟨s, hs, rfl⟩
-  exact le_antisymm (supₛ_le fun c hc => hc.2) (supₛ_le_supₛ fun c cs => ⟨hs c cs, le_supₛ cs⟩)
-#align Sup_compact_le_eq supₛ_compact_le_eq
+theorem sSup_compact_le_eq (b) :
+    sSup { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b := by
+  rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩
+  exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)
+#align Sup_compact_le_eq sSup_compact_le_eq
 
 @[simp]
-theorem supₛ_compact_eq_top : supₛ { a : α | CompleteLattice.IsCompactElement a } = ⊤ := by
-  refine' Eq.trans (congr rfl (Set.ext fun x => _)) (supₛ_compact_le_eq ⊤)
+theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ := by
+  refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq ⊤)
   exact (and_iff_left le_top).symm
-#align Sup_compact_eq_top supₛ_compact_eq_top
+#align Sup_compact_eq_top sSup_compact_eq_top
 
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
   ⟨fun ab c _ ca => le_trans ca ab, fun h => by
-    rw [← supₛ_compact_le_eq a, ← supₛ_compact_le_eq b]
-    exact supₛ_le_supₛ fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
+    rw [← sSup_compact_le_eq a, ← sSup_compact_le_eq b]
+    exact sSup_le_sSup fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
+theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
       by_cases hs : s.Nonempty
       · intro c hc hcinf
         rw [le_inf_iff] at hcinf
-        rw [CompleteLattice.isCompactElement_iff_le_of_directed_supₛ_le] at hc
+        rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc
         rcases hc s hs h hcinf.2 with ⟨d, ds, cd⟩
-        refine' (le_inf hcinf.1 cd).trans (le_trans _ (le_supᵢ₂ d ds))
+        refine' (le_inf hcinf.1 cd).trans (le_trans _ (le_iSup₂ d ds))
         rfl
       · rw [Set.not_nonempty_iff_eq_empty] at hs
         simp [hs])
-    supᵢ_inf_le_inf_supₛ
-#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eq
+    iSup_inf_le_inf_sSup
+#align directed_on.inf_Sup_eq DirectedOn.inf_sSup_eq
 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : supₛ s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
-  by simp_rw [@inf_comm _ _ _ a, h.inf_supₛ_eq]
-#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eq
+protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) : sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
+  by simp_rw [@inf_comm _ _ _ a, h.inf_sSup_eq]
+#align directed_on.Sup_inf_eq DirectedOn.sSup_inf_eq
 
-protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
-  by rw [supᵢ, h.directedOn_range.inf_supₛ_eq, supᵢ_range]
-#align directed.inf_supr_eq Directed.inf_supᵢ_eq
+protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
+  by rw [iSup, h.directedOn_range.inf_sSup_eq, iSup_range]
+#align directed.inf_supr_eq Directed.inf_iSup_eq
 
-protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
-  by rw [supᵢ, h.directedOn_range.supₛ_inf_eq, supᵢ_range]
-#align directed.supr_inf_eq Directed.supᵢ_inf_eq
+protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
+  by rw [iSup, h.directedOn_range.sSup_inf_eq, iSup_range]
+#align directed.supr_inf_eq Directed.iSup_inf_eq
 
-protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
-    Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
-  simp_rw [disjoint_iff, h.inf_supₛ_eq, supᵢ_eq_bot]
-#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_right
+protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :
+    Disjoint a (sSup s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
+  simp_rw [disjoint_iff, h.inf_sSup_eq, iSup_eq_bot]
+#align directed_on.disjoint_Sup_right DirectedOn.disjoint_sSup_right
 
-protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
-    Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
-  simp_rw [disjoint_iff, h.supₛ_inf_eq, supᵢ_eq_bot]
-#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_left
+protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
+    Disjoint (sSup s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
+  simp_rw [disjoint_iff, h.sSup_inf_eq, iSup_eq_bot]
+#align directed_on.disjoint_Sup_left DirectedOn.disjoint_sSup_left
 
-protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
+protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
     Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
-  simp_rw [disjoint_iff, h.inf_supᵢ_eq, supᵢ_eq_bot]
-#align directed.disjoint_supr_right Directed.disjoint_supᵢ_right
+  simp_rw [disjoint_iff, h.inf_iSup_eq, iSup_eq_bot]
+#align directed.disjoint_supr_right Directed.disjoint_iSup_right
 
-protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
+protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
     Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
-  simp_rw [disjoint_iff, h.supᵢ_inf_eq, supᵢ_eq_bot]
-#align directed.disjoint_supr_left Directed.disjoint_supᵢ_left
+  simp_rw [disjoint_iff, h.iSup_inf_eq, iSup_eq_bot]
+#align directed.disjoint_supr_left Directed.disjoint_iSup_left
 
 /-- This property is equivalent to `α` being upper continuous. -/
-theorem inf_supₛ_eq_supᵢ_inf_sup_finset :
-    a ⊓ supₛ s = ⨆ (t : Finset α) (_H : ↑t ⊆ s), a ⊓ t.sup id :=
+theorem inf_sSup_eq_iSup_inf_sup_finset :
+    a ⊓ sSup s = ⨆ (t : Finset α) (_H : ↑t ⊆ s), a ⊓ t.sup id :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
       intro c hc hcinf
       rw [le_inf_iff] at hcinf
       rcases hc s hcinf.2 with ⟨t, ht1, ht2⟩
-      refine' (le_inf hcinf.1 ht2).trans (le_trans _ (le_supᵢ₂ t ht1))
+      refine' (le_inf hcinf.1 ht2).trans (le_trans _ (le_iSup₂ t ht1))
       rfl)
-    (supᵢ_le fun t =>
-      supᵢ_le fun h => inf_le_inf_left _ ((Finset.sup_id_eq_supₛ t).symm ▸ supₛ_le_supₛ h))
-#align inf_Sup_eq_supr_inf_sup_finset inf_supₛ_eq_supᵢ_inf_sup_finset
+    (iSup_le fun t =>
+      iSup_le fun h => inf_le_inf_left _ ((Finset.sup_id_eq_sSup t).symm ▸ sSup_le_sSup h))
+#align inf_Sup_eq_supr_inf_sup_finset inf_sSup_eq_iSup_inf_sup_finset
 
 theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
     CompleteLattice.SetIndependent s ↔
       ∀ t : Finset α, ↑t ⊆ s → CompleteLattice.SetIndependent (↑t : Set α) :=
   ⟨fun hs t ht => hs.mono ht, fun h a ha => by
-    rw [disjoint_iff, inf_supₛ_eq_supᵢ_inf_sup_finset, supᵢ_eq_bot]
+    rw [disjoint_iff, inf_sSup_eq_iSup_inf_sup_finset, iSup_eq_bot]
     intro t
-    rw [supᵢ_eq_bot, Finset.sup_id_eq_supₛ]
+    rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
     intro ht
     classical
       have h' := (h (insert a t) ?_ (t.mem_insert_self a)).eq_bot
@@ -440,26 +440,26 @@ theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
         exact ⟨ha, Set.Subset.trans ht (Set.diff_subset _ _)⟩⟩
 #align complete_lattice.set_independent_iff_finite CompleteLattice.setIndependent_iff_finite
 
-theorem CompleteLattice.setIndependent_unionᵢ_of_directed {η : Type _} {s : η → Set α}
+theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type _} {s : η → Set α}
     (hs : Directed (· ⊆ ·) s) (h : ∀ i, CompleteLattice.SetIndependent (s i)) :
     CompleteLattice.SetIndependent (⋃ i, s i) := by
   by_cases hη : Nonempty η
   · rw [CompleteLattice.setIndependent_iff_finite]
     intro t ht
-    obtain ⟨I, fi, hI⟩ := Set.finite_subset_unionᵢ t.finite_toSet ht
+    obtain ⟨I, fi, hI⟩ := Set.finite_subset_iUnion t.finite_toSet ht
     obtain ⟨i, hi⟩ := hs.finset_le fi.toFinset
     exact (h i).mono
-        (Set.Subset.trans hI <| Set.unionᵢ₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))
+        (Set.Subset.trans hI <| Set.iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))
   · rintro a ⟨_, ⟨i, _⟩, _⟩
     exfalso
     exact hη ⟨i⟩
-#align complete_lattice.set_independent_Union_of_directed CompleteLattice.setIndependent_unionᵢ_of_directed
+#align complete_lattice.set_independent_Union_of_directed CompleteLattice.setIndependent_iUnion_of_directed
 
-theorem CompleteLattice.independent_unionₛ_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
+theorem CompleteLattice.independent_sUnion_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
     (h : ∀ a ∈ s, CompleteLattice.SetIndependent a) : CompleteLattice.SetIndependent (⋃₀ s) := by
-  rw [Set.unionₛ_eq_unionᵢ]
-  exact CompleteLattice.setIndependent_unionᵢ_of_directed hs.directed_val (by simpa using h)
-#align complete_lattice.independent_sUnion_of_directed CompleteLattice.independent_unionₛ_of_directed
+  rw [Set.sUnion_eq_iUnion]
+  exact CompleteLattice.setIndependent_iUnion_of_directed hs.directed_val (by simpa using h)
+#align complete_lattice.independent_sUnion_of_directed CompleteLattice.independent_sUnion_of_directed
 
 end
 
@@ -469,7 +469,7 @@ theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α →
     IsCompactlyGenerated α := by
   rw [wellFounded_iff_isSupFiniteCompact, isSupFiniteCompact_iff_all_elements_compact] at h
   -- x is the join of the set of compact elements {x}
-  exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, supₛ_singleton⟩⟩⟩
+  exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 #align complete_lattice.compactly_generated_of_well_founded CompleteLattice.isCompactlyGenerated_of_wellFounded
 
 /-- A compact element `k` has the property that any `b < k` lies below a "maximal element below
@@ -488,8 +488,8 @@ theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) :
     exact lt_irrefl _ hck
   · intro S SC cC I _
     by_cases hS : S.Nonempty
-    · refine' ⟨supₛ S, h.directed_supₛ_lt_of_lt hS cC.directedOn SC, _⟩
-      intro; apply le_supₛ
+    · refine' ⟨sSup S, h.directed_sSup_lt_of_lt hS cC.directedOn SC, _⟩
+      intro; apply le_sSup
     exact
       ⟨b, lt_of_le_of_ne hbk H, by
         simp only [Set.not_nonempty_iff_eq_empty.mp hS, Set.mem_empty_iff_false, forall_const,
@@ -510,7 +510,7 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
   ⟨fun b => by
     by_cases h : { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } ⊆ {⊥}
     · left
-      rw [← supₛ_compact_le_eq b, supₛ_eq_bot]
+      rw [← sSup_compact_le_eq b, sSup_eq_bot]
       exact h
     · rcases Set.not_subset.1 h with ⟨c, ⟨hc, hcb⟩, hcbot⟩
       right
@@ -532,17 +532,17 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
   ⟨fun b =>
     ⟨{ a | IsAtom a ∧ a ≤ b }, by
       symm
-      have hle : supₛ { a : α | IsAtom a ∧ a ≤ b } ≤ b := supₛ_le fun _ => And.right
+      have hle : sSup { a : α | IsAtom a ∧ a ≤ b } ≤ b := sSup_le fun _ => And.right
       apply (lt_or_eq_of_le hle).resolve_left _
       intro con
-      obtain ⟨c, hc⟩ := exists_isCompl (⟨supₛ { a : α | IsAtom a ∧ a ≤ b }, hle⟩ : Set.Iic b)
+      obtain ⟨c, hc⟩ := exists_isCompl (⟨sSup { a : α | IsAtom a ∧ a ≤ b }, hle⟩ : Set.Iic b)
       obtain rfl | ⟨a, ha, hac⟩ := eq_bot_or_exists_atom_le c
       · exact ne_of_lt con (Subtype.ext_iff.1 (eq_top_of_isCompl_bot hc))
       · apply ha.1
         rw [eq_bot_iff]
         apply le_trans (le_inf _ hac) hc.disjoint.le_bot
         rw [← Subtype.coe_le_coe, Subtype.coe_mk]
-        exact le_supₛ ⟨ha.of_isAtom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
+        exact le_sSup ⟨ha.of_isAtom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
 #align is_atomistic_of_complemented_lattice isAtomistic_of_complementedLattice
 
 /-!
@@ -552,33 +552,33 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
-theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
+theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
     ∃ s : Set α, CompleteLattice.SetIndependent s ∧
-    IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a := by
+    IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a := by
   -- porting note: `obtain` chokes on the placeholder.
   have := zorn_subset
-    {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (supₛ s) ∧ ∀ a ∈ s, IsAtom a}
+    {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a}
     fun c hc1 hc2 =>
       ⟨⋃₀ c,
-        ⟨CompleteLattice.independent_unionₛ_of_directed hc2.directedOn fun s hs => (hc1 hs).1, ?_,
+        ⟨CompleteLattice.independent_sUnion_of_directed hc2.directedOn fun s hs => (hc1 hs).1, ?_,
           fun a ⟨s, sc, as⟩ => (hc1 sc).2.2 a as⟩,
-        fun _ => Set.subset_unionₛ_of_mem⟩
+        fun _ => Set.subset_sUnion_of_mem⟩
   obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ := this
   swap
-  · rw [supₛ_unionₛ, ← supₛ_image, DirectedOn.disjoint_supₛ_right]
+  · rw [sSup_sUnion, ← sSup_image, DirectedOn.disjoint_sSup_right]
     · rintro _ ⟨s, hs, rfl⟩
       exact (hc1 hs).2.1
     · rw [directedOn_image]
-      exact hc2.directedOn.mono @fun s t => supₛ_le_supₛ
+      exact hc2.directedOn.mono @fun s t => sSup_le_sSup
   refine' ⟨s, s_ind, ⟨b_inf_Sup_s, _⟩, s_atoms⟩
-  rw [codisjoint_iff_le_sup, ← h, supₛ_le_iff]
+  rw [codisjoint_iff_le_sup, ← h, sSup_le_iff]
   intro a ha
   rw [← inf_eq_left]
   refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
   rw [← con, eq_comm, inf_eq_left]
-  refine' (le_supₛ _).trans le_sup_right
+  refine' (le_sSup _).trans le_sup_right
   rw [← disjoint_iff] at con
-  have a_dis_Sup_s : Disjoint a (supₛ s) := con.mono_right le_sup_right
+  have a_dis_Sup_s : Disjoint a (sSup s) := con.mono_right le_sup_right
   -- porting note: The two following `fun x hx => _` are no-op
   rw [← s_max (s ∪ {a}) ⟨fun x hx => _, _, fun x hx => _⟩ (Set.subset_union_left _ _)]
   · exact Set.mem_union_right _ (Set.mem_singleton _)
@@ -586,43 +586,43 @@ theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom
     rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain rfl | xa := eq_or_ne x a
     · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
-      exact con.mono_right ((supₛ_le_supₛ <| Set.diff_subset _ _).trans le_sup_right)
+      exact con.mono_right ((sSup_le_sSup <| Set.diff_subset _ _).trans le_sup_right)
     · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} := by
         simp only [Set.union_singleton]
         rw [Set.insert_diff_of_not_mem]
         rw [Set.mem_singleton_iff]
         exact Ne.symm xa
-      rw [h, supₛ_union, supₛ_singleton]
+      rw [h, sSup_union, sSup_singleton]
       apply
         (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
           (a_dis_Sup_s.mono_right _).symm
-      rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
-  · rw [supₛ_union, supₛ_singleton]
+      rw [← sSup_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
+  · rw [sSup_union, sSup_singleton]
     exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left con.symm
   · intro x hx
     rw [Set.mem_union, Set.mem_singleton_iff] at hx
     obtain hx | rfl := hx
     · exact s_atoms x hx
     · exact ha
-#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atoms
+#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_sSup_atoms
 
-theorem exists_setIndependent_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
-    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ supₛ s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
-  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h ⊥
+theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_sSup_atoms h ⊥
   ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
-#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_top
+#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_sSup_atoms_eq_top
 
 /-- See [Theorem 6.6][calugareanu]. -/
-theorem complementedLattice_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
+theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
     ComplementedLattice α :=
   ⟨fun b =>
-    let ⟨s, _, s_top, _⟩ := exists_setIndependent_isCompl_supₛ_atoms h b
-    ⟨supₛ s, s_top⟩⟩
-#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_top
+    let ⟨s, _, s_top, _⟩ := exists_setIndependent_isCompl_sSup_atoms h b
+    ⟨sSup s, s_top⟩⟩
+#align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_sSup_atoms_eq_top
 
 /-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_isAtomistic [IsAtomistic α] : ComplementedLattice α :=
-  complementedLattice_of_supₛ_atoms_eq_top supₛ_atoms_eq_top
+  complementedLattice_of_sSup_atoms_eq_top sSup_atoms_eq_top
 #align complemented_lattice_of_is_atomistic complementedLattice_of_isAtomistic
 
 theorem complementedLattice_iff_isAtomistic : ComplementedLattice α ↔ IsAtomistic α := by

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -122,8 +122,7 @@ theorem isCompactElement_iff_le_of_directed_supₛ_le (k : α) :
       -- Consider the set of finite joins of elements of the (plain) set s.
       let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }
       -- S is directed, nonempty, and still has sup above k.
-      have dir_US : DirectedOn (· ≤ ·) S :=
-        by
+      have dir_US : DirectedOn (· ≤ ·) S := by
         rintro x ⟨c, hc⟩ y ⟨d, hd⟩
         use x ⊔ y
         constructor
@@ -155,8 +154,7 @@ theorem IsCompactElement.exists_finset_of_le_supᵢ {k : α} (hk : IsCompactElem
     (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by
   classical
     let g : Finset ι → α := fun s => ⨆ i ∈ s, f i
-    have h1 : DirectedOn (· ≤ ·) (Set.range g) :=
-      by
+    have h1 : DirectedOn (· ≤ ·) (Set.range g) := by
       rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩
       exact
         ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, supᵢ_le_supᵢ_of_subset (Finset.subset_union_left s t),
@@ -203,8 +201,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
-    IsSupFiniteCompact α := fun s =>
-  by
+    IsSupFiniteCompact α := fun s => by
   let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
   obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := h.has_min S ⟨⊥, ∅, by simp⟩
   refine' ⟨t, ht₁, (supₛ_le _ _ fun y hy => _).antisymm _⟩
@@ -216,8 +213,8 @@ theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α 
     exact supₛ_le_supₛ ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
 
-theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSupClosedCompact α :=
-  by
+theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
+    IsSupClosedCompact α := by
   intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h
   cases' t.eq_empty_or_nonempty with h h
   · subst h
@@ -231,11 +228,9 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) : IsSu
 theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
     WellFounded ((· > ·) : α → α → Prop) := by
   refine' RelEmbedding.wellFounded_iff_no_descending_seq.mpr ⟨fun a => _⟩
-  suffices supₛ (Set.range a) ∈ Set.range a
-    by
+  suffices supₛ (Set.range a) ∈ Set.range a by
     obtain ⟨n, hn⟩ := Set.mem_range.mp this
-    have h' : supₛ (Set.range a) < a (n + 1) :=
-      by
+    have h' : supₛ (Set.range a) < a (n + 1) := by
       change _ > _
       simp [← hn, a.map_rel_iff]
     apply lt_irrefl (a (n + 1))
@@ -258,8 +253,7 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
     use t, hts
     rwa [← htsup]
   · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (supₛ s) s (by rfl)
-    have : supₛ s = t.sup id :=
-      by
+    have : supₛ s = t.sup id := by
       suffices t.sup id ≤ supₛ s by apply le_antisymm <;> assumption
       simp only [id.def, Finset.sup_le_iff]
       intro x hx
@@ -346,8 +340,8 @@ section
 variable [CompleteLattice α] [IsCompactlyGenerated α] {a b : α} {s : Set α}
 
 @[simp]
-theorem supₛ_compact_le_eq (b) : supₛ { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b :=
-  by
+theorem supₛ_compact_le_eq (b) :
+    supₛ { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b := by
   rcases IsCompactlyGenerated.exists_supₛ_eq b with ⟨s, hs, rfl⟩
   exact le_antisymm (supₛ_le fun c hc => hc.2) (supₛ_le_supₛ fun c cs => ⟨hs c cs, le_supₛ cs⟩)
 #align Sup_compact_le_eq supₛ_compact_le_eq
@@ -360,8 +354,7 @@ theorem supₛ_compact_eq_top : supₛ { a : α | CompleteLattice.IsCompactEleme
 
 theorem le_iff_compact_le_imp {a b : α} :
     a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
-  ⟨fun ab c _ ca => le_trans ca ab, fun h =>
-    by
+  ⟨fun ab c _ ca => le_trans ca ab, fun h => by
     rw [← supₛ_compact_le_eq a, ← supₛ_compact_le_eq b]
     exact supₛ_le_supₛ fun c hc => ⟨hc.1, h c hc.1 hc.2⟩⟩
 #align le_iff_compact_le_imp le_iff_compact_le_imp
@@ -560,8 +553,8 @@ Most explicitly, every element is the complement of a supremum of indepedendent
 /-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
 of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
 theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
-    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
-  by
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧
+    IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a := by
   -- porting note: `obtain` chokes on the placeholder.
   have := zorn_subset
     {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (supₛ s) ∧ ∀ a ∈ s, IsAtom a}
@@ -594,8 +587,7 @@ theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom
     obtain rfl | xa := eq_or_ne x a
     · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
       exact con.mono_right ((supₛ_le_supₛ <| Set.diff_subset _ _).trans le_sup_right)
-    · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
-        by
+    · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} := by
         simp only [Set.union_singleton]
         rw [Set.insert_diff_of_not_mem]
         rw [Set.mem_singleton_iff]

Match https://github.com/leanprover-community/mathlib/pull/18893

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit e8cf0cfec5fcab9baf46dc17d30c5e22048468be
+! leanprover-community/mathlib commit c813ed7de0f5115f956239124e9b30f3a621966f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -192,7 +192,7 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
     rw [isCompactElement_iff_le_of_directed_supₛ_le]
     intro d hemp hdir hsup
     rw [← Function.comp.left_id f]
-    rw [← Finset.sup_finset_image]
+    rw [← Finset.sup_image]
     apply Finset.sup_le_of_le_directed d hemp hdir
     rintro x hx
     obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx

Match https://github.com/leanprover-community/mathlib/pull/8475

• order.compactly_generated@210657c4ea4a4a7b234392f70a3a2a83346dfa90..e8cf0cfec5fcab9baf46dc17d30c5e22048468be
• order.directed@485b24ed47b1b7978d38a1e445158c6224c3f42c..e8cf0cfec5fcab9baf46dc17d30c5e22048468be

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
+! leanprover-community/mathlib commit e8cf0cfec5fcab9baf46dc17d30c5e22048468be
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -55,8 +55,7 @@ This is demonstrated by means of the following four lemmas:
 complete lattice, well-founded, compact
 -/
 
-
-variable {α : Type _} [CompleteLattice α]
+variable {ι : Sort _} {α : Type _} [CompleteLattice α] {f : ι → α}
 
 namespace CompleteLattice
 
@@ -368,7 +367,7 @@ theorem le_iff_compact_le_imp {a b : α} :
 #align le_iff_compact_le_imp le_iff_compact_le_imp
 
 /-- This property is sometimes referred to as `α` being upper continuous. -/
-theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
+theorem DirectedOn.inf_supₛ_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ supₛ s = ⨆ b ∈ s, a ⊓ b :=
   le_antisymm
     (by
       rw [le_iff_compact_le_imp]
@@ -382,7 +381,40 @@ theorem inf_supₛ_eq_of_directedOn (h : DirectedOn (· ≤ ·) s) : a ⊓ sup
       · rw [Set.not_nonempty_iff_eq_empty] at hs
         simp [hs])
     supᵢ_inf_le_inf_supₛ
-#align inf_Sup_eq_of_directed_on inf_supₛ_eq_of_directedOn
+#align directed_on.inf_Sup_eq DirectedOn.inf_supₛ_eq
+
+/-- This property is sometimes referred to as `α` being upper continuous. -/
+protected theorem DirectedOn.supₛ_inf_eq (h : DirectedOn (· ≤ ·) s) : supₛ s ⊓ a = ⨆ b ∈ s, b ⊓ a :=
+  by simp_rw [@inf_comm _ _ _ a, h.inf_supₛ_eq]
+#align directed_on.Sup_inf_eq DirectedOn.supₛ_inf_eq
+
+protected theorem Directed.inf_supᵢ_eq (h : Directed (· ≤ ·) f) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i :=
+  by rw [supᵢ, h.directedOn_range.inf_supₛ_eq, supᵢ_range]
+#align directed.inf_supr_eq Directed.inf_supᵢ_eq
+
+protected theorem Directed.supᵢ_inf_eq (h : Directed (· ≤ ·) f) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a :=
+  by rw [supᵢ, h.directedOn_range.supₛ_inf_eq, supᵢ_range]
+#align directed.supr_inf_eq Directed.supᵢ_inf_eq
+
+protected theorem DirectedOn.disjoint_supₛ_right (h : DirectedOn (· ≤ ·) s) :
+    Disjoint a (supₛ s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b := by
+  simp_rw [disjoint_iff, h.inf_supₛ_eq, supᵢ_eq_bot]
+#align directed_on.disjoint_Sup_right DirectedOn.disjoint_supₛ_right
+
+protected theorem DirectedOn.disjoint_supₛ_left (h : DirectedOn (· ≤ ·) s) :
+    Disjoint (supₛ s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a := by
+  simp_rw [disjoint_iff, h.supₛ_inf_eq, supᵢ_eq_bot]
+#align directed_on.disjoint_Sup_left DirectedOn.disjoint_supₛ_left
+
+protected theorem Directed.disjoint_supᵢ_right (h : Directed (· ≤ ·) f) :
+    Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by
+  simp_rw [disjoint_iff, h.inf_supᵢ_eq, supᵢ_eq_bot]
+#align directed.disjoint_supr_right Directed.disjoint_supᵢ_right
+
+protected theorem Directed.disjoint_supᵢ_left (h : Directed (· ≤ ·) f) :
+    Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a := by
+  simp_rw [disjoint_iff, h.supᵢ_inf_eq, supᵢ_eq_bot]
+#align directed.disjoint_supr_left Directed.disjoint_supᵢ_left
 
 /-- This property is equivalent to `α` being upper continuous. -/
 theorem inf_supₛ_eq_supᵢ_inf_sup_finset :
@@ -501,7 +533,7 @@ instance (priority := 100) isAtomic_of_complementedLattice [ComplementedLattice
       exact ⟨a, ha.of_isAtom_coe_Iic, hac.trans hcb⟩⟩
 #align is_atomic_of_complemented_lattice isAtomic_of_complementedLattice
 
-/-- See Lemma 5.1, Călugăreanu -/
+/-- See [Lemma 5.1][calugareanu]. -/
 instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLattice α] :
     IsAtomistic α :=
   ⟨fun b =>
@@ -520,70 +552,83 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
         exact le_supₛ ⟨ha.of_isAtom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
 #align is_atomistic_of_complemented_lattice isAtomistic_of_complementedLattice
 
-/-- See Theorem 6.6, Călugăreanu -/
+/-!
+Now we will prove that a compactly generated modular atomistic lattice is a complemented lattice.
+Most explicitly, every element is the complement of a supremum of indepedendent atoms.
+-/
+
+/-- In an atomic lattice, every element `b` has a complement of the form `Sup s`, where each element
+of `s` is an atom. See also `complemented_lattice_of_Sup_atoms_eq_top`. -/
+theorem exists_setIndependent_isCompl_supₛ_atoms (h : supₛ { a : α | IsAtom a } = ⊤) (b : α) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ IsCompl b (supₛ s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  by
+  -- porting note: `obtain` chokes on the placeholder.
+  have := zorn_subset
+    {s : Set α | CompleteLattice.SetIndependent s ∧ Disjoint b (supₛ s) ∧ ∀ a ∈ s, IsAtom a}
+    fun c hc1 hc2 =>
+      ⟨⋃₀ c,
+        ⟨CompleteLattice.independent_unionₛ_of_directed hc2.directedOn fun s hs => (hc1 hs).1, ?_,
+          fun a ⟨s, sc, as⟩ => (hc1 sc).2.2 a as⟩,
+        fun _ => Set.subset_unionₛ_of_mem⟩
+  obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ := this
+  swap
+  · rw [supₛ_unionₛ, ← supₛ_image, DirectedOn.disjoint_supₛ_right]
+    · rintro _ ⟨s, hs, rfl⟩
+      exact (hc1 hs).2.1
+    · rw [directedOn_image]
+      exact hc2.directedOn.mono @fun s t => supₛ_le_supₛ
+  refine' ⟨s, s_ind, ⟨b_inf_Sup_s, _⟩, s_atoms⟩
+  rw [codisjoint_iff_le_sup, ← h, supₛ_le_iff]
+  intro a ha
+  rw [← inf_eq_left]
+  refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
+  rw [← con, eq_comm, inf_eq_left]
+  refine' (le_supₛ _).trans le_sup_right
+  rw [← disjoint_iff] at con
+  have a_dis_Sup_s : Disjoint a (supₛ s) := con.mono_right le_sup_right
+  -- porting note: The two following `fun x hx => _` are no-op
+  rw [← s_max (s ∪ {a}) ⟨fun x hx => _, _, fun x hx => _⟩ (Set.subset_union_left _ _)]
+  · exact Set.mem_union_right _ (Set.mem_singleton _)
+  · intro x hx
+    rw [Set.mem_union, Set.mem_singleton_iff] at hx
+    obtain rfl | xa := eq_or_ne x a
+    · simp only [Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
+      exact con.mono_right ((supₛ_le_supₛ <| Set.diff_subset _ _).trans le_sup_right)
+    · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
+        by
+        simp only [Set.union_singleton]
+        rw [Set.insert_diff_of_not_mem]
+        rw [Set.mem_singleton_iff]
+        exact Ne.symm xa
+      rw [h, supₛ_union, supₛ_singleton]
+      apply
+        (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
+          (a_dis_Sup_s.mono_right _).symm
+      rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
+  · rw [supₛ_union, supₛ_singleton]
+    exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left con.symm
+  · intro x hx
+    rw [Set.mem_union, Set.mem_singleton_iff] at hx
+    obtain hx | rfl := hx
+    · exact s_atoms x hx
+    · exact ha
+#align exists_set_independent_is_compl_Sup_atoms exists_setIndependent_isCompl_supₛ_atoms
+
+theorem exists_setIndependent_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
+    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ supₛ s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a :=
+  let ⟨s, s_ind, s_top, s_atoms⟩ := exists_setIndependent_isCompl_supₛ_atoms h ⊥
+  ⟨s, s_ind, eq_top_of_isCompl_bot s_top.symm, s_atoms⟩
+#align exists_set_independent_of_Sup_atoms_eq_top exists_setIndependent_of_supₛ_atoms_eq_top
+
+/-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_supₛ_atoms_eq_top (h : supₛ { a : α | IsAtom a } = ⊤) :
     ComplementedLattice α :=
-  ⟨fun b => by
-    have H : ?_ := ?_ -- Porting note: this is an ugly hack, but `?H` on the next line fails
-    obtain ⟨s, ⟨s_ind, b_inf_Sup_s, s_atoms⟩, s_max⟩ :=
-      zorn_subset
-        { s : Set α | CompleteLattice.SetIndependent s ∧ b ⊓ supₛ s = ⊥ ∧ ∀ a ∈ s, IsAtom a } H
-    · refine'
-        ⟨supₛ s, disjoint_iff.mpr b_inf_Sup_s,
-          codisjoint_iff_le_sup.mpr <| h.symm.trans_le <| supₛ_le_iff.2 fun a ha => _⟩
-      rw [← inf_eq_left]
-      refine' (ha.le_iff.mp inf_le_left).resolve_left fun con => ha.1 _
-      rw [eq_bot_iff, ← con]
-      refine' le_inf (le_refl a) ((le_supₛ _).trans le_sup_right)
-      rw [← disjoint_iff] at *
-      have a_dis_Sup_s : Disjoint a (supₛ s) := con.mono_right le_sup_right
-      rw [← s_max (s ∪ {a}) ⟨_, ⟨_, _⟩⟩ (Set.subset_union_left _ _)]
-      · exact Set.mem_union_right _ (Set.mem_singleton _)
-      · intro x hx
-        rw [Set.mem_union, Set.mem_singleton_iff] at hx
-        by_cases xa : x = a
-        · simp only [xa, Set.mem_singleton, Set.insert_diff_of_mem, Set.union_singleton]
-          exact con.mono_right (le_trans (supₛ_le_supₛ (Set.diff_subset s {a})) le_sup_right)
-        · have h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a} :=
-            by
-            simp only [Set.union_singleton]
-            rw [Set.insert_diff_of_not_mem]
-            rw [Set.mem_singleton_iff]
-            exact Ne.symm xa
-          rw [h, supₛ_union, supₛ_singleton]
-          apply
-            (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
-              (a_dis_Sup_s.mono_right _).symm
-          rw [← supₛ_insert, Set.insert_diff_singleton, Set.insert_eq_of_mem (hx.resolve_right xa)]
-      · rw [supₛ_union, supₛ_singleton, ← disjoint_iff]
-        exact b_inf_Sup_s.disjoint_sup_right_of_disjoint_sup_left con.symm
-      · intro x hx
-        rw [Set.mem_union, Set.mem_singleton_iff] at hx
-        cases' hx with hx hx
-        · exact s_atoms x hx
-        · rw [hx]
-          exact ha
-    · intro c hc1 hc2
-      refine'
-        ⟨⋃₀ c,
-          ⟨CompleteLattice.independent_unionₛ_of_directed hc2.directedOn fun s hs => (hc1 hs).1, _,
-            fun a ha => _⟩,
-          fun _ => Set.subset_unionₛ_of_mem⟩
-      · rw [supₛ_unionₛ, ← supₛ_image, inf_supₛ_eq_of_directedOn, supᵢ_eq_bot]
-        · intro i
-          rw [supᵢ_eq_bot]
-          intro hi
-          obtain ⟨x, xc, rfl⟩ := (Set.mem_image _ _ _).1 hi
-          exact (hc1 xc).2.1
-        · rw [directedOn_image]
-          refine' hc2.directedOn.mono _
-          intro s t
-          apply supₛ_le_supₛ
-      · rcases Set.mem_unionₛ.1 ha with ⟨s, sc, as⟩
-        exact (hc1 sc).2.2 a as⟩
+  ⟨fun b =>
+    let ⟨s, _, s_top, _⟩ := exists_setIndependent_isCompl_supₛ_atoms h b
+    ⟨supₛ s, s_top⟩⟩
 #align complemented_lattice_of_Sup_atoms_eq_top complementedLattice_of_supₛ_atoms_eq_top
 
-/-- See Theorem 6.6, Călugăreanu -/
+/-- See [Theorem 6.6][calugareanu]. -/
 theorem complementedLattice_of_isAtomistic [IsAtomistic α] : ComplementedLattice α :=
   complementedLattice_of_supₛ_atoms_eq_top supₛ_atoms_eq_top
 #align complemented_lattice_of_is_atomistic complementedLattice_of_isAtomistic

These are proofs not propositions so should be lowerCamelCase.

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

@@ -515,7 +515,7 @@ instance (priority := 100) isAtomistic_of_complementedLattice [ComplementedLatti
       · exact ne_of_lt con (Subtype.ext_iff.1 (eq_top_of_isCompl_bot hc))
       · apply ha.1
         rw [eq_bot_iff]
-        apply le_trans (le_inf _ hac) hc.Disjoint.le_bot
+        apply le_trans (le_inf _ hac) hc.disjoint.le_bot
         rw [← Subtype.coe_le_coe, Subtype.coe_mk]
         exact le_supₛ ⟨ha.of_isAtom_coe_Iic, a.2⟩, fun _ => And.left⟩⟩
 #align is_atomistic_of_complemented_lattice isAtomistic_of_complementedLattice

• order.basic@de87d5053a9fe5cbde723172c0fb7e27e7436473..210657c4ea4a4a7b234392f70a3a2a83346dfa90
• order.well_founded@1c521b4fb909320eca16b2bb6f8b5b0490b1cb5e..210657c4ea4a4a7b234392f70a3a2a83346dfa90
• order.order_iso_nat@6623e6af705e97002a9054c1c05a980180276fc1..210657c4ea4a4a7b234392f70a3a2a83346dfa90
• order.compactly_generated@861a26926586cd46ff80264d121cdb6fa0e35cc1..210657c4ea4a4a7b234392f70a3a2a83346dfa90
• ring_theory.noetherian@da420a8c6dd5bdfb85c4ced85c34388f633bc6ff..210657c4ea4a4a7b234392f70a3a2a83346dfa90

Mathlib 3: https://github.com/leanprover-community/mathlib/pull/15071

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module order.compactly_generated
-! leanprover-community/mathlib commit 861a26926586cd46ff80264d121cdb6fa0e35cc1
+! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -204,30 +204,16 @@ theorem finset_sup_compact_of_compact {α β : Type _} [CompleteLattice α] {f :
 #align complete_lattice.finset_sup_compact_of_compact CompleteLattice.finset_sup_compact_of_compact
 
 theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
-    IsSupFiniteCompact α := by
-  intro s
-  let p : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
-  have hp : p.Nonempty := by
-    use ⊥, ∅
+    IsSupFiniteCompact α := fun s =>
+  by
+  let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
+  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := h.has_min S ⟨⊥, ∅, by simp⟩
+  refine' ⟨t, ht₁, (supₛ_le _ _ fun y hy => _).antisymm _⟩
+  · classical
+    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
+        (hm _ ⟨insert y t, by simp [Set.insert_subset, hy, ht₁]⟩)]
     simp
-  obtain ⟨m, ⟨t, ⟨ht₁, ht₂⟩⟩, hm⟩ := WellFounded.wellFounded_iff_has_max'.mp h p hp
-  use t
-  simp only [ht₁, ht₂, true_and_iff]
-  apply le_antisymm
-  · apply supₛ_le
-    intro y hy
-    classical
-      have hy' : (insert y t).sup id ∈ p := by
-        use insert y t
-        simp
-        rw [Set.insert_subset]
-        exact ⟨hy, ht₁⟩
-      have hm' : m ≤ (insert y t).sup id := by
-        rw [← ht₂]
-        exact Finset.sup_mono (t.subset_insert y)
-      rw [← hm _ hy' hm']
-      simp
-  · rw [← ht₂, Finset.sup_id_eq_supₛ]
+  · rw [Finset.sup_id_eq_supₛ]
     exact supₛ_le_supₛ ht₁
 #align complete_lattice.well_founded.is_Sup_finite_compact CompleteLattice.WellFounded.isSupFiniteCompact
 

@@ -16,7 +16,7 @@ import Mathlib.Order.Zorn
 import Mathlib.Data.Finset.Order
 import Mathlib.Data.Set.Intervals.OrderIso
 import Mathlib.Data.Finite.Set
-import Mathlib.Data.List.TFAE
+import Mathlib.Tactic.TFAE
 
 /-!
 # Compactness properties for complete lattices
@@ -286,23 +286,15 @@ open List in
 theorem wellFounded_characterisations : List.TFAE
     [WellFounded (( · > · ) : α → α → Prop),
       IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] := by
-  have h12 := WellFounded.isSupFiniteCompact α
-  have h23 := IsSupFiniteCompact.isSupClosedCompact α
-  have h31 := IsSupClosedCompact.wellFounded α
-  have h24 := isSupFiniteCompact_iff_all_elements_compact α
-  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil] <;> dsimp only [ilast']
-  · rw [← h24]; exact h12 ∘ h31
-  · rw [← h24]; exact h31 ∘ h23
-  -- Porting note: proof using `tfae`
-  -- tfae_have 1 → 2
-  -- · exact WellFounded.isSupFiniteCompact α
-  -- tfae_have 2 → 3
-  -- · exact IsSupFiniteCompact.isSupClosedCompact α
-  -- tfae_have 3 → 1
-  -- · exact IsSupClosedCompact.wellFounded α
-  -- tfae_have 2 ↔ 4
-  -- · exact isSupFiniteCompact_iff_all_elements_compact α
-  -- tfae_finish
+  tfae_have 1 → 2
+  · exact WellFounded.isSupFiniteCompact α
+  tfae_have 2 → 3
+  · exact IsSupFiniteCompact.isSupClosedCompact α
+  tfae_have 3 → 1
+  · exact IsSupClosedCompact.wellFounded α
+  tfae_have 2 ↔ 4
+  · exact isSupFiniteCompact_iff_all_elements_compact α
+  tfae_finish
 #align complete_lattice.well_founded_characterisations CompleteLattice.wellFounded_characterisations
 
 theorem wellFounded_iff_isSupFiniteCompact :

Missing tactics:

• Tfae
• change .. with

Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>