order.partial_sups ⟷ Mathlib.Order.PartialSups

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
-import Mathbin.Data.Finset.Lattice
-import Mathbin.Order.Hom.Basic
-import Mathbin.Order.ConditionallyCompleteLattice.Finset
+import Data.Finset.Lattice
+import Order.Hom.Basic
+import Order.ConditionallyCompleteLattice.Finset
 
 #align_import order.partial_sups from "leanprover-community/mathlib"@"a11f9106a169dd302a285019e5165f8ab32ff433"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module order.partial_sups
-! leanprover-community/mathlib commit a11f9106a169dd302a285019e5165f8ab32ff433
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Finset.Lattice
 import Mathbin.Order.Hom.Basic
 import Mathbin.Order.ConditionallyCompleteLattice.Finset
 
+#align_import order.partial_sups from "leanprover-community/mathlib"@"a11f9106a169dd302a285019e5165f8ab32ff433"
+
 /-!
 # The monotone sequence of partial supremums of a sequence
 

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

@@ -63,11 +63,13 @@ theorem partialSups_zero (f : ℕ → α) : partialSups f 0 = f 0 :=
 #align partial_sups_zero partialSups_zero
 -/
 
+#print partialSups_succ /-
 @[simp]
 theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
     partialSups f (n + 1) = partialSups f n ⊔ f (n + 1) :=
   rfl
 #align partial_sups_succ partialSups_succ
+-/
 
 #print le_partialSups_of_le /-
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
@@ -188,13 +190,16 @@ section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α]
 
+#print partialSups_eq_ciSup_Iic /-
 theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i :=
   by
   have : Set.Iio (n + 1) = Set.Iic n := Set.ext fun _ => Nat.lt_succ_iff
   rw [partialSups_eq_sup'_range, Finset.sup'_eq_csSup_image, Finset.coe_range, iSup, Set.range_comp,
     Subtype.range_coe, this]
 #align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iic
+-/
 
+#print ciSup_partialSups_eq /-
 @[simp]
 theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
     (⨆ n, partialSups f n) = ⨆ n, f n :=
@@ -204,6 +209,7 @@ theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
     exact ciSup_le fun i => le_ciSup h _
   · rwa [bddAbove_range_partialSups]
 #align csupr_partial_sups_eq ciSup_partialSups_eq
+-/
 
 end ConditionallyCompleteLattice
 
@@ -211,26 +217,34 @@ section CompleteLattice
 
 variable [CompleteLattice α]
 
+#print partialSups_eq_biSup /-
 theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
   simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f n
 #align partial_sups_eq_bsupr partialSups_eq_biSup
+-/
 
+#print iSup_partialSups_eq /-
 @[simp]
 theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
   ciSup_partialSups_eq <| OrderTop.bddAbove _
 #align supr_partial_sups_eq iSup_partialSups_eq
+-/
 
+#print iSup_le_iSup_of_partialSups_le_partialSups /-
 theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
     (h : partialSups f ≤ partialSups g) : (⨆ n, f n) ≤ ⨆ n, g n :=
   by
   rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g]
   exact iSup_mono h
 #align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSups
+-/
 
+#print iSup_eq_iSup_of_partialSups_eq_partialSups /-
 theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ℕ → α}
     (h : partialSups f = partialSups g) : (⨆ n, f n) = ⨆ n, g n := by
   simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]
 #align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSups
+-/
 
 end CompleteLattice
 

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

@@ -135,7 +135,7 @@ def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α
     where
   choice f h :=
     ⟨f, by
-      convert(partialSups f).Monotone
+      convert (partialSups f).Monotone
       exact (le_partialSups f).antisymm h⟩
   gc f g := by
     refine' ⟨(le_partialSups f).trans, fun h => _⟩

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

@@ -152,7 +152,7 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
   by
   induction' n with n ih
   · simp
-  · dsimp [partialSups] at ih⊢
+  · dsimp [partialSups] at ih ⊢
     simp_rw [@Finset.range_succ n.succ]
     rw [ih, Finset.sup'_insert, sup_comm]
 #align partial_sups_eq_sup'_range partialSups_eq_sup'_range
@@ -166,7 +166,7 @@ theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ →
   by
   induction' n with n ih
   · simp
-  · dsimp [partialSups] at ih⊢
+  · dsimp [partialSups] at ih ⊢
     rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
 -/

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

@@ -69,6 +69,7 @@ theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
   rfl
 #align partial_sups_succ partialSups_succ
 
+#print le_partialSups_of_le /-
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
   by
   induction' n with n ih
@@ -77,16 +78,21 @@ theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m 
     · exact le_sup_right
     · exact (ih h).trans le_sup_left
 #align le_partial_sups_of_le le_partialSups_of_le
+-/
 
+#print le_partialSups /-
 theorem le_partialSups (f : ℕ → α) : f ≤ partialSups f := fun n => le_partialSups_of_le f le_rfl
 #align le_partial_sups le_partialSups
+-/
 
+#print partialSups_le /-
 theorem partialSups_le (f : ℕ → α) (n : ℕ) (a : α) (w : ∀ m, m ≤ n → f m ≤ a) :
     partialSups f n ≤ a := by
   induction' n with n ih
   · apply w 0 le_rfl
   · exact sup_le (ih fun m p => w m (Nat.le_succ_of_le p)) (w (n + 1) le_rfl)
 #align partial_sups_le partialSups_le
+-/
 
 #print bddAbove_range_partialSups /-
 @[simp]
@@ -154,6 +160,7 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
 
 end SemilatticeSup
 
+#print partialSups_eq_sup_range /-
 theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ → α) (n : ℕ) :
     partialSups f n = (Finset.range (n + 1)).sup f :=
   by
@@ -162,7 +169,9 @@ theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ →
   · dsimp [partialSups] at ih⊢
     rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
+-/
 
+#print partialSups_disjoint_of_disjoint /-
 /- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as
 submodules. -/
 theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f : ℕ → α)
@@ -173,6 +182,7 @@ theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f :
   · rw [partialSups_succ, disjoint_sup_left]
     exact ⟨ih (Nat.lt_of_succ_lt hmn), h hmn.ne⟩
 #align partial_sups_disjoint_of_disjoint partialSups_disjoint_of_disjoint
+-/
 
 section ConditionallyCompleteLattice
 

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

@@ -63,24 +63,12 @@ theorem partialSups_zero (f : ℕ → α) : partialSups f 0 = f 0 :=
 #align partial_sups_zero partialSups_zero
 -/
 
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-Case conversion may be inaccurate. Consider using '#align partial_sups_succ partialSups_succₓ'. -/
 @[simp]
 theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
     partialSups f (n + 1) = partialSups f n ⊔ f (n + 1) :=
   rfl
 #align partial_sups_succ partialSups_succ
 
-/- warning: le_partial_sups_of_le -> le_partialSups_of_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align le_partial_sups_of_le le_partialSups_of_leₓ'. -/
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
   by
   induction' n with n ih
@@ -90,21 +78,9 @@ theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m 
     · exact (ih h).trans le_sup_left
 #align le_partial_sups_of_le le_partialSups_of_le
 
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-Case conversion may be inaccurate. Consider using '#align le_partial_sups le_partialSupsₓ'. -/
 theorem le_partialSups (f : ℕ → α) : f ≤ partialSups f := fun n => le_partialSups_of_le f le_rfl
 #align le_partial_sups le_partialSups
 
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-Case conversion may be inaccurate. Consider using '#align partial_sups_le partialSups_leₓ'. -/
 theorem partialSups_le (f : ℕ → α) (n : ℕ) (a : α) (w : ∀ m, m ≤ n → f m ≤ a) :
     partialSups f n ≤ a := by
   induction' n with n ih
@@ -178,12 +154,6 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
 
 end SemilatticeSup
 
-/- warning: partial_sups_eq_sup_range -> partialSups_eq_sup_range is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align partial_sups_eq_sup_range partialSups_eq_sup_rangeₓ'. -/
 theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ → α) (n : ℕ) :
     partialSups f n = (Finset.range (n + 1)).sup f :=
   by
@@ -193,12 +163,6 @@ theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ →
     rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
 
-/- warning: partial_sups_disjoint_of_disjoint -> partialSups_disjoint_of_disjoint is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align partial_sups_disjoint_of_disjoint partialSups_disjoint_of_disjointₓ'. -/
 /- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as
 submodules. -/
 theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f : ℕ → α)
@@ -214,12 +178,6 @@ section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α]
 
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-Case conversion may be inaccurate. Consider using '#align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iicₓ'. -/
 theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i :=
   by
   have : Set.Iio (n + 1) = Set.Iic n := Set.ext fun _ => Nat.lt_succ_iff
@@ -227,12 +185,6 @@ theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n =
     Subtype.range_coe, this]
 #align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iic
 
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-Case conversion may be inaccurate. Consider using '#align csupr_partial_sups_eq ciSup_partialSups_eqₓ'. -/
 @[simp]
 theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
     (⨆ n, partialSups f n) = ⨆ n, f n :=
@@ -249,33 +201,15 @@ section CompleteLattice
 
 variable [CompleteLattice α]
 
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-Case conversion may be inaccurate. Consider using '#align partial_sups_eq_bsupr partialSups_eq_biSupₓ'. -/
 theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
   simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f n
 #align partial_sups_eq_bsupr partialSups_eq_biSup
 
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-Case conversion may be inaccurate. Consider using '#align supr_partial_sups_eq iSup_partialSups_eqₓ'. -/
 @[simp]
 theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
   ciSup_partialSups_eq <| OrderTop.bddAbove _
 #align supr_partial_sups_eq iSup_partialSups_eq
 
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-Case conversion may be inaccurate. Consider using '#align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSupsₓ'. -/
 theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
     (h : partialSups f ≤ partialSups g) : (⨆ n, f n) ≤ ⨆ n, g n :=
   by
@@ -283,12 +217,6 @@ theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
   exact iSup_mono h
 #align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSups
 
-/- warning: supr_eq_supr_of_partial_sups_eq_partial_sups -> iSup_eq_iSup_of_partialSups_eq_partialSups is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSupsₓ'. -/
 theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ℕ → α}
     (h : partialSups f = partialSups g) : (⨆ n, f n) = ⨆ n, g n := by
   simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]

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

@@ -84,8 +84,7 @@ Case conversion may be inaccurate. Consider using '#align le_partial_sups_of_le
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
   by
   induction' n with n ih
-  · cases h
-    exact le_rfl
+  · cases h; exact le_rfl
   · cases' h with h h
     · exact le_sup_right
     · exact (ih h).trans le_sup_left

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

@@ -75,7 +75,12 @@ theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
   rfl
 #align partial_sups_succ partialSups_succ
 
-#print le_partialSups_of_le /-
+/- warning: le_partial_sups_of_le -> le_partialSups_of_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) {m : Nat} {n : Nat}, (LE.le.{0} Nat Nat.hasLe m n) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (f m) (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f) n))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) {m : Nat} {n : Nat}, (LE.le.{0} Nat instLENat m n) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (f m) (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f) n))
+Case conversion may be inaccurate. Consider using '#align le_partial_sups_of_le le_partialSups_of_leₓ'. -/
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
   by
   induction' n with n ih
@@ -85,21 +90,28 @@ theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m 
     · exact le_sup_right
     · exact (ih h).trans le_sup_left
 #align le_partial_sups_of_le le_partialSups_of_le
--/
 
-#print le_partialSups /-
+/- warning: le_partial_sups -> le_partialSups is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α), LE.le.{u1} (Nat -> α) (Pi.hasLe.{0, u1} Nat (fun (ᾰ : Nat) => α) (fun (i : Nat) => Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))) f (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α), LE.le.{u1} (Nat -> α) (Pi.hasLe.{0, u1} Nat (fun (ᾰ : Nat) => α) (fun (i : Nat) => Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))) f (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f))
+Case conversion may be inaccurate. Consider using '#align le_partial_sups le_partialSupsₓ'. -/
 theorem le_partialSups (f : ℕ → α) : f ≤ partialSups f := fun n => le_partialSups_of_le f le_rfl
 #align le_partial_sups le_partialSups
--/
 
-#print partialSups_le /-
+/- warning: partial_sups_le -> partialSups_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) (n : Nat) (a : α), (forall (m : Nat), (LE.le.{0} Nat Nat.hasLe m n) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (f m) a)) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f) n) a)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) (n : Nat) (a : α), (forall (m : Nat), (LE.le.{0} Nat instLENat m n) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (f m) a)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f) n) a)
+Case conversion may be inaccurate. Consider using '#align partial_sups_le partialSups_leₓ'. -/
 theorem partialSups_le (f : ℕ → α) (n : ℕ) (a : α) (w : ∀ m, m ≤ n → f m ≤ a) :
     partialSups f n ≤ a := by
   induction' n with n ih
   · apply w 0 le_rfl
   · exact sup_le (ih fun m p => w m (Nat.le_succ_of_le p)) (w (n + 1) le_rfl)
 #align partial_sups_le partialSups_le
--/
 
 #print bddAbove_range_partialSups /-
 @[simp]
@@ -167,7 +179,12 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
 
 end SemilatticeSup
 
-#print partialSups_eq_sup_range /-
+/- warning: partial_sups_eq_sup_range -> partialSups_eq_sup_range is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f) n) (Finset.sup.{u1, 0} α Nat _inst_1 _inst_2 (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) f)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f) n) (Finset.sup.{u1, 0} α Nat _inst_1 _inst_2 (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) f)
+Case conversion may be inaccurate. Consider using '#align partial_sups_eq_sup_range partialSups_eq_sup_rangeₓ'. -/
 theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ → α) (n : ℕ) :
     partialSups f n = (Finset.range (n + 1)).sup f :=
   by
@@ -176,9 +193,13 @@ theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ →
   · dsimp [partialSups] at ih⊢
     rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
--/
 
-#print partialSups_disjoint_of_disjoint /-
+/- warning: partial_sups_disjoint_of_disjoint -> partialSups_disjoint_of_disjoint is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : DistribLattice.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1)))))] (f : Nat -> α), (Pairwise.{0} Nat (Function.onFun.{1, succ u1, 1} Nat α Prop (Disjoint.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))) _inst_2) f)) -> (forall {m : Nat} {n : Nat}, (LT.lt.{0} Nat Nat.hasLt m n) -> (Disjoint.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))) _inst_2 (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1)) f) m) (f n)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : DistribLattice.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1)))))] (f : Nat -> α), (Pairwise.{0} Nat (Function.onFun.{1, succ u1, 1} Nat α Prop (Disjoint.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))) _inst_2) f)) -> (forall {m : Nat} {n : Nat}, (LT.lt.{0} Nat instLTNat m n) -> (Disjoint.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (DistribLattice.toLattice.{u1} α _inst_1))) _inst_2 (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1)))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (DistribLattice.toLattice.{u1} α _inst_1)) f) m) (f n)))
+Case conversion may be inaccurate. Consider using '#align partial_sups_disjoint_of_disjoint partialSups_disjoint_of_disjointₓ'. -/
 /- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as
 submodules. -/
 theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f : ℕ → α)
@@ -189,7 +210,6 @@ theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f :
   · rw [partialSups_succ, disjoint_sup_left]
     exact ⟨ih (Nat.lt_of_succ_lt hmn), h hmn.ne⟩
 #align partial_sups_disjoint_of_disjoint partialSups_disjoint_of_disjoint
--/
 
 section ConditionallyCompleteLattice
 
@@ -253,7 +273,7 @@ theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n,
 
 /- warning: supr_le_supr_of_partial_sups_le_partial_sups -> iSup_le_iSup_of_partialSups_le_partialSups is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.preorder.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toHasLe.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.preorder.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.instPreorderOrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
 Case conversion may be inaccurate. Consider using '#align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSupsₓ'. -/

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

@@ -195,34 +195,34 @@ section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α]
 
-/- warning: partial_sups_eq_csupr_Iic -> partialSups_eq_csupᵢ_Iic is a dubious translation:
+/- warning: partial_sups_eq_csupr_Iic -> partialSups_eq_ciSup_Iic is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) (fun (i : coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) => f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (coeSubtype.{1} Nat (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)))))) i)))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) (fun (i : coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) => f ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (CoeTCₓ.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)) Nat (coeSubtype.{1} Nat (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) n)))))) i)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) (Set.Elem.{0} Nat (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) (fun (i : Set.Elem.{0} Nat (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) => f (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) i)))
-Case conversion may be inaccurate. Consider using '#align partial_sups_eq_csupr_Iic partialSups_eq_csupᵢ_Iicₓ'. -/
-theorem partialSups_eq_csupᵢ_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i :=
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) (Set.Elem.{0} Nat (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) (fun (i : Set.Elem.{0} Nat (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) => f (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x (Set.Iic.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) n)) i)))
+Case conversion may be inaccurate. Consider using '#align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iicₓ'. -/
+theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i :=
   by
   have : Set.Iio (n + 1) = Set.Iic n := Set.ext fun _ => Nat.lt_succ_iff
-  rw [partialSups_eq_sup'_range, Finset.sup'_eq_csupₛ_image, Finset.coe_range, supᵢ, Set.range_comp,
+  rw [partialSups_eq_sup'_range, Finset.sup'_eq_csSup_image, Finset.coe_range, iSup, Set.range_comp,
     Subtype.range_coe, this]
-#align partial_sups_eq_csupr_Iic partialSups_eq_csupᵢ_Iic
+#align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iic
 
-/- warning: csupr_partial_sups_eq -> csupᵢ_partialSups_eq is a dubious translation:
+/- warning: csupr_partial_sups_eq -> ciSup_partialSups_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] {f : Nat -> α}, (BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (Set.range.{u1, 1} α Nat f)) -> (Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) Nat (fun (n : Nat) => coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) Nat (fun (n : Nat) => f n)))
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] {f : Nat -> α}, (BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (Set.range.{u1, 1} α Nat f)) -> (Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) Nat (fun (n : Nat) => coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α _inst_1) Nat (fun (n : Nat) => f n)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] {f : Nat -> α}, (BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (Set.range.{u1, 1} α Nat f)) -> (Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) Nat (fun (n : Nat) => OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) Nat (fun (n : Nat) => f n)))
-Case conversion may be inaccurate. Consider using '#align csupr_partial_sups_eq csupᵢ_partialSups_eqₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : ConditionallyCompleteLattice.{u1} α] {f : Nat -> α}, (BddAbove.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (Set.range.{u1, 1} α Nat f)) -> (Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) Nat (fun (n : Nat) => OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α _inst_1)) f) n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α _inst_1) Nat (fun (n : Nat) => f n)))
+Case conversion may be inaccurate. Consider using '#align csupr_partial_sups_eq ciSup_partialSups_eqₓ'. -/
 @[simp]
-theorem csupᵢ_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
+theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
     (⨆ n, partialSups f n) = ⨆ n, f n :=
   by
-  refine' (csupᵢ_le fun n => _).antisymm (csupᵢ_mono _ <| le_partialSups f)
-  · rw [partialSups_eq_csupᵢ_Iic]
-    exact csupᵢ_le fun i => le_csupᵢ h _
+  refine' (ciSup_le fun n => _).antisymm (ciSup_mono _ <| le_partialSups f)
+  · rw [partialSups_eq_ciSup_Iic]
+    exact ciSup_le fun i => le_ciSup h _
   · rwa [bddAbove_range_partialSups]
-#align csupr_partial_sups_eq csupᵢ_partialSups_eq
+#align csupr_partial_sups_eq ciSup_partialSups_eq
 
 end ConditionallyCompleteLattice
 
@@ -230,50 +230,50 @@ section CompleteLattice
 
 variable [CompleteLattice α]
 
-/- warning: partial_sups_eq_bsupr -> partialSups_eq_bsupᵢ is a dubious translation:
+/- warning: partial_sups_eq_bsupr -> partialSups_eq_biSup is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (i : Nat) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (LE.le.{0} Nat Nat.hasLe i n) (fun (H : LE.le.{0} Nat Nat.hasLe i n) => f i)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (i : Nat) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (LE.le.{0} Nat Nat.hasLe i n) (fun (H : LE.le.{0} Nat Nat.hasLe i n) => f i)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (i : Nat) => supᵢ.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (LE.le.{0} Nat instLENat i n) (fun (H : LE.le.{0} Nat instLENat i n) => f i)))
-Case conversion may be inaccurate. Consider using '#align partial_sups_eq_bsupr partialSups_eq_bsupᵢₓ'. -/
-theorem partialSups_eq_bsupᵢ (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
-  simpa only [supᵢ_subtype] using partialSups_eq_csupᵢ_Iic f n
-#align partial_sups_eq_bsupr partialSups_eq_bsupᵢ
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (i : Nat) => iSup.{u1, 0} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) (LE.le.{0} Nat instLENat i n) (fun (H : LE.le.{0} Nat instLENat i n) => f i)))
+Case conversion may be inaccurate. Consider using '#align partial_sups_eq_bsupr partialSups_eq_biSupₓ'. -/
+theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
+  simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f n
+#align partial_sups_eq_bsupr partialSups_eq_biSup
 
-/- warning: supr_partial_sups_eq -> supᵢ_partialSups_eq is a dubious translation:
+/- warning: supr_partial_sups_eq -> iSup_partialSups_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α), Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α), Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α), Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n))
-Case conversion may be inaccurate. Consider using '#align supr_partial_sups_eq supᵢ_partialSups_eqₓ'. -/
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] (f : Nat -> α), Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n))
+Case conversion may be inaccurate. Consider using '#align supr_partial_sups_eq iSup_partialSups_eqₓ'. -/
 @[simp]
-theorem supᵢ_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
-  csupᵢ_partialSups_eq <| OrderTop.bddAbove _
-#align supr_partial_sups_eq supᵢ_partialSups_eq
+theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
+  ciSup_partialSups_eq <| OrderTop.bddAbove _
+#align supr_partial_sups_eq iSup_partialSups_eq
 
-/- warning: supr_le_supr_of_partial_sups_le_partial_sups -> supᵢ_le_supᵢ_of_partialSups_le_partialSups is a dubious translation:
+/- warning: supr_le_supr_of_partial_sups_le_partial_sups -> iSup_le_iSup_of_partialSups_le_partialSups is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.preorder.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.preorder.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.instPreorderOrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
-Case conversion may be inaccurate. Consider using '#align supr_le_supr_of_partial_sups_le_partial_sups supᵢ_le_supᵢ_of_partialSups_le_partialSupsₓ'. -/
-theorem supᵢ_le_supᵢ_of_partialSups_le_partialSups {f g : ℕ → α}
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (LE.le.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (Preorder.toLE.{u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (OrderHom.instPreorderOrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (CompleteSemilatticeInf.toPartialOrder.{u1} α (CompleteLattice.toCompleteSemilatticeInf.{u1} α _inst_1)))) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
+Case conversion may be inaccurate. Consider using '#align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSupsₓ'. -/
+theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
     (h : partialSups f ≤ partialSups g) : (⨆ n, f n) ≤ ⨆ n, g n :=
   by
-  rw [← supᵢ_partialSups_eq f, ← supᵢ_partialSups_eq g]
-  exact supᵢ_mono h
-#align supr_le_supr_of_partial_sups_le_partial_sups supᵢ_le_supᵢ_of_partialSups_le_partialSups
+  rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g]
+  exact iSup_mono h
+#align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSups
 
-/- warning: supr_eq_supr_of_partial_sups_eq_partial_sups -> supᵢ_eq_supᵢ_of_partialSups_eq_partialSups is a dubious translation:
+/- warning: supr_eq_supr_of_partial_sups_eq_partial_sups -> iSup_eq_iSup_of_partialSups_eq_partialSups is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (Eq.{succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (Eq.{succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toHasSup.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (Eq.{succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (Eq.{succ u1} α (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (supᵢ.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
-Case conversion may be inaccurate. Consider using '#align supr_eq_supr_of_partial_sups_eq_partial_sups supᵢ_eq_supᵢ_of_partialSups_eq_partialSupsₓ'. -/
-theorem supᵢ_eq_supᵢ_of_partialSups_eq_partialSups {f g : ℕ → α}
+  forall {α : Type.{u1}} [_inst_1 : CompleteLattice.{u1} α] {f : Nat -> α} {g : Nat -> α}, (Eq.{succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)))))) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) f) (partialSups.{u1} α (Lattice.toSemilatticeSup.{u1} α (ConditionallyCompleteLattice.toLattice.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1))) g)) -> (Eq.{succ u1} α (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => f n)) (iSup.{u1, 1} α (ConditionallyCompleteLattice.toSupSet.{u1} α (CompleteLattice.toConditionallyCompleteLattice.{u1} α _inst_1)) Nat (fun (n : Nat) => g n)))
+Case conversion may be inaccurate. Consider using '#align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSupsₓ'. -/
+theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ℕ → α}
     (h : partialSups f = partialSups g) : (⨆ n, f n) = ⨆ n, g n := by
-  simp_rw [← supᵢ_partialSups_eq f, ← supᵢ_partialSups_eq g, h]
-#align supr_eq_supr_of_partial_sups_eq_partial_sups supᵢ_eq_supᵢ_of_partialSups_eq_partialSups
+  simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]
+#align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSups
 
 end CompleteLattice
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -142,7 +142,7 @@ def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α
     where
   choice f h :=
     ⟨f, by
-      convert (partialSups f).Monotone
+      convert(partialSups f).Monotone
       exact (le_partialSups f).antisymm h⟩
   gc f g := by
     refine' ⟨(le_partialSups f).trans, fun h => _⟩

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

@@ -63,13 +63,17 @@ theorem partialSups_zero (f : ℕ → α) : partialSups f 0 = f 0 :=
 #align partial_sups_zero partialSups_zero
 -/
 
-#print partialSups_succ /-
+/- warning: partial_sups_succ -> partialSups_succ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Sup.sup.{u1} α (SemilatticeSup.toHasSup.{u1} α _inst_1) (coeFn.{succ u1, succ u1} (OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (fun (_x : OrderHom.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) => Nat -> α) (OrderHom.hasCoeToFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))) (partialSups.{u1} α _inst_1 f) n) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : SemilatticeSup.{u1} α] (f : Nat -> α) (n : Nat), Eq.{succ u1} α (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Sup.sup.{u1} α (SemilatticeSup.toSup.{u1} α _inst_1) (OrderHom.toFun.{0, u1} Nat α (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)) (partialSups.{u1} α _inst_1 f) n) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))
+Case conversion may be inaccurate. Consider using '#align partial_sups_succ partialSups_succₓ'. -/
 @[simp]
 theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
     partialSups f (n + 1) = partialSups f n ⊔ f (n + 1) :=
   rfl
 #align partial_sups_succ partialSups_succ
--/
 
 #print le_partialSups_of_le /-
 theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=

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

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

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

@@ -85,7 +85,7 @@ theorem partialSups_le (f : ℕ → α) (n : ℕ) (a : α) (w : ∀ m, m ≤ n 
 lemma upperBounds_range_partialSups (f : ℕ → α) :
     upperBounds (Set.range (partialSups f)) = upperBounds (Set.range f) := by
   ext a
-  simp only [mem_upperBounds, Set.forall_range_iff, partialSups_le_iff]
+  simp only [mem_upperBounds, Set.forall_mem_range, partialSups_le_iff]
   exact ⟨fun h _ ↦ h _ _ le_rfl, fun h _ _ _ ↦ h _⟩
 
 @[simp]

@@ -61,7 +61,7 @@ theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
 #align partial_sups_succ partialSups_succ
 
 lemma partialSups_iff_forall {f : ℕ → α} (p : α → Prop)
-    (hp : ∀ {a b}, p (a ⊔ b) ↔ p a ∧ p b) : ∀  {n : ℕ}, p (partialSups f n) ↔ ∀ k ≤ n, p (f k)
+    (hp : ∀ {a b}, p (a ⊔ b) ↔ p a ∧ p b) : ∀ {n : ℕ}, p (partialSups f n) ↔ ∀ k ≤ n, p (f k)
   | 0 => by simp
   | (n + 1) => by simp [hp, partialSups_iff_forall, ← Nat.lt_succ_iff, ← Nat.forall_lt_succ]
 

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

@@ -180,7 +180,7 @@ theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆
   simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f n
 #align partial_sups_eq_bsupr partialSups_eq_biSup
 
--- Porting note: simp can prove this @[simp]
+-- Porting note (#10618): simp can prove this @[simp]
 theorem iSup_partialSups_eq (f : ℕ → α) : ⨆ n, partialSups f n = ⨆ n, f n :=
   ciSup_partialSups_eq <| OrderTop.bddAbove _
 #align supr_partial_sups_eq iSup_partialSups_eq

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -5,7 +5,8 @@ Authors: Scott Morrison
 -/
 import Mathlib.Data.Finset.Lattice
 import Mathlib.Order.Hom.Basic
-import Mathlib.Order.ConditionallyCompleteLattice.Finset
+import Mathlib.Data.Set.Finite
+import Mathlib.Order.ConditionallyCompleteLattice.Basic
 
 #align_import order.partial_sups from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
 

Also rename Filter.Tendsto.sup_right_nhds to Filter.Tendsto.sup_nhds etc.

@@ -122,6 +122,10 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
   eq_of_forall_ge_iff fun _ ↦ by simp [Nat.lt_succ_iff]
 #align partial_sups_eq_sup'_range partialSups_eq_sup'_range
 
+lemma partialSups_apply {ι : Type*} {π : ι → Type*} [(i : ι) → SemilatticeSup (π i)]
+    (f : ℕ → (i : ι) → π i) (n : ℕ) (i : ι) : partialSups f n i = partialSups (f · i) n := by
+  simp only [partialSups_eq_sup'_range, Finset.sup'_apply]
+
 end SemilatticeSup
 
 theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ → α) (n : ℕ) :

• Add partialSups_iff_forall and partialSups_le_iff.
• Use them to golf some proofs.
• Drop partialSups_apply_mono because it's just (partialSups f).mono.
• Add disjoint_partialSups_left and disjoint_partialSups_right.

Motivated by partialSups_le_iff from the Mandelbrot Set Connectedness Project.

Co-authored-by: @girving

@@ -59,34 +59,38 @@ theorem partialSups_succ (f : ℕ → α) (n : ℕ) :
   rfl
 #align partial_sups_succ partialSups_succ
 
-theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n := by
-  induction' n with n ih
-  · rw [nonpos_iff_eq_zero.mp h, partialSups_zero]
-  · cases' h with h h
-    · exact le_sup_right
-    · exact (ih h).trans le_sup_left
+lemma partialSups_iff_forall {f : ℕ → α} (p : α → Prop)
+    (hp : ∀ {a b}, p (a ⊔ b) ↔ p a ∧ p b) : ∀  {n : ℕ}, p (partialSups f n) ↔ ∀ k ≤ n, p (f k)
+  | 0 => by simp
+  | (n + 1) => by simp [hp, partialSups_iff_forall, ← Nat.lt_succ_iff, ← Nat.forall_lt_succ]
+
+@[simp]
+lemma partialSups_le_iff {f : ℕ → α} {n : ℕ} {a : α} : partialSups f n ≤ a ↔ ∀ k ≤ n, f k ≤ a :=
+  partialSups_iff_forall (· ≤ a) sup_le_iff
+
+theorem le_partialSups_of_le (f : ℕ → α) {m n : ℕ} (h : m ≤ n) : f m ≤ partialSups f n :=
+  partialSups_le_iff.1 le_rfl m h
 #align le_partial_sups_of_le le_partialSups_of_le
 
 theorem le_partialSups (f : ℕ → α) : f ≤ partialSups f := fun _n => le_partialSups_of_le f le_rfl
 #align le_partial_sups le_partialSups
 
 theorem partialSups_le (f : ℕ → α) (n : ℕ) (a : α) (w : ∀ m, m ≤ n → f m ≤ a) :
-    partialSups f n ≤ a := by
-  induction' n with n ih
-  · apply w 0 le_rfl
-  · exact sup_le (ih fun m p => w m (Nat.le_succ_of_le p)) (w (n + 1) le_rfl)
+    partialSups f n ≤ a :=
+  partialSups_le_iff.2 w
 #align partial_sups_le partialSups_le
 
+@[simp]
+lemma upperBounds_range_partialSups (f : ℕ → α) :
+    upperBounds (Set.range (partialSups f)) = upperBounds (Set.range f) := by
+  ext a
+  simp only [mem_upperBounds, Set.forall_range_iff, partialSups_le_iff]
+  exact ⟨fun h _ ↦ h _ _ le_rfl, fun h _ _ _ ↦ h _⟩
+
 @[simp]
 theorem bddAbove_range_partialSups {f : ℕ → α} :
-    BddAbove (Set.range (partialSups f)) ↔ BddAbove (Set.range f) := by
-  apply exists_congr fun a => _
-  intro a
-  constructor
-  · rintro h b ⟨i, rfl⟩
-    exact (le_partialSups _ _).trans (h (Set.mem_range_self i))
-  · rintro h b ⟨i, rfl⟩
-    exact partialSups_le _ _ _ fun _ _ => h (Set.mem_range_self _)
+    BddAbove (Set.range (partialSups f)) ↔ BddAbove (Set.range f) :=
+  .of_eq <| congr_arg Set.Nonempty <| upperBounds_range_partialSups f
 #align bdd_above_range_partial_sups bddAbove_range_partialSups
 
 theorem Monotone.partialSups_eq {f : ℕ → α} (hf : Monotone f) : (partialSups f : ℕ → α) = f := by
@@ -96,16 +100,10 @@ theorem Monotone.partialSups_eq {f : ℕ → α} (hf : Monotone f) : (partialSup
   · rw [partialSups_succ, ih, sup_eq_right.2 (hf (Nat.le_succ _))]
 #align monotone.partial_sups_eq Monotone.partialSups_eq
 
-theorem partialSups_mono : Monotone (partialSups : (ℕ → α) → ℕ →o α) := by
-  rintro f g h n
-  induction' n with n ih
-  · exact h 0
-  · exact sup_le_sup ih (h _)
+theorem partialSups_mono : Monotone (partialSups : (ℕ → α) → ℕ →o α) := fun _f _g h _n ↦
+  partialSups_le_iff.2 fun k hk ↦ (h k).trans (le_partialSups_of_le _ hk)
 #align partial_sups_mono partialSups_mono
 
-theorem partialSups_apply_mono (f : ℕ → α) : Monotone (partialSups f) :=
-  fun n _ hnm => partialSups_le f n _ (fun _ hm'n => le_partialSups_of_le _ (hm'n.trans hnm))
-
 /-- `partialSups` forms a Galois insertion with the coercion from monotone functions to functions.
 -/
 def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α) (↑) where
@@ -120,42 +118,42 @@ def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α
 #align partial_sups.gi partialSups.gi
 
 theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
-    partialSups f n = (Finset.range (n + 1)).sup' ⟨n, Finset.self_mem_range_succ n⟩ f := by
-  induction' n with n ih
-  · simp
-  · dsimp [partialSups] at ih ⊢
-    simp_rw [@Finset.range_succ n.succ]
-    rw [ih, Finset.sup'_insert, sup_comm]
+    partialSups f n = (Finset.range (n + 1)).sup' ⟨n, Finset.self_mem_range_succ n⟩ f :=
+  eq_of_forall_ge_iff fun _ ↦ by simp [Nat.lt_succ_iff]
 #align partial_sups_eq_sup'_range partialSups_eq_sup'_range
 
 end SemilatticeSup
 
 theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ → α) (n : ℕ) :
-    partialSups f n = (Finset.range (n + 1)).sup f := by
-  induction' n with n ih
-  · simp
-  · dsimp [partialSups] at ih ⊢
-    rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
+    partialSups f n = (Finset.range (n + 1)).sup f :=
+  eq_of_forall_ge_iff fun _ ↦ by simp [Nat.lt_succ_iff]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
 
+@[simp]
+lemma disjoint_partialSups_left [DistribLattice α] [OrderBot α] {f : ℕ → α} {n : ℕ} {x : α} :
+    Disjoint (partialSups f n) x ↔ ∀ k ≤ n, Disjoint (f k) x :=
+  partialSups_iff_forall (Disjoint · x) disjoint_sup_left
+
+@[simp]
+lemma disjoint_partialSups_right [DistribLattice α] [OrderBot α] {f : ℕ → α} {n : ℕ} {x : α} :
+    Disjoint x (partialSups f n) ↔ ∀ k ≤ n, Disjoint x (f k) :=
+  partialSups_iff_forall (Disjoint x) disjoint_sup_right
+
 /- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as
 submodules. -/
 theorem partialSups_disjoint_of_disjoint [DistribLattice α] [OrderBot α] (f : ℕ → α)
-    (h : Pairwise (Disjoint on f)) {m n : ℕ} (hmn : m < n) : Disjoint (partialSups f m) (f n) := by
-  induction' m with m ih
-  · exact h hmn.ne
-  · rw [partialSups_succ, disjoint_sup_left]
-    exact ⟨ih (Nat.lt_of_succ_lt hmn), h hmn.ne⟩
+    (h : Pairwise (Disjoint on f)) {m n : ℕ} (hmn : m < n) : Disjoint (partialSups f m) (f n) :=
+  disjoint_partialSups_left.2 fun _k hk ↦ h <| (hk.trans_lt hmn).ne
 #align partial_sups_disjoint_of_disjoint partialSups_disjoint_of_disjoint
 
 section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α]
 
-theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i := by
-  have : Set.Iio (n + 1) = Set.Iic n := Set.ext fun _ => Nat.lt_succ_iff
-  rw [partialSups_eq_sup'_range, Finset.sup'_eq_csSup_image, Finset.coe_range, iSup, this]
-  simp only [Set.range, Subtype.exists, Set.mem_Iic, exists_prop, (· '' ·)]
+theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i :=
+  eq_of_forall_ge_iff fun _ ↦ by
+    rw [ciSup_set_le_iff Set.nonempty_Iic ((Set.finite_le_nat _).image _).bddAbove,
+      partialSups_le_iff]; rfl
 #align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iic
 
 @[simp]

This will improve spaces in the mathlib4 docs.

@@ -13,7 +13,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Finset
 # The monotone sequence of partial supremums of a sequence
 
 We define `partialSups : (ℕ → α) → ℕ →o α` inductively. For `f : ℕ → α`, `partialSups f` is
-the sequence `f 0 `, `f 0 ⊔ f 1`, `f 0 ⊔ f 1 ⊔ f 2`, ... The point of this definition is that
+the sequence `f 0`, `f 0 ⊔ f 1`, `f 0 ⊔ f 1 ⊔ f 2`, ... The point of this definition is that
 * it doesn't need a `⨆`, as opposed to `⨆ (i ≤ n), f i` (which also means the wrong thing on
   `ConditionallyCompleteLattice`s).
 * it doesn't need a `⊥`, as opposed to `(Finset.range (n + 1)).sup f`.

Co-authored-by: Peter Pfaffelhuber

From the Kolmogorov extension theorem project.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -103,6 +103,9 @@ theorem partialSups_mono : Monotone (partialSups : (ℕ → α) → ℕ →o α)
   · exact sup_le_sup ih (h _)
 #align partial_sups_mono partialSups_mono
 
+theorem partialSups_apply_mono (f : ℕ → α) : Monotone (partialSups f) :=
+  fun n _ hnm => partialSups_le f n _ (fun _ hm'n => le_partialSups_of_le _ (hm'n.trans hnm))
+
 /-- `partialSups` forms a Galois insertion with the coercion from monotone functions to functions.
 -/
 def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α) (↑) where

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

This has nice performance benefits.

@@ -36,7 +36,7 @@ Necessary for the TODO in the module docstring of `Order.disjointed`.
 -/
 
 
-variable {α : Type _}
+variable {α : Type*}
 
 section SemilatticeSup
 

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module order.partial_sups
-! leanprover-community/mathlib commit d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Finset.Lattice
 import Mathlib.Order.Hom.Basic
 import Mathlib.Order.ConditionallyCompleteLattice.Finset
 
+#align_import order.partial_sups from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
+
 /-!
 # The monotone sequence of partial supremums of a sequence
 

@@ -160,7 +160,7 @@ theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n =
 
 @[simp]
 theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
-    (⨆ n, partialSups f n) = ⨆ n, f n := by
+    ⨆ n, partialSups f n = ⨆ n, f n := by
   refine' (ciSup_le fun n => _).antisymm (ciSup_mono _ <| le_partialSups f)
   · rw [partialSups_eq_ciSup_Iic]
     exact ciSup_le fun i => le_ciSup h _
@@ -178,18 +178,18 @@ theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆
 #align partial_sups_eq_bsupr partialSups_eq_biSup
 
 -- Porting note: simp can prove this @[simp]
-theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
+theorem iSup_partialSups_eq (f : ℕ → α) : ⨆ n, partialSups f n = ⨆ n, f n :=
   ciSup_partialSups_eq <| OrderTop.bddAbove _
 #align supr_partial_sups_eq iSup_partialSups_eq
 
 theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
-    (h : partialSups f ≤ partialSups g) : (⨆ n, f n) ≤ ⨆ n, g n := by
+    (h : partialSups f ≤ partialSups g) : ⨆ n, f n ≤ ⨆ n, g n := by
   rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g]
   exact iSup_mono h
 #align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSups
 
 theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ℕ → α}
-    (h : partialSups f = partialSups g) : (⨆ n, f n) = ⨆ n, g n := by
+    (h : partialSups f = partialSups g) : ⨆ n, f n = ⨆ n, g n := by
   simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]
 #align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSups
 

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -123,7 +123,7 @@ theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
     partialSups f n = (Finset.range (n + 1)).sup' ⟨n, Finset.self_mem_range_succ n⟩ f := by
   induction' n with n ih
   · simp
-  · dsimp [partialSups] at ih⊢
+  · dsimp [partialSups] at ih ⊢
     simp_rw [@Finset.range_succ n.succ]
     rw [ih, Finset.sup'_insert, sup_comm]
 #align partial_sups_eq_sup'_range partialSups_eq_sup'_range
@@ -134,7 +134,7 @@ theorem partialSups_eq_sup_range [SemilatticeSup α] [OrderBot α] (f : ℕ →
     partialSups f n = (Finset.range (n + 1)).sup f := by
   induction' n with n ih
   · simp
-  · dsimp [partialSups] at ih⊢
+  · dsimp [partialSups] at ih ⊢
     rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]
 #align partial_sups_eq_sup_range partialSups_eq_sup_range
 

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>

@@ -22,7 +22,7 @@ the sequence `f 0 `, `f 0 ⊔ f 1`, `f 0 ⊔ f 1 ⊔ f 2`, ... The point of this
 * it doesn't need a `⊥`, as opposed to `(Finset.range (n + 1)).sup f`.
 * it avoids needing to prove that `Finset.range (n + 1)` is nonempty to use `Finset.sup'`.
 
-Equivalence with those definitions is shown by `partialSups_eq_bsupᵢ`, `partialSups_eq_sup_range`,
+Equivalence with those definitions is shown by `partialSups_eq_biSup`, `partialSups_eq_sup_range`,
 and `partialSups_eq_sup'_range` respectively.
 
 ## Notes
@@ -152,20 +152,20 @@ section ConditionallyCompleteLattice
 
 variable [ConditionallyCompleteLattice α]
 
-theorem partialSups_eq_csupᵢ_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i := by
+theorem partialSups_eq_ciSup_Iic (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i : Set.Iic n, f i := by
   have : Set.Iio (n + 1) = Set.Iic n := Set.ext fun _ => Nat.lt_succ_iff
-  rw [partialSups_eq_sup'_range, Finset.sup'_eq_csupₛ_image, Finset.coe_range, supᵢ, this]
+  rw [partialSups_eq_sup'_range, Finset.sup'_eq_csSup_image, Finset.coe_range, iSup, this]
   simp only [Set.range, Subtype.exists, Set.mem_Iic, exists_prop, (· '' ·)]
-#align partial_sups_eq_csupr_Iic partialSups_eq_csupᵢ_Iic
+#align partial_sups_eq_csupr_Iic partialSups_eq_ciSup_Iic
 
 @[simp]
-theorem csupᵢ_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
+theorem ciSup_partialSups_eq {f : ℕ → α} (h : BddAbove (Set.range f)) :
     (⨆ n, partialSups f n) = ⨆ n, f n := by
-  refine' (csupᵢ_le fun n => _).antisymm (csupᵢ_mono _ <| le_partialSups f)
-  · rw [partialSups_eq_csupᵢ_Iic]
-    exact csupᵢ_le fun i => le_csupᵢ h _
+  refine' (ciSup_le fun n => _).antisymm (ciSup_mono _ <| le_partialSups f)
+  · rw [partialSups_eq_ciSup_Iic]
+    exact ciSup_le fun i => le_ciSup h _
   · rwa [bddAbove_range_partialSups]
-#align csupr_partial_sups_eq csupᵢ_partialSups_eq
+#align csupr_partial_sups_eq ciSup_partialSups_eq
 
 end ConditionallyCompleteLattice
 
@@ -173,24 +173,24 @@ section CompleteLattice
 
 variable [CompleteLattice α]
 
-theorem partialSups_eq_bsupᵢ (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
-  simpa only [supᵢ_subtype] using partialSups_eq_csupᵢ_Iic f n
-#align partial_sups_eq_bsupr partialSups_eq_bsupᵢ
+theorem partialSups_eq_biSup (f : ℕ → α) (n : ℕ) : partialSups f n = ⨆ i ≤ n, f i := by
+  simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f n
+#align partial_sups_eq_bsupr partialSups_eq_biSup
 
 -- Porting note: simp can prove this @[simp]
-theorem supᵢ_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
-  csupᵢ_partialSups_eq <| OrderTop.bddAbove _
-#align supr_partial_sups_eq supᵢ_partialSups_eq
+theorem iSup_partialSups_eq (f : ℕ → α) : (⨆ n, partialSups f n) = ⨆ n, f n :=
+  ciSup_partialSups_eq <| OrderTop.bddAbove _
+#align supr_partial_sups_eq iSup_partialSups_eq
 
-theorem supᵢ_le_supᵢ_of_partialSups_le_partialSups {f g : ℕ → α}
+theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ℕ → α}
     (h : partialSups f ≤ partialSups g) : (⨆ n, f n) ≤ ⨆ n, g n := by
-  rw [← supᵢ_partialSups_eq f, ← supᵢ_partialSups_eq g]
-  exact supᵢ_mono h
-#align supr_le_supr_of_partial_sups_le_partial_sups supᵢ_le_supᵢ_of_partialSups_le_partialSups
+  rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g]
+  exact iSup_mono h
+#align supr_le_supr_of_partial_sups_le_partial_sups iSup_le_iSup_of_partialSups_le_partialSups
 
-theorem supᵢ_eq_supᵢ_of_partialSups_eq_partialSups {f g : ℕ → α}
+theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ℕ → α}
     (h : partialSups f = partialSups g) : (⨆ n, f n) = ⨆ n, g n := by
-  simp_rw [← supᵢ_partialSups_eq f, ← supᵢ_partialSups_eq g, h]
-#align supr_eq_supr_of_partial_sups_eq_partial_sups supᵢ_eq_supᵢ_of_partialSups_eq_partialSups
+  simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]
+#align supr_eq_supr_of_partial_sups_eq_partial_sups iSup_eq_iSup_of_partialSups_eq_partialSups
 
 end CompleteLattice

This introduces a tactic congr! that is an analogue to mathlib 3's congr'. It is a more insistent version of congr that makes use of more congruence lemmas (including user congruence lemmas), propext, funext, and Subsingleton instances. It also has a feature to lift reflexive relations to equalities. Along with funext, the tactic does intros, allowing congr! to get access to function bodies; the introduced variables can be named using rename_i if needed.

This also modifies convert to use congr! rather than congr, which makes it work more like the mathlib3 version of the tactic.

@@ -110,7 +110,7 @@ theorem partialSups_mono : Monotone (partialSups : (ℕ → α) → ℕ →o α)
 -/
 def partialSups.gi : GaloisInsertion (partialSups : (ℕ → α) → ℕ →o α) (↑) where
   choice f h :=
-    ⟨f, by convert (partialSups f).monotone; exact (le_partialSups f).antisymm h⟩
+    ⟨f, by convert (partialSups f).monotone using 1; exact (le_partialSups f).antisymm h⟩
   gc f g := by
     refine' ⟨(le_partialSups f).trans, fun h => _⟩
     convert partialSups_mono h

port of order.partial.sups

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