group_theory.subsemigroup.membership ⟷ Mathlib.GroupTheory.Subsemigroup.Membership

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

@@ -139,7 +139,7 @@ then it holds for all elements of the supremum of `S`. -/
 theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx : x ∈ ⨆ i, S i)
     (hp : ∀ (i), ∀ x ∈ S i, C x) (hmul : ∀ x y, C x → C y → C (x * y)) : C x :=
   by
-  rw [supr_eq_closure] at hx 
+  rw [supr_eq_closure] at hx
   refine' closure_induction hx (fun x hx => _) hmul
   obtain ⟨i, hi⟩ := set.mem_Union.mp hx
   exact hp _ _ hi

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

@@ -52,7 +52,7 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
   by
   refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
-    simpa only [closure_Union_of_finite, closure_eq (S _)] using this
+    simpa only [closure_iUnion_of_finite, closure_eq (S _)] using this
   refine' fun hx => closure_induction hx (fun y hy => mem_Union.mp hy) _
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩

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

@@ -3,7 +3,7 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathbin.GroupTheory.Subsemigroup.Basic
+import GroupTheory.Subsemigroup.Basic
 
 #align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"baba818b9acea366489e8ba32d2cc0fcaf50a1f7"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -52,7 +52,7 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
   by
   refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
-    simpa only [closure_iUnion, closure_eq (S _)] using this
+    simpa only [closure_Union_of_finite, closure_eq (S _)] using this
   refine' fun hx => closure_induction hx (fun y hy => mem_Union.mp hy) _
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩

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

@@ -56,7 +56,7 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
   refine' fun hx => closure_induction hx (fun y hy => mem_Union.mp hy) _
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩
-    exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
+    exact ⟨k, (S k).hMul_mem (hki hi) (hkj hj)⟩
 #align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
 #align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
 -/
@@ -107,7 +107,7 @@ theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S
 #print Subsemigroup.mul_mem_sup /-
 @[to_additive]
 theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=
-  mul_mem (mem_sup_left hx) (mem_sup_right hy)
+  hMul_mem (mem_sup_left hx) (mem_sup_right hy)
 #align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_sup
 #align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
 -/
@@ -152,7 +152,7 @@ theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx
 @[elab_as_elim, to_additive "A dependent version of `add_subsemigroup.supr_induction`. "]
 theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
     (hp : ∀ (i), ∀ x ∈ S i, C x (mem_iSup_of_mem i ‹_›))
-    (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x : M}
+    (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (hMul_mem ‹_› ‹_›)) {x : M}
     (hx : x ∈ ⨆ i, S i) : C x hx :=
   by
   refine' Exists.elim _ fun (hx : x ∈ ⨆ i, S i) (hc : C x hx) => hc

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module group_theory.subsemigroup.membership
-! leanprover-community/mathlib commit baba818b9acea366489e8ba32d2cc0fcaf50a1f7
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.GroupTheory.Subsemigroup.Basic
 
+#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"baba818b9acea366489e8ba32d2cc0fcaf50a1f7"
+
 /-!
 # Subsemigroups: membership criteria
 

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

@@ -46,6 +46,7 @@ open Set
 
 namespace Subsemigroup
 
+#print Subsemigroup.mem_iSup_of_directed /-
 -- TODO: this section can be generalized to `[mul_mem_class B M] [complete_lattice B]`
 -- such that `complete_lattice.le` coincides with `set_like.le`
 @[to_additive]
@@ -61,59 +62,77 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
     exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
 #align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
 #align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
+-/
 
+#print Subsemigroup.coe_iSup_of_directed /-
 @[to_additive]
 theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
     ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
   Set.ext fun x => by simp [mem_supr_of_directed hS]
 #align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directed
 #align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_iSup_of_directed
+-/
 
+#print Subsemigroup.mem_sSup_of_directed_on /-
 @[to_additive]
 theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
     x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
   simp only [sSup_eq_iSup', mem_supr_of_directed hS.directed_coe, SetCoe.exists, Subtype.coe_mk]
 #align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_on
 #align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_sSup_of_directed_on
+-/
 
+#print Subsemigroup.coe_sSup_of_directed_on /-
 @[to_additive]
 theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
     (↑(sSup S) : Set M) = ⋃ s ∈ S, ↑s :=
   Set.ext fun x => by simp [mem_Sup_of_directed_on hS]
 #align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_on
 #align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_sSup_of_directed_on
+-/
 
+#print Subsemigroup.mem_sup_left /-
 @[to_additive]
 theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T :=
   show S ≤ S ⊔ T from le_sup_left
 #align subsemigroup.mem_sup_left Subsemigroup.mem_sup_left
 #align add_subsemigroup.mem_sup_left AddSubsemigroup.mem_sup_left
+-/
 
+#print Subsemigroup.mem_sup_right /-
 @[to_additive]
 theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T :=
   show T ≤ S ⊔ T from le_sup_right
 #align subsemigroup.mem_sup_right Subsemigroup.mem_sup_right
 #align add_subsemigroup.mem_sup_right AddSubsemigroup.mem_sup_right
+-/
 
+#print Subsemigroup.mul_mem_sup /-
 @[to_additive]
 theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=
   mul_mem (mem_sup_left hx) (mem_sup_right hy)
 #align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_sup
 #align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
+-/
 
+#print Subsemigroup.mem_iSup_of_mem /-
 @[to_additive]
 theorem mem_iSup_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ iSup S :=
   show S i ≤ iSup S from le_iSup _ _
 #align subsemigroup.mem_supr_of_mem Subsemigroup.mem_iSup_of_mem
 #align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_iSup_of_mem
+-/
 
+#print Subsemigroup.mem_sSup_of_mem /-
 @[to_additive]
 theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
     ∀ {x : M}, x ∈ s → x ∈ sSup S :=
   show s ≤ sSup S from le_sSup hs
 #align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_sSup_of_mem
 #align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_sSup_of_mem
+-/
 
+#print Subsemigroup.iSup_induction /-
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
 then it holds for all elements of the supremum of `S`. -/
@@ -129,7 +148,9 @@ theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx
   exact hp _ _ hi
 #align subsemigroup.supr_induction Subsemigroup.iSup_induction
 #align add_subsemigroup.supr_induction AddSubsemigroup.iSup_induction
+-/
 
+#print Subsemigroup.iSup_induction' /-
 /-- A dependent version of `subsemigroup.supr_induction`. -/
 @[elab_as_elim, to_additive "A dependent version of `add_subsemigroup.supr_induction`. "]
 theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
@@ -144,6 +165,7 @@ theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S
     exact ⟨_, hmul _ _ _ _ Cx Cy⟩
 #align subsemigroup.supr_induction' Subsemigroup.iSup_induction'
 #align add_subsemigroup.supr_induction' AddSubsemigroup.iSup_induction'
+-/
 
 end Subsemigroup
 

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

@@ -123,7 +123,7 @@ then it holds for all elements of the supremum of `S`. -/
 theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx : x ∈ ⨆ i, S i)
     (hp : ∀ (i), ∀ x ∈ S i, C x) (hmul : ∀ x y, C x → C y → C (x * y)) : C x :=
   by
-  rw [supr_eq_closure] at hx
+  rw [supr_eq_closure] at hx 
   refine' closure_induction hx (fun x hx => _) hmul
   obtain ⟨i, hi⟩ := set.mem_Union.mp hx
   exact hp _ _ hi

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

@@ -46,12 +46,6 @@ open Set
 
 namespace Subsemigroup
 
-/- warning: subsemigroup.mem_supr_of_directed -> Subsemigroup.mem_iSup_of_directed is a dubious translation:
-lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toHasLe.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i))))
-but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42) S) -> (forall {x : M}, Iff (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directedₓ'. -/
 -- TODO: this section can be generalized to `[mul_mem_class B M] [complete_lattice B]`
 -- such that `complete_lattice.le` coincides with `set_like.le`
 @[to_additive]
@@ -68,12 +62,6 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
 #align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
 #align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
 
-/- warning: subsemigroup.coe_supr_of_directed -> Subsemigroup.coe_iSup_of_directed is a dubious translation:
-lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toHasLe.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (S i))))
-but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269) S) -> (Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (S i))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directedₓ'. -/
 @[to_additive]
 theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
     ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
@@ -81,12 +69,6 @@ theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
 #align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directed
 #align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_iSup_of_directed
 
-/- warning: subsemigroup.mem_Sup_of_directed_on -> Subsemigroup.mem_sSup_of_directed_on is a dubious translation:
-lean 3 declaration is
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 @[to_additive]
 theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
     x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
@@ -94,12 +76,6 @@ theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (·
 #align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_on
 #align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_sSup_of_directed_on
 
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 @[to_additive]
 theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
     (↑(sSup S) : Set M) = ⋃ s ∈ S, ↑s :=
@@ -107,60 +83,30 @@ theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (·
 #align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_on
 #align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_sSup_of_directed_on
 
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 @[to_additive]
 theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T :=
   show S ≤ S ⊔ T from le_sup_left
 #align subsemigroup.mem_sup_left Subsemigroup.mem_sup_left
 #align add_subsemigroup.mem_sup_left AddSubsemigroup.mem_sup_left
 
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 @[to_additive]
 theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T :=
   show T ≤ S ⊔ T from le_sup_right
 #align subsemigroup.mem_sup_right Subsemigroup.mem_sup_right
 #align add_subsemigroup.mem_sup_right AddSubsemigroup.mem_sup_right
 
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 @[to_additive]
 theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=
   mul_mem (mem_sup_left hx) (mem_sup_right hy)
 #align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_sup
 #align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
 
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 @[to_additive]
 theorem mem_iSup_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ iSup S :=
   show S i ≤ iSup S from le_iSup _ _
 #align subsemigroup.mem_supr_of_mem Subsemigroup.mem_iSup_of_mem
 #align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_iSup_of_mem
 
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 @[to_additive]
 theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
     ∀ {x : M}, x ∈ s → x ∈ sSup S :=
@@ -168,12 +114,6 @@ theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s
 #align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_sSup_of_mem
 #align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_sSup_of_mem
 
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-Case conversion may be inaccurate. Consider using '#align subsemigroup.supr_induction Subsemigroup.iSup_inductionₓ'. -/
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
 then it holds for all elements of the supremum of `S`. -/
@@ -190,12 +130,6 @@ theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx
 #align subsemigroup.supr_induction Subsemigroup.iSup_induction
 #align add_subsemigroup.supr_induction AddSubsemigroup.iSup_induction
 
-/- warning: subsemigroup.supr_induction' -> Subsemigroup.iSup_induction' is a dubious translation:
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 /-- A dependent version of `subsemigroup.supr_induction`. -/
 @[elab_as_elim, to_additive "A dependent version of `add_subsemigroup.supr_induction`. "]
 theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}

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

@@ -48,7 +48,7 @@ namespace Subsemigroup
 
 /- warning: subsemigroup.mem_supr_of_directed -> Subsemigroup.mem_iSup_of_directed is a dubious translation:
 lean 3 declaration is
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+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toHasLe.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i))))
 but is expected to have type
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 Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directedₓ'. -/
@@ -70,7 +70,7 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
 
 /- warning: subsemigroup.coe_supr_of_directed -> Subsemigroup.coe_iSup_of_directed is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (S i))))
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toHasLe.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (S i))))
 but is expected to have type
   forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269) S) -> (Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (S i))))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directedₓ'. -/
@@ -83,7 +83,7 @@ theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
 
 /- warning: subsemigroup.mem_Sup_of_directed_on -> Subsemigroup.mem_sSup_of_directed_on is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Exists.{0} (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s))))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toHasLe.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Exists.{0} (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s))))
 but is expected to have type
   forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359) S) -> (forall {x : M}, Iff (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => And (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x s))))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_onₓ'. -/
@@ -96,7 +96,7 @@ theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (·
 
 /- warning: subsemigroup.coe_Sup_of_directed_on -> Subsemigroup.coe_sSup_of_directed_on is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (Eq.{succ u1} (Set.{u1} M) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Set.iUnion.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.iUnion.{u1, 0} M (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => (fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) s))))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toHasLe.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (Eq.{succ u1} (Set.{u1} M) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Set.iUnion.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.iUnion.{u1, 0} M (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => (fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) s))))
 but is expected to have type
   forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430) S) -> (Eq.{succ u1} (Set.{u1} M) (SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Set.iUnion.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.iUnion.{u1, 0} M (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) s))))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_onₓ'. -/

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

@@ -46,66 +46,66 @@ open Set
 
 namespace Subsemigroup
 
-/- warning: subsemigroup.mem_supr_of_directed -> Subsemigroup.mem_supᵢ_of_directed is a dubious translation:
+/- warning: subsemigroup.mem_supr_of_directed -> Subsemigroup.mem_iSup_of_directed is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i))))
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i))))
 but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42) S) -> (forall {x : M}, Iff (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_supᵢ_of_directedₓ'. -/
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.40 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.42) S) -> (forall {x : M}, Iff (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Exists.{u1} ι (fun (i : ι) => Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i))))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directedₓ'. -/
 -- TODO: this section can be generalized to `[mul_mem_class B M] [complete_lattice B]`
 -- such that `complete_lattice.le` coincides with `set_like.le`
 @[to_additive]
-theorem mem_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :
+theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :
     (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i :=
   by
-  refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_supᵢ S i) hi⟩
+  refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
-    simpa only [closure_unionᵢ, closure_eq (S _)] using this
+    simpa only [closure_iUnion, closure_eq (S _)] using this
   refine' fun hx => closure_induction hx (fun y hy => mem_Union.mp hy) _
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩
     exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
-#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_supᵢ_of_directed
-#align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_supᵢ_of_directed
+#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
+#align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
 
-/- warning: subsemigroup.coe_supr_of_directed -> Subsemigroup.coe_supᵢ_of_directed is a dubious translation:
+/- warning: subsemigroup.coe_supr_of_directed -> Subsemigroup.coe_iSup_of_directed is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Set.unionᵢ.{u2, u1} M ι (fun (i : ι) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (S i))))
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (SetLike.partialOrder.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1))))) S) -> (Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => (fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Subsemigroup.{u2} M _inst_1) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)))) (S i))))
 but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269) S) -> (Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Set.unionᵢ.{u2, u1} M ι (fun (i : ι) => SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (S i))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_supᵢ_of_directedₓ'. -/
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)}, (Directed.{u2, u1} (Subsemigroup.{u2} M _inst_1) ι (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 : Subsemigroup.{u2} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269 : Subsemigroup.{u2} M _inst_1) => LE.le.{u2} (Subsemigroup.{u2} M _inst_1) (Preorder.toLE.{u2} (Subsemigroup.{u2} M _inst_1) (PartialOrder.toPreorder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.267 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.269) S) -> (Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (Set.iUnion.{u2, u1} M ι (fun (i : ι) => SetLike.coe.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (S i))))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directedₓ'. -/
 @[to_additive]
-theorem coe_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
+theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
     ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
   Set.ext fun x => by simp [mem_supr_of_directed hS]
-#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_supᵢ_of_directed
-#align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_supᵢ_of_directed
+#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directed
+#align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_iSup_of_directed
 
-/- warning: subsemigroup.mem_Sup_of_directed_on -> Subsemigroup.mem_supₛ_of_directed_on is a dubious translation:
+/- warning: subsemigroup.mem_Sup_of_directed_on -> Subsemigroup.mem_sSup_of_directed_on is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Exists.{0} (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s))))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (forall {x : M}, Iff (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Exists.{0} (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s))))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359) S) -> (forall {x : M}, Iff (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => And (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x s))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_supₛ_of_directed_onₓ'. -/
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.357 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.359) S) -> (forall {x : M}, Iff (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Exists.{succ u1} (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => And (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x s))))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_onₓ'. -/
 @[to_additive]
-theorem mem_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
-    x ∈ supₛ S ↔ ∃ s ∈ S, x ∈ s := by
-  simp only [supₛ_eq_supᵢ', mem_supr_of_directed hS.directed_coe, SetCoe.exists, Subtype.coe_mk]
-#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_supₛ_of_directed_on
-#align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_supₛ_of_directed_on
+theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
+    x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
+  simp only [sSup_eq_iSup', mem_supr_of_directed hS.directed_coe, SetCoe.exists, Subtype.coe_mk]
+#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_on
+#align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_sSup_of_directed_on
 
-/- warning: subsemigroup.coe_Sup_of_directed_on -> Subsemigroup.coe_supₛ_of_directed_on is a dubious translation:
+/- warning: subsemigroup.coe_Sup_of_directed_on -> Subsemigroup.coe_sSup_of_directed_on is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (Eq.{succ u1} (Set.{u1} M) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Set.unionᵢ.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.unionᵢ.{u1, 0} M (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => (fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) s))))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (SetLike.partialOrder.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1))))) S) -> (Eq.{succ u1} (Set.{u1} M) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)) (Set.iUnion.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.iUnion.{u1, 0} M (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => (fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)))) s))))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430) S) -> (Eq.{succ u1} (Set.{u1} M) (SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Set.unionᵢ.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.unionᵢ.{u1, 0} M (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) s))))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_supₛ_of_directed_onₓ'. -/
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)}, (DirectedOn.{u1} (Subsemigroup.{u1} M _inst_1) (fun (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 : Subsemigroup.{u1} M _inst_1) (x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430 : Subsemigroup.{u1} M _inst_1) => LE.le.{u1} (Subsemigroup.{u1} M _inst_1) (Preorder.toLE.{u1} (Subsemigroup.{u1} M _inst_1) (PartialOrder.toPreorder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeInf.toPartialOrder.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1))))) x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.428 x._@.Mathlib.GroupTheory.Subsemigroup.Membership._hyg.430) S) -> (Eq.{succ u1} (Set.{u1} M) (SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)) (Set.iUnion.{u1, succ u1} M (Subsemigroup.{u1} M _inst_1) (fun (s : Subsemigroup.{u1} M _inst_1) => Set.iUnion.{u1, 0} M (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) (fun (H : Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) => SetLike.coe.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1) s))))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_onₓ'. -/
 @[to_additive]
-theorem coe_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
-    (↑(supₛ S) : Set M) = ⋃ s ∈ S, ↑s :=
+theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
+    (↑(sSup S) : Set M) = ⋃ s ∈ S, ↑s :=
   Set.ext fun x => by simp [mem_Sup_of_directed_on hS]
-#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_supₛ_of_directed_on
-#align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_supₛ_of_directed_on
+#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_on
+#align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_sSup_of_directed_on
 
 /- warning: subsemigroup.mem_sup_left -> Subsemigroup.mem_sup_left is a dubious translation:
 lean 3 declaration is
@@ -143,63 +143,63 @@ theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈
 #align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_sup
 #align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
 
-/- warning: subsemigroup.mem_supr_of_mem -> Subsemigroup.mem_supᵢ_of_mem is a dubious translation:
+/- warning: subsemigroup.mem_supr_of_mem -> Subsemigroup.mem_iSup_of_mem is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)} (i : ι) {x : M}, (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)) -> (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι S))
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)} (i : ι) {x : M}, (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)) -> (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι S))
 but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)} (i : ι) {x : M}, (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)) -> (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι S))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_supᵢ_of_memₓ'. -/
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] {S : ι -> (Subsemigroup.{u2} M _inst_1)} (i : ι) {x : M}, (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)) -> (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι S))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_iSup_of_memₓ'. -/
 @[to_additive]
-theorem mem_supᵢ_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ supᵢ S :=
-  show S i ≤ supᵢ S from le_supᵢ _ _
-#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_supᵢ_of_mem
-#align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_supᵢ_of_mem
+theorem mem_iSup_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ iSup S :=
+  show S i ≤ iSup S from le_iSup _ _
+#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_iSup_of_mem
+#align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_iSup_of_mem
 
-/- warning: subsemigroup.mem_Sup_of_mem -> Subsemigroup.mem_supₛ_of_mem is a dubious translation:
+/- warning: subsemigroup.mem_Sup_of_mem -> Subsemigroup.mem_sSup_of_mem is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)} {s : Subsemigroup.{u1} M _inst_1}, (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) -> (forall {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)} {s : Subsemigroup.{u1} M _inst_1}, (Membership.Mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.hasMem.{u1} (Subsemigroup.{u1} M _inst_1)) s S) -> (forall {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x s) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1))) S)))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)} {s : Subsemigroup.{u1} M _inst_1}, (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) -> (forall {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x s) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (SupSet.supₛ.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)))
-Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_supₛ_of_memₓ'. -/
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Set.{u1} (Subsemigroup.{u1} M _inst_1)} {s : Subsemigroup.{u1} M _inst_1}, (Membership.mem.{u1, u1} (Subsemigroup.{u1} M _inst_1) (Set.{u1} (Subsemigroup.{u1} M _inst_1)) (Set.instMembershipSet.{u1} (Subsemigroup.{u1} M _inst_1)) s S) -> (forall {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x s) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (SupSet.sSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toSupSet.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)) S)))
+Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_sSup_of_memₓ'. -/
 @[to_additive]
-theorem mem_supₛ_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
-    ∀ {x : M}, x ∈ s → x ∈ supₛ S :=
-  show s ≤ supₛ S from le_supₛ hs
-#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_supₛ_of_mem
-#align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_supₛ_of_mem
+theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
+    ∀ {x : M}, x ∈ s → x ∈ sSup S :=
+  show s ≤ sSup S from le_sSup hs
+#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_sSup_of_mem
+#align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_sSup_of_mem
 
-/- warning: subsemigroup.supr_induction -> Subsemigroup.supᵢ_induction is a dubious translation:
+/- warning: subsemigroup.supr_induction -> Subsemigroup.iSup_induction is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : M -> Prop} {x : M}, (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) -> (forall (i : ι) (x : M), (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)) -> (C x)) -> (forall (x : M) (y : M), (C x) -> (C y) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y))) -> (C x)
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : M -> Prop} {x : M}, (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) -> (forall (i : ι) (x : M), (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)) -> (C x)) -> (forall (x : M) (y : M), (C x) -> (C y) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y))) -> (C x)
 but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : M -> Prop} {x : M}, (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) -> (forall (i : ι) (x : M), (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)) -> (C x)) -> (forall (x : M) (y : M), (C x) -> (C y) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y))) -> (C x)
-Case conversion may be inaccurate. Consider using '#align subsemigroup.supr_induction Subsemigroup.supᵢ_inductionₓ'. -/
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : M -> Prop} {x : M}, (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) -> (forall (i : ι) (x : M), (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)) -> (C x)) -> (forall (x : M) (y : M), (C x) -> (C y) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y))) -> (C x)
+Case conversion may be inaccurate. Consider using '#align subsemigroup.supr_induction Subsemigroup.iSup_inductionₓ'. -/
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
 then it holds for all elements of the supremum of `S`. -/
 @[elab_as_elim,
   to_additive
       " An induction principle for elements of `⨆ i, S i`.\nIf `C` holds all elements of `S i` for all `i`, and is preserved under addition,\nthen it holds for all elements of the supremum of `S`. "]
-theorem supᵢ_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx : x ∈ ⨆ i, S i)
+theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x : M} (hx : x ∈ ⨆ i, S i)
     (hp : ∀ (i), ∀ x ∈ S i, C x) (hmul : ∀ x y, C x → C y → C (x * y)) : C x :=
   by
   rw [supr_eq_closure] at hx
   refine' closure_induction hx (fun x hx => _) hmul
   obtain ⟨i, hi⟩ := set.mem_Union.mp hx
   exact hp _ _ hi
-#align subsemigroup.supr_induction Subsemigroup.supᵢ_induction
-#align add_subsemigroup.supr_induction AddSubsemigroup.supᵢ_induction
+#align subsemigroup.supr_induction Subsemigroup.iSup_induction
+#align add_subsemigroup.supr_induction AddSubsemigroup.iSup_induction
 
-/- warning: subsemigroup.supr_induction' -> Subsemigroup.supᵢ_induction' is a dubious translation:
+/- warning: subsemigroup.supr_induction' -> Subsemigroup.iSup_induction' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : forall (x : M), (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) -> Prop}, (forall (i : ι) (x : M) (H : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)), C x (Subsemigroup.mem_supᵢ_of_mem.{u1, u2} ι M _inst_1 (fun (i : ι) => S i) i x H)) -> (forall (x : M) (y : M) (hx : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (hy : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) y (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))), (C x hx) -> (C y hy) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y) (MulMemClass.mul_mem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M _inst_1 (Subsemigroup.setLike.{u2} M _inst_1) (Subsemigroup.mulMemClass.{u2} M _inst_1) (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i)) x y hx hy))) -> (forall {x : M} (hx : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))), C x hx)
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : forall (x : M), (Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) -> Prop}, (forall (i : ι) (x : M) (H : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (S i)), C x (Subsemigroup.mem_iSup_of_mem.{u1, u2} ι M _inst_1 (fun (i : ι) => S i) i x H)) -> (forall (x : M) (y : M) (hx : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))) (hy : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) y (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))), (C x hx) -> (C y hy) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y) (MulMemClass.mul_mem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M _inst_1 (Subsemigroup.setLike.{u2} M _inst_1) (Subsemigroup.mulMemClass.{u2} M _inst_1) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i)) x y hx hy))) -> (forall {x : M} (hx : Membership.Mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.hasMem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.setLike.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u2} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.completeLattice.{u2} M _inst_1))) ι (fun (i : ι) => S i))), C x hx)
 but is expected to have type
-  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : forall (x : M), (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) -> Prop}, (forall (i : ι) (x : M) (H : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)), C x (Subsemigroup.mem_supᵢ_of_mem.{u1, u2} ι M _inst_1 (fun (i : ι) => S i) i x H)) -> (forall (x : M) (y : M) (hx : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (hy : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) y (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))), (C x hx) -> (C y hy) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y) (MulMemClass.mul_mem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M _inst_1 (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (Subsemigroup.instMulMemClassSubsemigroupInstSetLikeSubsemigroup.{u2} M _inst_1) (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i)) x y hx hy))) -> (forall {x : M} (hx : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (supᵢ.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))), C x hx)
-Case conversion may be inaccurate. Consider using '#align subsemigroup.supr_induction' Subsemigroup.supᵢ_induction'ₓ'. -/
+  forall {ι : Sort.{u1}} {M : Type.{u2}} [_inst_1 : Mul.{u2} M] (S : ι -> (Subsemigroup.{u2} M _inst_1)) {C : forall (x : M), (Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) -> Prop}, (forall (i : ι) (x : M) (H : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (S i)), C x (Subsemigroup.mem_iSup_of_mem.{u1, u2} ι M _inst_1 (fun (i : ι) => S i) i x H)) -> (forall (x : M) (y : M) (hx : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))) (hy : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) y (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))), (C x hx) -> (C y hy) -> (C (HMul.hMul.{u2, u2, u2} M M M (instHMul.{u2} M _inst_1) x y) (MulMemClass.mul_mem.{u2, u2} (Subsemigroup.{u2} M _inst_1) M _inst_1 (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1) (Subsemigroup.instMulMemClassSubsemigroupInstSetLikeSubsemigroup.{u2} M _inst_1) (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i)) x y hx hy))) -> (forall {x : M} (hx : Membership.mem.{u2, u2} M (Subsemigroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subsemigroup.{u2} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u2} M _inst_1)) x (iSup.{u2, u1} (Subsemigroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subsemigroup.{u2} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u2} M _inst_1)) ι (fun (i : ι) => S i))), C x hx)
+Case conversion may be inaccurate. Consider using '#align subsemigroup.supr_induction' Subsemigroup.iSup_induction'ₓ'. -/
 /-- A dependent version of `subsemigroup.supr_induction`. -/
 @[elab_as_elim, to_additive "A dependent version of `add_subsemigroup.supr_induction`. "]
-theorem supᵢ_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
-    (hp : ∀ (i), ∀ x ∈ S i, C x (mem_supᵢ_of_mem i ‹_›))
+theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
+    (hp : ∀ (i), ∀ x ∈ S i, C x (mem_iSup_of_mem i ‹_›))
     (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x : M}
     (hx : x ∈ ⨆ i, S i) : C x hx :=
   by
@@ -208,8 +208,8 @@ theorem supᵢ_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i,
   · exact ⟨_, hp _ _ hx⟩
   · rintro ⟨_, Cx⟩ ⟨_, Cy⟩
     exact ⟨_, hmul _ _ _ _ Cx Cy⟩
-#align subsemigroup.supr_induction' Subsemigroup.supᵢ_induction'
-#align add_subsemigroup.supr_induction' AddSubsemigroup.supᵢ_induction'
+#align subsemigroup.supr_induction' Subsemigroup.iSup_induction'
+#align add_subsemigroup.supr_induction' AddSubsemigroup.iSup_induction'
 
 end Subsemigroup
 

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

@@ -109,9 +109,9 @@ theorem coe_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (
 
 /- warning: subsemigroup.mem_sup_left -> Subsemigroup.mem_sup_left is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x S) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x S) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x S) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x S) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_sup_left Subsemigroup.mem_sup_leftₓ'. -/
 @[to_additive]
 theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T :=
@@ -121,9 +121,9 @@ theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S 
 
 /- warning: subsemigroup.mem_sup_right -> Subsemigroup.mem_sup_right is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x T) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x T) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x T) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x T) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.mem_sup_right Subsemigroup.mem_sup_rightₓ'. -/
 @[to_additive]
 theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T :=
@@ -133,9 +133,9 @@ theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S
 
 /- warning: subsemigroup.mul_mem_sup -> Subsemigroup.mul_mem_sup is a dubious translation:
 lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M} {y : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x S) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) y T) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M _inst_1) x y) (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M} {y : M}, (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) x S) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) y T) -> (Membership.Mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.hasMem.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.setLike.{u1} M _inst_1)) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M _inst_1) x y) (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.completeLattice.{u1} M _inst_1)))) S T))
 but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M} {y : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x S) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) y T) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M _inst_1) x y) (HasSup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
+  forall {M : Type.{u1}} [_inst_1 : Mul.{u1} M] {S : Subsemigroup.{u1} M _inst_1} {T : Subsemigroup.{u1} M _inst_1} {x : M} {y : M}, (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) x S) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) y T) -> (Membership.mem.{u1, u1} M (Subsemigroup.{u1} M _inst_1) (SetLike.instMembership.{u1, u1} (Subsemigroup.{u1} M _inst_1) M (Subsemigroup.instSetLikeSubsemigroup.{u1} M _inst_1)) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M _inst_1) x y) (Sup.sup.{u1} (Subsemigroup.{u1} M _inst_1) (SemilatticeSup.toSup.{u1} (Subsemigroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subsemigroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subsemigroup.{u1} M _inst_1) (Subsemigroup.instCompleteLatticeSubsemigroup.{u1} M _inst_1)))) S T))
 Case conversion may be inaccurate. Consider using '#align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_supₓ'. -/
 @[to_additive]
 theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=

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

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

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

@@ -48,9 +48,9 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
     simpa only [closure_iUnion, closure_eq (S _)] using this
   refine fun hx ↦ closure_induction hx (fun y hy ↦ mem_iUnion.mp hy) ?_
-  · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
-    rcases hS i j with ⟨k, hki, hkj⟩
-    exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
+  rintro x y ⟨i, hi⟩ ⟨j, hj⟩
+  rcases hS i j with ⟨k, hki, hkj⟩
+  exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
 #align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
 #align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
 

The additive version are still incorrectly named, but these can easily be tracked down later (#11462) by searching for to_additive (attr := elab_as_elim).

@@ -119,11 +119,11 @@ then it holds for all elements of the supremum of `S`. -/
 elements of `S i` for all `i`, and is preserved under addition, then it holds for all elements of
 the supremum of `S`."]
 theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
-    (hp : ∀ i, ∀ x₂ ∈ S i, C x₂) (hmul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by
+    (mem : ∀ i, ∀ x₂ ∈ S i, C x₂) (mul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by
   rw [iSup_eq_closure] at hx₁
-  refine' closure_induction hx₁ (fun x₂ hx₂ => _) hmul
+  refine' closure_induction hx₁ (fun x₂ hx₂ => _) mul
   obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx₂
-  exact hp _ _ hi
+  exact mem _ _ hi
 #align subsemigroup.supr_induction Subsemigroup.iSup_induction
 #align add_subsemigroup.supr_induction AddSubsemigroup.iSup_induction
 
@@ -131,15 +131,15 @@ theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (
 @[to_additive (attr := elab_as_elim)
 "A dependent version of `AddSubsemigroup.iSup_induction`."]
 theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
-    (hp : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_iSup_of_mem i ‹_›))
-    (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x₁ : M}
+    (mem : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_iSup_of_mem i ‹_›))
+    (mul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x₁ : M}
     (hx₁ : x₁ ∈ ⨆ i, S i) : C x₁ hx₁ := by
   refine Exists.elim ?_ fun (hx₁' : x₁ ∈ ⨆ i, S i) (hc : C x₁ hx₁') => hc
   refine @iSup_induction _ _ _ S (fun x' => ∃ hx'', C x' hx'') _ hx₁
       (fun i x₂ hx₂ => ?_) fun x₃ y => ?_
-  · exact ⟨_, hp _ _ hx₂⟩
+  · exact ⟨_, mem _ _ hx₂⟩
   · rintro ⟨_, Cx⟩ ⟨_, Cy⟩
-    exact ⟨_, hmul _ _ _ _ Cx Cy⟩
+    exact ⟨_, mul _ _ _ _ Cx Cy⟩
 #align subsemigroup.supr_induction' Subsemigroup.iSup_induction'
 #align add_subsemigroup.supr_induction' AddSubsemigroup.iSup_induction'
 

Changes in this PR shouldn't change the public API. The only changes about ∃ x ∈ s, _ is inside a proof.

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

@@ -119,7 +119,7 @@ then it holds for all elements of the supremum of `S`. -/
 elements of `S i` for all `i`, and is preserved under addition, then it holds for all elements of
 the supremum of `S`."]
 theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
-    (hp : ∀ (i) (x₂ : M) (_hxS : x₂ ∈ S i), C x₂) (hmul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by
+    (hp : ∀ i, ∀ x₂ ∈ S i, C x₂) (hmul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by
   rw [iSup_eq_closure] at hx₁
   refine' closure_induction hx₁ (fun x₂ hx₂ => _) hmul
   obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx₂

@@ -44,10 +44,10 @@ namespace Subsemigroup
 @[to_additive]
 theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :
     (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by
-  refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩
+  refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
     simpa only [closure_iUnion, closure_eq (S _)] using this
-  refine' fun hx => closure_induction hx (fun y hy => mem_iUnion.mp hy) _
+  refine fun hx ↦ closure_induction hx (fun y hy ↦ mem_iUnion.mp hy) ?_
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩
     exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
@@ -56,7 +56,7 @@ theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤
 
 @[to_additive]
 theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
-    ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
+    ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, S i :=
   Set.ext fun x => by simp [mem_iSup_of_directed hS]
 #align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directed
 #align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_iSup_of_directed

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

This has nice performance benefits.

@@ -29,7 +29,7 @@ stub and only provides rudimentary support.
 subsemigroup
 -/
 
-variable {ι : Sort _} {M A B : Type _}
+variable {ι : Sort*} {M A B : Type*}
 
 section NonAssoc
 

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
-
-! This file was ported from Lean 3 source module group_theory.subsemigroup.membership
-! leanprover-community/mathlib commit 6cb77a8eaff0ddd100e87b1591c6d3ad319514ff
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.GroupTheory.Subsemigroup.Basic
 
+#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
+
 /-!
 # Subsemigroups: membership criteria
 

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

@@ -42,7 +42,7 @@ open Set
 
 namespace Subsemigroup
 
--- TODO: this section can be generalized to `[mul_mem_class B M] [complete_lattice B]`
+-- TODO: this section can be generalized to `[MulMemClass B M] [CompleteLattice B]`
 -- such that `complete_lattice.le` coincides with `set_like.le`
 @[to_additive]
 theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :

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>

@@ -17,8 +17,8 @@ In this file we prove various facts about membership in a subsemigroup.
 The intent is to mimic `GroupTheory/Submonoid/Membership`, but currently this file is mostly a
 stub and only provides rudimentary support.
 
-* `mem_supᵢ_of_directed`, `coe_supᵢ_of_directed`, `mem_supₛ_of_directed_on`,
-  `coe_supₛ_of_directed_on`: the supremum of a directed collection of subsemigroup is their union.
+* `mem_iSup_of_directed`, `coe_iSup_of_directed`, `mem_sSup_of_directed_on`,
+  `coe_sSup_of_directed_on`: the supremum of a directed collection of subsemigroup is their union.
 
 ## TODO
 
@@ -45,39 +45,39 @@ namespace Subsemigroup
 -- TODO: this section can be generalized to `[mul_mem_class B M] [complete_lattice B]`
 -- such that `complete_lattice.le` coincides with `set_like.le`
 @[to_additive]
-theorem mem_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :
+theorem mem_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) {x : M} :
     (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by
-  refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_supᵢ S i) hi⟩
+  refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩
   suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
-    simpa only [closure_unionᵢ, closure_eq (S _)] using this
-  refine' fun hx => closure_induction hx (fun y hy => mem_unionᵢ.mp hy) _
+    simpa only [closure_iUnion, closure_eq (S _)] using this
+  refine' fun hx => closure_induction hx (fun y hy => mem_iUnion.mp hy) _
   · rintro x y ⟨i, hi⟩ ⟨j, hj⟩
     rcases hS i j with ⟨k, hki, hkj⟩
     exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
-#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_supᵢ_of_directed
-#align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_supᵢ_of_directed
+#align subsemigroup.mem_supr_of_directed Subsemigroup.mem_iSup_of_directed
+#align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_iSup_of_directed
 
 @[to_additive]
-theorem coe_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
+theorem coe_iSup_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
     ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
-  Set.ext fun x => by simp [mem_supᵢ_of_directed hS]
-#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_supᵢ_of_directed
-#align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_supᵢ_of_directed
+  Set.ext fun x => by simp [mem_iSup_of_directed hS]
+#align subsemigroup.coe_supr_of_directed Subsemigroup.coe_iSup_of_directed
+#align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_iSup_of_directed
 
 @[to_additive]
-theorem mem_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
-    x ∈ supₛ S ↔ ∃ s ∈ S, x ∈ s := by
-  simp only [supₛ_eq_supᵢ', mem_supᵢ_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
+theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
+    x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by
+  simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
     exists_prop]
-#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_supₛ_of_directed_on
-#align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_supₛ_of_directed_on
+#align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_sSup_of_directed_on
+#align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_sSup_of_directed_on
 
 @[to_additive]
-theorem coe_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
-    (↑(supₛ S) : Set M) = ⋃ s ∈ S, ↑s :=
-  Set.ext fun x => by simp [mem_supₛ_of_directed_on hS]
-#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_supₛ_of_directed_on
-#align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_supₛ_of_directed_on
+theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
+    (↑(sSup S) : Set M) = ⋃ s ∈ S, ↑s :=
+  Set.ext fun x => by simp [mem_sSup_of_directed_on hS]
+#align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_sSup_of_directed_on
+#align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_sSup_of_directed_on
 
 @[to_additive]
 theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
@@ -100,19 +100,19 @@ theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈
 #align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
 
 @[to_additive]
-theorem mem_supᵢ_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ supᵢ S := by
-  have : S i ≤ supᵢ S := le_supᵢ _ _
+theorem mem_iSup_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ iSup S := by
+  have : S i ≤ iSup S := le_iSup _ _
   tauto
-#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_supᵢ_of_mem
-#align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_supᵢ_of_mem
+#align subsemigroup.mem_supr_of_mem Subsemigroup.mem_iSup_of_mem
+#align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_iSup_of_mem
 
 @[to_additive]
-theorem mem_supₛ_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
-    ∀ {x : M}, x ∈ s → x ∈ supₛ S := by
-  have : s ≤ supₛ S := le_supₛ hs
+theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
+    ∀ {x : M}, x ∈ s → x ∈ sSup S := by
+  have : s ≤ sSup S := le_sSup hs
   tauto
-#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_supₛ_of_mem
-#align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_supₛ_of_mem
+#align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_sSup_of_mem
+#align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_sSup_of_mem
 
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
@@ -121,30 +121,30 @@ then it holds for all elements of the supremum of `S`. -/
 "An induction principle for elements of `⨆ i, S i`. If `C` holds all
 elements of `S i` for all `i`, and is preserved under addition, then it holds for all elements of
 the supremum of `S`."]
-theorem supᵢ_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
+theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
     (hp : ∀ (i) (x₂ : M) (_hxS : x₂ ∈ S i), C x₂) (hmul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by
-  rw [supᵢ_eq_closure] at hx₁
+  rw [iSup_eq_closure] at hx₁
   refine' closure_induction hx₁ (fun x₂ hx₂ => _) hmul
-  obtain ⟨i, hi⟩ := Set.mem_unionᵢ.mp hx₂
+  obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx₂
   exact hp _ _ hi
-#align subsemigroup.supr_induction Subsemigroup.supᵢ_induction
-#align add_subsemigroup.supr_induction AddSubsemigroup.supᵢ_induction
+#align subsemigroup.supr_induction Subsemigroup.iSup_induction
+#align add_subsemigroup.supr_induction AddSubsemigroup.iSup_induction
 
-/-- A dependent version of `Subsemigroup.supᵢ_induction`. -/
+/-- A dependent version of `Subsemigroup.iSup_induction`. -/
 @[to_additive (attr := elab_as_elim)
-"A dependent version of `AddSubsemigroup.supᵢ_induction`."]
-theorem supᵢ_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
-    (hp : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_supᵢ_of_mem i ‹_›))
+"A dependent version of `AddSubsemigroup.iSup_induction`."]
+theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
+    (hp : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_iSup_of_mem i ‹_›))
     (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x₁ : M}
     (hx₁ : x₁ ∈ ⨆ i, S i) : C x₁ hx₁ := by
   refine Exists.elim ?_ fun (hx₁' : x₁ ∈ ⨆ i, S i) (hc : C x₁ hx₁') => hc
-  refine @supᵢ_induction _ _ _ S (fun x' => ∃ hx'', C x' hx'') _ hx₁
+  refine @iSup_induction _ _ _ S (fun x' => ∃ hx'', C x' hx'') _ hx₁
       (fun i x₂ hx₂ => ?_) fun x₃ y => ?_
   · exact ⟨_, hp _ _ hx₂⟩
   · rintro ⟨_, Cx⟩ ⟨_, Cy⟩
     exact ⟨_, hmul _ _ _ _ Cx Cy⟩
-#align subsemigroup.supr_induction' Subsemigroup.supᵢ_induction'
-#align add_subsemigroup.supr_induction' AddSubsemigroup.supᵢ_induction'
+#align subsemigroup.supr_induction' Subsemigroup.iSup_induction'
+#align add_subsemigroup.supr_induction' AddSubsemigroup.iSup_induction'
 
 end Subsemigroup
 

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>

@@ -55,12 +55,14 @@ theorem mem_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· 
     rcases hS i j with ⟨k, hki, hkj⟩
     exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
 #align subsemigroup.mem_supr_of_directed Subsemigroup.mem_supᵢ_of_directed
+#align add_subsemigroup.mem_supr_of_directed AddSubsemigroup.mem_supᵢ_of_directed
 
 @[to_additive]
 theorem coe_supᵢ_of_directed {S : ι → Subsemigroup M} (hS : Directed (· ≤ ·) S) :
     ((⨆ i, S i : Subsemigroup M) : Set M) = ⋃ i, ↑(S i) :=
   Set.ext fun x => by simp [mem_supᵢ_of_directed hS]
 #align subsemigroup.coe_supr_of_directed Subsemigroup.coe_supᵢ_of_directed
+#align add_subsemigroup.coe_supr_of_directed AddSubsemigroup.coe_supᵢ_of_directed
 
 @[to_additive]
 theorem mem_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
@@ -68,35 +70,41 @@ theorem mem_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (
   simp only [supₛ_eq_supᵢ', mem_supᵢ_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
     exists_prop]
 #align subsemigroup.mem_Sup_of_directed_on Subsemigroup.mem_supₛ_of_directed_on
+#align add_subsemigroup.mem_Sup_of_directed_on AddSubsemigroup.mem_supₛ_of_directed_on
 
 @[to_additive]
 theorem coe_supₛ_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
     (↑(supₛ S) : Set M) = ⋃ s ∈ S, ↑s :=
   Set.ext fun x => by simp [mem_supₛ_of_directed_on hS]
 #align subsemigroup.coe_Sup_of_directed_on Subsemigroup.coe_supₛ_of_directed_on
+#align add_subsemigroup.coe_Sup_of_directed_on AddSubsemigroup.coe_supₛ_of_directed_on
 
 @[to_additive]
 theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
   have : S ≤ S ⊔ T := le_sup_left
   tauto
 #align subsemigroup.mem_sup_left Subsemigroup.mem_sup_left
+#align add_subsemigroup.mem_sup_left AddSubsemigroup.mem_sup_left
 
 @[to_additive]
 theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by
   have : T ≤ S ⊔ T := le_sup_right
   tauto
 #align subsemigroup.mem_sup_right Subsemigroup.mem_sup_right
+#align add_subsemigroup.mem_sup_right AddSubsemigroup.mem_sup_right
 
 @[to_additive]
 theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=
   mul_mem (mem_sup_left hx) (mem_sup_right hy)
 #align subsemigroup.mul_mem_sup Subsemigroup.mul_mem_sup
+#align add_subsemigroup.add_mem_sup AddSubsemigroup.add_mem_sup
 
 @[to_additive]
 theorem mem_supᵢ_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ supᵢ S := by
   have : S i ≤ supᵢ S := le_supᵢ _ _
   tauto
 #align subsemigroup.mem_supr_of_mem Subsemigroup.mem_supᵢ_of_mem
+#align add_subsemigroup.mem_supr_of_mem AddSubsemigroup.mem_supᵢ_of_mem
 
 @[to_additive]
 theorem mem_supₛ_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
@@ -104,6 +112,7 @@ theorem mem_supₛ_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs :
   have : s ≤ supₛ S := le_supₛ hs
   tauto
 #align subsemigroup.mem_Sup_of_mem Subsemigroup.mem_supₛ_of_mem
+#align add_subsemigroup.mem_Sup_of_mem AddSubsemigroup.mem_supₛ_of_mem
 
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
@@ -119,6 +128,7 @@ theorem supᵢ_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M}
   obtain ⟨i, hi⟩ := Set.mem_unionᵢ.mp hx₂
   exact hp _ _ hi
 #align subsemigroup.supr_induction Subsemigroup.supᵢ_induction
+#align add_subsemigroup.supr_induction AddSubsemigroup.supᵢ_induction
 
 /-- A dependent version of `Subsemigroup.supᵢ_induction`. -/
 @[to_additive (attr := elab_as_elim)
@@ -134,6 +144,7 @@ theorem supᵢ_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i,
   · rintro ⟨_, Cx⟩ ⟨_, Cy⟩
     exact ⟨_, hmul _ _ _ _ Cx Cy⟩
 #align subsemigroup.supr_induction' Subsemigroup.supᵢ_induction'
+#align add_subsemigroup.supr_induction' AddSubsemigroup.supᵢ_induction'
 
 end Subsemigroup
 

• The new syntax for any attributes that need to be copied by to_additive is @[to_additive (attrs := simp, ext, simps)]
• Adds the auxiliary declarations generated by the simp and simps attributes to the to_additive-dictionary.
• Future issue: Does not yet translate auxiliary declarations for other attributes (including custom simp-attributes). In particular it's possible that norm_cast might generate some auxiliary declarations.
• Fixes #950
• Fixes #953
• Fixes #1149
• This moves the interaction between to_additive and simps from the Simps file to the toAdditive file for uniformity.
• Make the same changes to @[reassoc]

Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -108,7 +108,8 @@ theorem mem_supₛ_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs :
 /-- An induction principle for elements of `⨆ i, S i`.
 If `C` holds all elements of `S i` for all `i`, and is preserved under multiplication,
 then it holds for all elements of the supremum of `S`. -/
-@[elab_as_elim, to_additive "An induction principle for elements of `⨆ i, S i`. If `C` holds all
+@[to_additive (attr := elab_as_elim)
+"An induction principle for elements of `⨆ i, S i`. If `C` holds all
 elements of `S i` for all `i`, and is preserved under addition, then it holds for all elements of
 the supremum of `S`."]
 theorem supᵢ_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
@@ -120,7 +121,8 @@ theorem supᵢ_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M}
 #align subsemigroup.supr_induction Subsemigroup.supᵢ_induction
 
 /-- A dependent version of `Subsemigroup.supᵢ_induction`. -/
-@[elab_as_elim, to_additive "A dependent version of `AddSubsemigroup.supᵢ_induction`."]
+@[to_additive (attr := elab_as_elim)
+"A dependent version of `AddSubsemigroup.supᵢ_induction`."]
 theorem supᵢ_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
     (hp : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_supᵢ_of_mem i ‹_›))
     (hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x₁ : M}

Co-authored-by: Moritz Firsching <firsching@google.com> Co-authored-by: Frédéric Dupuis <dupuisf@iro.umontreal.ca>