group_theory.torsion ⟷ Mathlib.GroupTheory.Torsion

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

@@ -84,7 +84,7 @@ noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
     inv := fun g => g ^ (orderOf g - 1)
     hMul_left_inv := fun g =>
       by
-      erw [← pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]
+      erw [← pow_succ, tsub_add_cancel_of_le, pow_orderOf_eq_one]
       exact IsOfFinOrder.orderOf_pos (tG g) }
 #align is_torsion.group IsTorsion.group
 #align is_torsion.add_group IsTorsion.addGroup

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

@@ -285,7 +285,7 @@ theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠
     Disjoint (CommMonoid.primaryComponent G p) (CommMonoid.primaryComponent G p') :=
   Submonoid.disjoint_def.mpr <| by
     rintro g ⟨_ | n, hn⟩ ⟨n', hn'⟩
-    · rwa [pow_zero, orderOf_eq_one_iff] at hn 
+    · rwa [pow_zero, orderOf_eq_one_iff] at hn
     ·
       exact
         absurd (eq_of_prime_pow_eq hp.out.prime hp'.out.prime n.succ_pos (hn.symm.trans hn')) hne

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

@@ -163,7 +163,7 @@ theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g =
       "The group exponent exists for any bounded additive torsion group."]
 theorem IsTorsion.exponentExists (tG : IsTorsion G)
     (bounded : (Set.range fun g : G => orderOf g).Finite) : ExponentExists G :=
-  exponentExists_iff_ne_zero.mpr <|
+  exponent_ne_zero.mpr <|
     (exponent_ne_zero_iff_range_orderOf_finite fun g => IsOfFinOrder.orderOf_pos (tG g)).mpr bounded
 #align is_torsion.exponent_exists IsTorsion.exponentExists
 #align is_add_torsion.exponent_exists IsAddTorsion.exponentExists
@@ -173,7 +173,7 @@ theorem IsTorsion.exponentExists (tG : IsTorsion G)
 /-- Finite groups are torsion groups. -/
 @[to_additive is_add_torsion_of_finite "Finite additive groups are additive torsion groups."]
 theorem isTorsion_of_finite [Finite G] : IsTorsion G :=
-  ExponentExists.isTorsion <| exponentExists_iff_ne_zero.mpr exponent_ne_zero_of_finite
+  ExponentExists.isTorsion <| exponent_ne_zero.mpr exponent_ne_zero_of_finite
 #align is_torsion_of_finite isTorsion_of_finite
 #align is_add_torsion_of_finite is_add_torsion_of_finite
 -/

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

@@ -85,7 +85,7 @@ noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
     hMul_left_inv := fun g =>
       by
       erw [← pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]
-      exact orderOf_pos' (tG g) }
+      exact IsOfFinOrder.orderOf_pos (tG g) }
 #align is_torsion.group IsTorsion.group
 #align is_torsion.add_group IsTorsion.addGroup
 -/
@@ -98,7 +98,7 @@ variable [Group G] {N : Subgroup G} [Group H]
 /-- Subgroups of torsion groups are torsion groups. -/
 @[to_additive "Subgroups of additive torsion groups are additive torsion groups."]
 theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H := fun h =>
-  (isOfFinOrder_iff_coe H.toSubmonoid h).mpr <| tG h
+  (Submonoid.isOfFinOrder_coe H.toSubmonoid h).mpr <| tG h
 #align is_torsion.subgroup IsTorsion.subgroup
 #align is_torsion.add_subgroup IsTorsion.addSubgroup
 -/
@@ -164,7 +164,7 @@ theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g =
 theorem IsTorsion.exponentExists (tG : IsTorsion G)
     (bounded : (Set.range fun g : G => orderOf g).Finite) : ExponentExists G :=
   exponentExists_iff_ne_zero.mpr <|
-    (exponent_ne_zero_iff_range_orderOf_finite fun g => orderOf_pos' (tG g)).mpr bounded
+    (exponent_ne_zero_iff_range_orderOf_finite fun g => IsOfFinOrder.orderOf_pos (tG g)).mpr bounded
 #align is_torsion.exponent_exists IsTorsion.exponentExists
 #align is_add_torsion.exponent_exists IsAddTorsion.exponentExists
 -/
@@ -466,7 +466,7 @@ theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬Is
 @[to_additive "Subgroups of additive torsion-free groups are additively torsion-free."]
 theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsionFree H :=
   fun h hne =>
-  (isOfFinOrder_iff_coe H.toSubmonoid h).Not.mpr <|
+  (Submonoid.isOfFinOrder_coe H.toSubmonoid h).Not.mpr <|
     tG h <| by norm_cast <;> simp [hne, not_false_iff]
 #align is_torsion_free.subgroup IsTorsionFree.subgroup
 #align is_torsion_free.add_subgroup IsTorsionFree.addSubgroup

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

@@ -67,7 +67,7 @@ def IsTorsion :=
 @[simp,
   to_additive "An additive monoid is not a torsion monoid if it has an element of infinite order."]
 theorem not_isTorsion_iff : ¬IsTorsion G ↔ ∃ g : G, ¬IsOfFinOrder g := by
-  rw [is_torsion, not_forall]
+  rw [is_torsion, Classical.not_forall]
 #align monoid.not_is_torsion_iff Monoid.not_isTorsion_iff
 #align add_monoid.not_is_torsion_iff AddMonoid.not_isTorsion_iff
 -/
@@ -422,7 +422,7 @@ def IsTorsionFree :=
 @[simp,
   to_additive "An additive monoid is not torsion free if any nontrivial element has finite order."]
 theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsOfFinOrder g := by
-  simp_rw [is_torsion_free, Ne.def, not_forall, Classical.not_not, exists_prop]
+  simp_rw [is_torsion_free, Ne.def, Classical.not_forall, Classical.not_not, exists_prop]
 #align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iff
 #align add_monoid.not_is_torsion_free_iff AddMonoid.not_isTorsionFree_iff
 -/

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

@@ -3,11 +3,11 @@ Copyright (c) 2022 Julian Berman. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Julian Berman
 -/
-import Mathbin.GroupTheory.Exponent
-import Mathbin.GroupTheory.OrderOfElement
-import Mathbin.GroupTheory.PGroup
-import Mathbin.GroupTheory.QuotientGroup
-import Mathbin.GroupTheory.Submonoid.Operations
+import GroupTheory.Exponent
+import GroupTheory.OrderOfElement
+import GroupTheory.PGroup
+import GroupTheory.QuotientGroup
+import GroupTheory.Submonoid.Operations
 
 #align_import group_theory.torsion from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
 

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

@@ -82,7 +82,7 @@ open Monoid
 noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
   { ‹Monoid G› with
     inv := fun g => g ^ (orderOf g - 1)
-    mul_left_inv := fun g =>
+    hMul_left_inv := fun g =>
       by
       erw [← pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]
       exact orderOf_pos' (tG g) }
@@ -224,7 +224,7 @@ namespace CommMonoid
 def torsion : Submonoid G where
   carrier := {x | IsOfFinOrder x}
   one_mem' := isOfFinOrder_one
-  mul_mem' _ _ hx hy := hx.mul hy
+  hMul_mem' _ _ hx hy := hx.mul hy
 #align comm_monoid.torsion CommMonoid.torsion
 #align add_comm_monoid.add_torsion AddCommMonoid.addTorsion
 -/
@@ -254,14 +254,14 @@ def primaryComponent : Submonoid G
     where
   carrier := {g | ∃ n : ℕ, orderOf g = p ^ n}
   one_mem' := ⟨0, by rw [pow_zero, orderOf_one]⟩
-  mul_mem' g₁ g₂ hg₁ hg₂ :=
+  hMul_mem' g₁ g₂ hg₁ hg₂ :=
     exists_orderOf_eq_prime_pow_iff.mpr <|
       by
       obtain ⟨m, hm⟩ := exists_order_of_eq_prime_pow_iff.mp hg₁
       obtain ⟨n, hn⟩ := exists_order_of_eq_prime_pow_iff.mp hg₂
       exact
         ⟨m + n, by
-          rw [mul_pow, pow_add, pow_mul, hm, one_pow, Monoid.one_mul, mul_comm, pow_mul, hn,
+          rw [mul_pow, pow_add, pow_mul, hm, one_pow, Monoid.one_hMul, mul_comm, pow_mul, hn,
             one_pow]⟩
 #align comm_monoid.primary_component CommMonoid.primaryComponent
 #align add_comm_monoid.primary_component AddCommMonoid.primaryComponent

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Julian Berman. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Julian Berman
-
-! This file was ported from Lean 3 source module group_theory.torsion
-! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.GroupTheory.Exponent
 import Mathbin.GroupTheory.OrderOfElement
@@ -14,6 +9,8 @@ import Mathbin.GroupTheory.PGroup
 import Mathbin.GroupTheory.QuotientGroup
 import Mathbin.GroupTheory.Submonoid.Operations
 
+#align_import group_theory.torsion from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-!
 # Torsion groups
 

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

@@ -97,13 +97,16 @@ section Group
 
 variable [Group G] {N : Subgroup G} [Group H]
 
+#print IsTorsion.subgroup /-
 /-- Subgroups of torsion groups are torsion groups. -/
 @[to_additive "Subgroups of additive torsion groups are additive torsion groups."]
 theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H := fun h =>
   (isOfFinOrder_iff_coe H.toSubmonoid h).mpr <| tG h
 #align is_torsion.subgroup IsTorsion.subgroup
 #align is_torsion.add_subgroup IsTorsion.addSubgroup
+-/
 
+#print IsTorsion.of_surjective /-
 /-- The image of a surjective torsion group homomorphism is torsion. -/
 @[to_additive AddIsTorsion.of_surjective
       "The image of a surjective additive torsion group homomorphism is torsion."]
@@ -114,7 +117,9 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
   exact f.is_of_fin_order (tG g)
 #align is_torsion.of_surjective IsTorsion.of_surjective
 #align add_is_torsion.of_surjective AddIsTorsion.of_surjective
+-/
 
+#print IsTorsion.extension_closed /-
 /-- Torsion groups are closed under extensions. -/
 @[to_additive AddIsTorsion.extension_closed "Additive torsion groups are closed under extensions."]
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
@@ -130,7 +135,9 @@ theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsi
         rw [pow_mul, ← h, ← Subgroup.coe_pow, hnn, Subgroup.coe_one]⟩
 #align is_torsion.extension_closed IsTorsion.extension_closed
 #align add_is_torsion.extension_closed AddIsTorsion.extension_closed
+-/
 
+#print IsTorsion.quotient_iff /-
 /-- The image of a quotient is torsion iff the group is torsion. -/
 @[to_additive AddIsTorsion.quotient_iff
       "The image of a quotient is additively torsion iff the group is torsion."]
@@ -139,6 +146,7 @@ theorem IsTorsion.quotient_iff {f : G →* H} (hf : Function.Surjective f) (hN :
   ⟨fun tH => IsTorsion.extension_closed hN tH tN, fun tG => IsTorsion.of_surjective hf tG⟩
 #align is_torsion.quotient_iff IsTorsion.quotient_iff
 #align add_is_torsion.quotient_iff AddIsTorsion.quotient_iff
+-/
 
 #print ExponentExists.isTorsion /-
 /-- If a group exponent exists, the group is torsion. -/
@@ -182,6 +190,7 @@ variable (R M : Type _) [AddCommMonoid M]
 
 namespace AddMonoid
 
+#print AddMonoid.IsTorsion.module_of_torsion /-
 /-- A module whose scalars are additively torsion is additively torsion. -/
 theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R) : IsTorsion M :=
   fun f =>
@@ -190,11 +199,14 @@ theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R)
     obtain ⟨n, npos, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero _).mp (tR 1)
     exact ⟨n, npos, by simp only [nsmul_eq_smul_cast R _ f, ← nsmul_one, hn, zero_smul]⟩
 #align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsion
+-/
 
+#print AddMonoid.IsTorsion.module_of_finite /-
 /-- A module with a finite ring of scalars is additively torsion. -/
 theorem IsTorsion.module_of_finite [Ring R] [Finite R] [Module R M] : IsTorsion M :=
   (is_add_torsion_of_finite : IsTorsion R).module_of_torsion _ _
 #align add_monoid.is_torsion.module_of_finite AddMonoid.IsTorsion.module_of_finite
+-/
 
 end AddMonoid
 
@@ -222,6 +234,7 @@ def torsion : Submonoid G where
 
 variable {G}
 
+#print CommMonoid.torsion.isTorsion /-
 /-- Torsion submonoids are torsion. -/
 @[to_additive "Additive torsion submonoids are additively torsion."]
 theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =>
@@ -231,11 +244,10 @@ theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =
         (isPeriodicPt_mul_iff_pow_eq_one _).mp hn]⟩
 #align comm_monoid.torsion.is_torsion CommMonoid.torsion.isTorsion
 #align add_comm_monoid.add_torsion.is_torsion AddCommMonoid.addTorsion.isTorsion
+-/
 
 variable (G) (p : ℕ) [hp : Fact p.Prime]
 
-include hp
-
 #print CommMonoid.primaryComponent /-
 /-- The `p`-primary component is the submonoid of elements with order prime-power of `p`. -/
 @[to_additive
@@ -260,13 +272,16 @@ def primaryComponent : Submonoid G
 
 variable {G} {p}
 
+#print CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow /-
 /-- Elements of the `p`-primary component have order `p^n` for some `n`. -/
 @[to_additive "Elements of the `p`-primary component have additive order `p^n` for some `n`"]
 theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComponent G p) :
     ∃ n : ℕ, orderOf g = p ^ n := by simpa [primary_component] using g.property
 #align comm_monoid.primary_component.exists_order_of_eq_prime_pow CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow
 #align add_comm_monoid.primary_component.exists_order_of_eq_prime_nsmul AddCommMonoid.primaryComponent.exists_orderOf_eq_prime_nsmul
+-/
 
+#print CommMonoid.primaryComponent.disjoint /-
 /-- The `p`- and `q`-primary components are disjoint for `p ≠ q`. -/
 @[to_additive "The `p`- and `q`-primary components are disjoint for `p ≠ q`."]
 theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠ p') :
@@ -279,6 +294,7 @@ theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠
         absurd (eq_of_prime_pow_eq hp.out.prime hp'.out.prime n.succ_pos (hn.symm.trans hn')) hne
 #align comm_monoid.primary_component.disjoint CommMonoid.primaryComponent.disjoint
 #align add_comm_monoid.primary_component.disjoint AddCommMonoid.primaryComponent.disjoint
+-/
 
 end CommMonoid
 
@@ -288,35 +304,44 @@ namespace Monoid.IsTorsion
 
 variable {G}
 
+#print Monoid.IsTorsion.torsion_eq_top /-
 /-- The torsion submonoid of a torsion monoid is `⊤`. -/
 @[simp, to_additive "The additive torsion submonoid of an additive torsion monoid is `⊤`."]
 theorem torsion_eq_top (tG : IsTorsion G) : torsion G = ⊤ := by ext <;> tauto
 #align monoid.is_torsion.torsion_eq_top Monoid.IsTorsion.torsion_eq_top
 #align add_monoid.is_torsion.torsion_eq_top AddMonoid.IsTorsion.torsion_eq_top
+-/
 
+#print Monoid.IsTorsion.torsionMulEquiv /-
 /-- A torsion monoid is isomorphic to its torsion submonoid. -/
 @[to_additive "An additive torsion monoid is isomorphic to its torsion submonoid."]
 def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
   (MulEquiv.submonoidCongr tG.torsion_eq_top).trans Submonoid.topEquiv
 #align monoid.is_torsion.torsion_mul_equiv Monoid.IsTorsion.torsionMulEquiv
 #align add_monoid.is_torsion.torsion_add_equiv AddMonoid.IsTorsion.torsionAddEquiv
+-/
 
+#print Monoid.IsTorsion.torsionMulEquiv_apply /-
 @[to_additive]
 theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
     tG.torsionMulEquiv a = MulEquiv.submonoidCongr tG.torsion_eq_top a :=
   rfl
 #align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_apply
 #align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsionAddEquiv_apply
+-/
 
+#print Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe /-
 @[to_additive]
 theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
     tG.torsionMulEquiv.symm a = ⟨Submonoid.topEquiv.symm a, tG _⟩ :=
   rfl
 #align monoid.is_torsion.torsion_mul_equiv_symm_apply_coe Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe
 #align add_monoid.is_torsion.torsion_add_equiv_symm_apply_coe AddMonoid.IsTorsion.torsionAddEquiv_symm_apply_coe
+-/
 
 end Monoid.IsTorsion
 
+#print Torsion.ofTorsion /-
 /-- Torsion submonoids of a torsion submonoid are isomorphic to the submonoid. -/
 @[simp,
   to_additive AddCommMonoid.Torsion.ofTorsion
@@ -325,6 +350,7 @@ def Torsion.ofTorsion : torsion (torsion G) ≃* torsion G :=
   Monoid.IsTorsion.torsionMulEquiv CommMonoid.torsion.isTorsion
 #align torsion.of_torsion Torsion.ofTorsion
 #align add_comm_monoid.torsion.of_torsion AddCommMonoid.Torsion.ofTorsion
+-/
 
 end CommMonoid
 
@@ -355,8 +381,6 @@ theorem torsion_eq_torsion_submonoid : CommMonoid.torsion G = (torsion G).toSubm
 
 variable (p : ℕ) [hp : Fact p.Prime]
 
-include hp
-
 #print CommGroup.primaryComponent /-
 /-- The `p`-primary component is the subgroup of elements with order prime-power of `p`. -/
 @[to_additive
@@ -371,11 +395,13 @@ def primaryComponent : Subgroup G :=
 
 variable {G} {p}
 
+#print CommGroup.primaryComponent.isPGroup /-
 /-- The `p`-primary component is a `p` group. -/
 theorem primaryComponent.isPGroup : IsPGroup p <| primaryComponent G p := fun g =>
   (propext exists_orderOf_eq_prime_pow_iff.symm).mpr
     (CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow g)
 #align comm_group.primary_component.is_p_group CommGroup.primaryComponent.isPGroup
+-/
 
 end CommGroup
 
@@ -394,6 +420,7 @@ def IsTorsionFree :=
 #align add_monoid.is_torsion_free AddMonoid.IsTorsionFree
 -/
 
+#print Monoid.not_isTorsionFree_iff /-
 /-- A nontrivial monoid is not torsion-free if any nontrivial element has finite order. -/
 @[simp,
   to_additive "An additive monoid is not torsion free if any nontrivial element has finite order."]
@@ -401,6 +428,7 @@ theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsO
   simp_rw [is_torsion_free, Ne.def, not_forall, Classical.not_not, exists_prop]
 #align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iff
 #align add_monoid.not_is_torsion_free_iff AddMonoid.not_isTorsionFree_iff
+-/
 
 end Monoid
 
@@ -436,6 +464,7 @@ theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬Is
 #align add_monoid.is_torsion_free.not_torsion AddMonoid.IsTorsionFree.not_torsion
 -/
 
+#print IsTorsionFree.subgroup /-
 /-- Subgroups of torsion-free groups are torsion-free. -/
 @[to_additive "Subgroups of additive torsion-free groups are additively torsion-free."]
 theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsionFree H :=
@@ -444,7 +473,9 @@ theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsi
     tG h <| by norm_cast <;> simp [hne, not_false_iff]
 #align is_torsion_free.subgroup IsTorsionFree.subgroup
 #align is_torsion_free.add_subgroup IsTorsionFree.addSubgroup
+-/
 
+#print IsTorsionFree.prod /-
 /-- Direct products of torsion free groups are torsion free. -/
 @[to_additive AddMonoid.IsTorsionFree.prod
       "Direct products of additive torsion free groups are torsion free."]
@@ -454,6 +485,7 @@ theorem IsTorsionFree.prod {η : Type _} {Gs : η → Type _} [∀ i, Group (Gs
     funext fun i => Classical.not_not.mp <| mt (tfGs i (w i)) <| Classical.not_not.mpr <| h.apply i
 #align is_torsion_free.prod IsTorsionFree.prod
 #align add_monoid.is_torsion_free.prod AddMonoid.IsTorsionFree.prod
+-/
 
 end Group
 

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

@@ -213,7 +213,7 @@ namespace CommMonoid
 -/
 @[to_additive add_torsion "The torsion submonoid of an additive commutative monoid."]
 def torsion : Submonoid G where
-  carrier := { x | IsOfFinOrder x }
+  carrier := {x | IsOfFinOrder x}
   one_mem' := isOfFinOrder_one
   mul_mem' _ _ hx hy := hx.mul hy
 #align comm_monoid.torsion CommMonoid.torsion
@@ -243,7 +243,7 @@ include hp
   simps]
 def primaryComponent : Submonoid G
     where
-  carrier := { g | ∃ n : ℕ, orderOf g = p ^ n }
+  carrier := {g | ∃ n : ℕ, orderOf g = p ^ n}
   one_mem' := ⟨0, by rw [pow_zero, orderOf_one]⟩
   mul_mem' g₁ g₂ hg₁ hg₂ :=
     exists_orderOf_eq_prime_pow_iff.mpr <|

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

@@ -273,7 +273,7 @@ theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠
     Disjoint (CommMonoid.primaryComponent G p) (CommMonoid.primaryComponent G p') :=
   Submonoid.disjoint_def.mpr <| by
     rintro g ⟨_ | n, hn⟩ ⟨n', hn'⟩
-    · rwa [pow_zero, orderOf_eq_one_iff] at hn
+    · rwa [pow_zero, orderOf_eq_one_iff] at hn 
     ·
       exact
         absurd (eq_of_prime_pow_eq hp.out.prime hp'.out.prime n.succ_pos (hn.symm.trans hn')) hne

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

@@ -227,7 +227,7 @@ variable {G}
 theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =>
   ⟨n, npos,
     Subtype.ext <| by
-      rw [mul_left_iterate, _root_.mul_one, [anonymous], Subtype.coe_mk, Submonoid.coe_one,
+      rw [mul_left_iterate, _root_.mul_one, Submonoid.coe_pow, Subtype.coe_mk, Submonoid.coe_one,
         (isPeriodicPt_mul_iff_pow_eq_one _).mp hn]⟩
 #align comm_monoid.torsion.is_torsion CommMonoid.torsion.isTorsion
 #align add_comm_monoid.add_torsion.is_torsion AddCommMonoid.addTorsion.isTorsion

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

@@ -97,12 +97,6 @@ section Group
 
 variable [Group G] {N : Subgroup G} [Group H]
 
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 /-- Subgroups of torsion groups are torsion groups. -/
 @[to_additive "Subgroups of additive torsion groups are additive torsion groups."]
 theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H := fun h =>
@@ -110,12 +104,6 @@ theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H :=
 #align is_torsion.subgroup IsTorsion.subgroup
 #align is_torsion.add_subgroup IsTorsion.addSubgroup
 
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 /-- The image of a surjective torsion group homomorphism is torsion. -/
 @[to_additive AddIsTorsion.of_surjective
       "The image of a surjective additive torsion group homomorphism is torsion."]
@@ -127,12 +115,6 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
 #align is_torsion.of_surjective IsTorsion.of_surjective
 #align add_is_torsion.of_surjective AddIsTorsion.of_surjective
 
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 /-- Torsion groups are closed under extensions. -/
 @[to_additive AddIsTorsion.extension_closed "Additive torsion groups are closed under extensions."]
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
@@ -149,12 +131,6 @@ theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsi
 #align is_torsion.extension_closed IsTorsion.extension_closed
 #align add_is_torsion.extension_closed AddIsTorsion.extension_closed
 
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 /-- The image of a quotient is torsion iff the group is torsion. -/
 @[to_additive AddIsTorsion.quotient_iff
       "The image of a quotient is additively torsion iff the group is torsion."]
@@ -206,12 +182,6 @@ variable (R M : Type _) [AddCommMonoid M]
 
 namespace AddMonoid
 
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 /-- A module whose scalars are additively torsion is additively torsion. -/
 theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R) : IsTorsion M :=
   fun f =>
@@ -221,12 +191,6 @@ theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R)
     exact ⟨n, npos, by simp only [nsmul_eq_smul_cast R _ f, ← nsmul_one, hn, zero_smul]⟩
 #align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsion
 
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 /-- A module with a finite ring of scalars is additively torsion. -/
 theorem IsTorsion.module_of_finite [Ring R] [Finite R] [Module R M] : IsTorsion M :=
   (is_add_torsion_of_finite : IsTorsion R).module_of_torsion _ _
@@ -258,12 +222,6 @@ def torsion : Submonoid G where
 
 variable {G}
 
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 /-- Torsion submonoids are torsion. -/
 @[to_additive "Additive torsion submonoids are additively torsion."]
 theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =>
@@ -302,12 +260,6 @@ def primaryComponent : Submonoid G
 
 variable {G} {p}
 
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 /-- Elements of the `p`-primary component have order `p^n` for some `n`. -/
 @[to_additive "Elements of the `p`-primary component have additive order `p^n` for some `n`"]
 theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComponent G p) :
@@ -315,12 +267,6 @@ theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComp
 #align comm_monoid.primary_component.exists_order_of_eq_prime_pow CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow
 #align add_comm_monoid.primary_component.exists_order_of_eq_prime_nsmul AddCommMonoid.primaryComponent.exists_orderOf_eq_prime_nsmul
 
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 /-- The `p`- and `q`-primary components are disjoint for `p ≠ q`. -/
 @[to_additive "The `p`- and `q`-primary components are disjoint for `p ≠ q`."]
 theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠ p') :
@@ -342,24 +288,12 @@ namespace Monoid.IsTorsion
 
 variable {G}
 
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 /-- The torsion submonoid of a torsion monoid is `⊤`. -/
 @[simp, to_additive "The additive torsion submonoid of an additive torsion monoid is `⊤`."]
 theorem torsion_eq_top (tG : IsTorsion G) : torsion G = ⊤ := by ext <;> tauto
 #align monoid.is_torsion.torsion_eq_top Monoid.IsTorsion.torsion_eq_top
 #align add_monoid.is_torsion.torsion_eq_top AddMonoid.IsTorsion.torsion_eq_top
 
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 /-- A torsion monoid is isomorphic to its torsion submonoid. -/
 @[to_additive "An additive torsion monoid is isomorphic to its torsion submonoid."]
 def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
@@ -367,9 +301,6 @@ def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
 #align monoid.is_torsion.torsion_mul_equiv Monoid.IsTorsion.torsionMulEquiv
 #align add_monoid.is_torsion.torsion_add_equiv AddMonoid.IsTorsion.torsionAddEquiv
 
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-<too large>
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 @[to_additive]
 theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
     tG.torsionMulEquiv a = MulEquiv.submonoidCongr tG.torsion_eq_top a :=
@@ -377,9 +308,6 @@ theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
 #align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_apply
 #align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsionAddEquiv_apply
 
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 @[to_additive]
 theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
     tG.torsionMulEquiv.symm a = ⟨Submonoid.topEquiv.symm a, tG _⟩ :=
@@ -389,9 +317,6 @@ theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
 
 end Monoid.IsTorsion
 
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 /-- Torsion submonoids of a torsion submonoid are isomorphic to the submonoid. -/
 @[simp,
   to_additive AddCommMonoid.Torsion.ofTorsion
@@ -446,12 +371,6 @@ def primaryComponent : Subgroup G :=
 
 variable {G} {p}
 
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-Case conversion may be inaccurate. Consider using '#align comm_group.primary_component.is_p_group CommGroup.primaryComponent.isPGroupₓ'. -/
 /-- The `p`-primary component is a `p` group. -/
 theorem primaryComponent.isPGroup : IsPGroup p <| primaryComponent G p := fun g =>
   (propext exists_orderOf_eq_prime_pow_iff.symm).mpr
@@ -475,12 +394,6 @@ def IsTorsionFree :=
 #align add_monoid.is_torsion_free AddMonoid.IsTorsionFree
 -/
 
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-Case conversion may be inaccurate. Consider using '#align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iffₓ'. -/
 /-- A nontrivial monoid is not torsion-free if any nontrivial element has finite order. -/
 @[simp,
   to_additive "An additive monoid is not torsion free if any nontrivial element has finite order."]
@@ -523,12 +436,6 @@ theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬Is
 #align add_monoid.is_torsion_free.not_torsion AddMonoid.IsTorsionFree.not_torsion
 -/
 
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 /-- Subgroups of torsion-free groups are torsion-free. -/
 @[to_additive "Subgroups of additive torsion-free groups are additively torsion-free."]
 theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsionFree H :=
@@ -538,12 +445,6 @@ theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsi
 #align is_torsion_free.subgroup IsTorsionFree.subgroup
 #align is_torsion_free.add_subgroup IsTorsionFree.addSubgroup
 
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 /-- Direct products of torsion free groups are torsion free. -/
 @[to_additive AddMonoid.IsTorsionFree.prod
       "Direct products of additive torsion free groups are torsion free."]

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Julian Berman
 
 ! This file was ported from Lean 3 source module group_theory.torsion
-! leanprover-community/mathlib commit 1f4705ccdfe1e557fc54a0ce081a05e33d2e6240
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.GroupTheory.Submonoid.Operations
 /-!
 # Torsion groups
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines torsion groups, i.e. groups where all elements have finite order.
 
 ## Main definitions
@@ -365,10 +368,7 @@ def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
 #align add_monoid.is_torsion.torsion_add_equiv AddMonoid.IsTorsion.torsionAddEquiv
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_applyₓ'. -/
 @[to_additive]
 theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
@@ -378,10 +378,7 @@ theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
 #align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsionAddEquiv_apply
 
 /- warning: monoid.is_torsion.torsion_mul_equiv_symm_apply_coe -> Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_mul_equiv_symm_apply_coe Monoid.IsTorsion.torsionMulEquiv_symm_apply_coeₓ'. -/
 @[to_additive]
 theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
@@ -393,10 +390,7 @@ theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
 end Monoid.IsTorsion
 
 /- warning: torsion.of_torsion -> Torsion.ofTorsion is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align torsion.of_torsion Torsion.ofTorsionₓ'. -/
 /-- Torsion submonoids of a torsion submonoid are isomorphic to the submonoid. -/
 @[simp,

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -53,13 +53,16 @@ namespace Monoid
 
 variable (G) [Monoid G]
 
+#print Monoid.IsTorsion /-
 /-- A predicate on a monoid saying that all elements are of finite order. -/
 @[to_additive "A predicate on an additive monoid saying that all elements are of finite order."]
 def IsTorsion :=
   ∀ g : G, IsOfFinOrder g
 #align monoid.is_torsion Monoid.IsTorsion
 #align add_monoid.is_torsion AddMonoid.IsTorsion
+-/
 
+#print Monoid.not_isTorsion_iff /-
 /-- A monoid is not a torsion monoid if it has an element of infinite order. -/
 @[simp,
   to_additive "An additive monoid is not a torsion monoid if it has an element of infinite order."]
@@ -67,11 +70,13 @@ theorem not_isTorsion_iff : ¬IsTorsion G ↔ ∃ g : G, ¬IsOfFinOrder g := by
   rw [is_torsion, not_forall]
 #align monoid.not_is_torsion_iff Monoid.not_isTorsion_iff
 #align add_monoid.not_is_torsion_iff AddMonoid.not_isTorsion_iff
+-/
 
 end Monoid
 
 open Monoid
 
+#print IsTorsion.group /-
 /-- Torsion monoids are really groups. -/
 @[to_additive "Torsion additive monoids are really additive groups"]
 noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
@@ -83,18 +88,31 @@ noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
       exact orderOf_pos' (tG g) }
 #align is_torsion.group IsTorsion.group
 #align is_torsion.add_group IsTorsion.addGroup
+-/
 
 section Group
 
 variable [Group G] {N : Subgroup G} [Group H]
 
+/- warning: is_torsion.subgroup -> IsTorsion.subgroup is a dubious translation:
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+  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G], (Monoid.IsTorsion.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) -> (forall (H : Subgroup.{u1} G _inst_1), Monoid.IsTorsion.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (Subgroup.toGroup.{u1} G _inst_1 H))))
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+Case conversion may be inaccurate. Consider using '#align is_torsion.subgroup IsTorsion.subgroupₓ'. -/
 /-- Subgroups of torsion groups are torsion groups. -/
 @[to_additive "Subgroups of additive torsion groups are additive torsion groups."]
 theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H := fun h =>
   (isOfFinOrder_iff_coe H.toSubmonoid h).mpr <| tG h
 #align is_torsion.subgroup IsTorsion.subgroup
-#align is_torsion.add_subgroup IsTorsion.add_subgroup
-
+#align is_torsion.add_subgroup IsTorsion.addSubgroup
+
+/- warning: is_torsion.of_surjective -> IsTorsion.of_surjective is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_torsion.of_surjective IsTorsion.of_surjectiveₓ'. -/
 /-- The image of a surjective torsion group homomorphism is torsion. -/
 @[to_additive AddIsTorsion.of_surjective
       "The image of a surjective additive torsion group homomorphism is torsion."]
@@ -106,6 +124,12 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
 #align is_torsion.of_surjective IsTorsion.of_surjective
 #align add_is_torsion.of_surjective AddIsTorsion.of_surjective
 
+/- warning: is_torsion.extension_closed -> IsTorsion.extension_closed is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_torsion.extension_closed IsTorsion.extension_closedₓ'. -/
 /-- Torsion groups are closed under extensions. -/
 @[to_additive AddIsTorsion.extension_closed "Additive torsion groups are closed under extensions."]
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
@@ -122,6 +146,12 @@ theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsi
 #align is_torsion.extension_closed IsTorsion.extension_closed
 #align add_is_torsion.extension_closed AddIsTorsion.extension_closed
 
+/- warning: is_torsion.quotient_iff -> IsTorsion.quotient_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_torsion.quotient_iff IsTorsion.quotient_iffₓ'. -/
 /-- The image of a quotient is torsion iff the group is torsion. -/
 @[to_additive AddIsTorsion.quotient_iff
       "The image of a quotient is additively torsion iff the group is torsion."]
@@ -131,6 +161,7 @@ theorem IsTorsion.quotient_iff {f : G →* H} (hf : Function.Surjective f) (hN :
 #align is_torsion.quotient_iff IsTorsion.quotient_iff
 #align add_is_torsion.quotient_iff AddIsTorsion.quotient_iff
 
+#print ExponentExists.isTorsion /-
 /-- If a group exponent exists, the group is torsion. -/
 @[to_additive ExponentExists.is_add_torsion
       "If a group exponent exists, the group is additively torsion."]
@@ -140,7 +171,9 @@ theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g =
   exact (isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, npos, hn g⟩
 #align exponent_exists.is_torsion ExponentExists.isTorsion
 #align exponent_exists.is_add_torsion ExponentExists.is_add_torsion
+-/
 
+#print IsTorsion.exponentExists /-
 /-- The group exponent exists for any bounded torsion group. -/
 @[to_additive IsAddTorsion.exponentExists
       "The group exponent exists for any bounded additive torsion group."]
@@ -150,13 +183,16 @@ theorem IsTorsion.exponentExists (tG : IsTorsion G)
     (exponent_ne_zero_iff_range_orderOf_finite fun g => orderOf_pos' (tG g)).mpr bounded
 #align is_torsion.exponent_exists IsTorsion.exponentExists
 #align is_add_torsion.exponent_exists IsAddTorsion.exponentExists
+-/
 
+#print isTorsion_of_finite /-
 /-- Finite groups are torsion groups. -/
 @[to_additive is_add_torsion_of_finite "Finite additive groups are additive torsion groups."]
 theorem isTorsion_of_finite [Finite G] : IsTorsion G :=
   ExponentExists.isTorsion <| exponentExists_iff_ne_zero.mpr exponent_ne_zero_of_finite
 #align is_torsion_of_finite isTorsion_of_finite
 #align is_add_torsion_of_finite is_add_torsion_of_finite
+-/
 
 end Group
 
@@ -167,6 +203,12 @@ variable (R M : Type _) [AddCommMonoid M]
 
 namespace AddMonoid
 
+/- warning: add_monoid.is_torsion.module_of_torsion -> AddMonoid.IsTorsion.module_of_torsion is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : Semiring.{u1} R] [_inst_3 : Module.{u1, u2} R M _inst_2 _inst_1], (AddMonoid.IsTorsion.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_2))))) -> (AddMonoid.IsTorsion.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))
+but is expected to have type
+  forall (R : Type.{u2}) (M : Type.{u1}) [_inst_1 : AddCommMonoid.{u1} M] [_inst_2 : Semiring.{u2} R] [_inst_3 : Module.{u2, u1} R M _inst_2 _inst_1], (AddMonoid.IsTorsion.{u2} R (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))))) -> (AddMonoid.IsTorsion.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))
+Case conversion may be inaccurate. Consider using '#align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsionₓ'. -/
 /-- A module whose scalars are additively torsion is additively torsion. -/
 theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R) : IsTorsion M :=
   fun f =>
@@ -176,6 +218,12 @@ theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R)
     exact ⟨n, npos, by simp only [nsmul_eq_smul_cast R _ f, ← nsmul_one, hn, zero_smul]⟩
 #align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsion
 
+/- warning: add_monoid.is_torsion.module_of_finite -> AddMonoid.IsTorsion.module_of_finite is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : Ring.{u1} R] [_inst_3 : Finite.{succ u1} R] [_inst_4 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) _inst_1], AddMonoid.IsTorsion.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)
+but is expected to have type
+  forall (R : Type.{u2}) (M : Type.{u1}) [_inst_1 : AddCommMonoid.{u1} M] [_inst_2 : Ring.{u2} R] [_inst_3 : Finite.{succ u2} R] [_inst_4 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_2) _inst_1], AddMonoid.IsTorsion.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)
+Case conversion may be inaccurate. Consider using '#align add_monoid.is_torsion.module_of_finite AddMonoid.IsTorsion.module_of_finiteₓ'. -/
 /-- A module with a finite ring of scalars is additively torsion. -/
 theorem IsTorsion.module_of_finite [Ring R] [Finite R] [Module R M] : IsTorsion M :=
   (is_add_torsion_of_finite : IsTorsion R).module_of_torsion _ _
@@ -191,6 +239,7 @@ variable (G) [CommMonoid G]
 
 namespace CommMonoid
 
+#print CommMonoid.torsion /-
 /-- The torsion submonoid of a commutative monoid.
 
 (Note that by `monoid.is_torsion.group` torsion monoids are truthfully groups.)
@@ -202,9 +251,16 @@ def torsion : Submonoid G where
   mul_mem' _ _ hx hy := hx.mul hy
 #align comm_monoid.torsion CommMonoid.torsion
 #align add_comm_monoid.add_torsion AddCommMonoid.addTorsion
+-/
 
 variable {G}
 
+/- warning: comm_monoid.torsion.is_torsion -> CommMonoid.torsion.isTorsion is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], Monoid.IsTorsion.{u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) (CommMonoid.torsion.{u1} G _inst_1)) (Submonoid.toMonoid.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1) (CommMonoid.torsion.{u1} G _inst_1))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], Monoid.IsTorsion.{u1} (Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.instSetLikeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) x (CommMonoid.torsion.{u1} G _inst_1))) (Submonoid.toMonoid.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1) (CommMonoid.torsion.{u1} G _inst_1))
+Case conversion may be inaccurate. Consider using '#align comm_monoid.torsion.is_torsion CommMonoid.torsion.isTorsionₓ'. -/
 /-- Torsion submonoids are torsion. -/
 @[to_additive "Additive torsion submonoids are additively torsion."]
 theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =>
@@ -213,12 +269,13 @@ theorem torsion.isTorsion : IsTorsion <| torsion G := fun ⟨_, n, npos, hn⟩ =
       rw [mul_left_iterate, _root_.mul_one, [anonymous], Subtype.coe_mk, Submonoid.coe_one,
         (isPeriodicPt_mul_iff_pow_eq_one _).mp hn]⟩
 #align comm_monoid.torsion.is_torsion CommMonoid.torsion.isTorsion
-#align add_comm_monoid.add_torsion.is_torsion AddCommMonoid.addTorsion.is_torsion
+#align add_comm_monoid.add_torsion.is_torsion AddCommMonoid.addTorsion.isTorsion
 
 variable (G) (p : ℕ) [hp : Fact p.Prime]
 
 include hp
 
+#print CommMonoid.primaryComponent /-
 /-- The `p`-primary component is the submonoid of elements with order prime-power of `p`. -/
 @[to_additive
       "The `p`-primary component is the submonoid of elements with additive order prime-power of `p`.",
@@ -238,16 +295,29 @@ def primaryComponent : Submonoid G
             one_pow]⟩
 #align comm_monoid.primary_component CommMonoid.primaryComponent
 #align add_comm_monoid.primary_component AddCommMonoid.primaryComponent
+-/
 
 variable {G} {p}
 
+/- warning: comm_monoid.primary_component.exists_order_of_eq_prime_pow -> CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)] (g : coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp)), Exists.{1} Nat (fun (n : Nat) => Eq.{1} Nat (orderOf.{u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp)) (Submonoid.toMonoid.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp)) g) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.Pow.{0} Nat Nat.monoid)) p n))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)] (g : Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.instSetLikeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) x (CommMonoid.primaryComponent.{u1} G _inst_1 p hp))), Exists.{1} Nat (fun (n : Nat) => Eq.{1} Nat (orderOf.{u1} (Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.instSetLikeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) x (CommMonoid.primaryComponent.{u1} G _inst_1 p hp))) (Submonoid.toMonoid.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp)) g) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat instPowNat) p n))
+Case conversion may be inaccurate. Consider using '#align comm_monoid.primary_component.exists_order_of_eq_prime_pow CommMonoid.primaryComponent.exists_orderOf_eq_prime_powₓ'. -/
 /-- Elements of the `p`-primary component have order `p^n` for some `n`. -/
 @[to_additive "Elements of the `p`-primary component have additive order `p^n` for some `n`"]
 theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComponent G p) :
     ∃ n : ℕ, orderOf g = p ^ n := by simpa [primary_component] using g.property
 #align comm_monoid.primary_component.exists_order_of_eq_prime_pow CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow
-#align add_comm_monoid.primary_component.exists_order_of_eq_prime_nsmul AddCommMonoid.primaryComponent.exists_order_of_eq_prime_nsmul
-
+#align add_comm_monoid.primary_component.exists_order_of_eq_prime_nsmul AddCommMonoid.primaryComponent.exists_orderOf_eq_prime_nsmul
+
+/- warning: comm_monoid.primary_component.disjoint -> CommMonoid.primaryComponent.disjoint is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)] {p' : Nat} [hp' : Fact (Nat.Prime p')], (Ne.{1} Nat p p') -> (Disjoint.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.partialOrder.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) (BoundedOrder.toOrderBot.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Preorder.toHasLe.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (PartialOrder.toPreorder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.partialOrder.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))))) (CompleteLattice.toBoundedOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.completeLattice.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp) (CommMonoid.primaryComponent.{u1} G _inst_1 p' hp'))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)] {p' : Nat} [hp' : Fact (Nat.Prime p')], (Ne.{1} Nat p p') -> (Disjoint.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.instCompleteLatticeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))) (BoundedOrder.toOrderBot.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Preorder.toLE.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (PartialOrder.toPreorder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.instCompleteLatticeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))))) (CompleteLattice.toBoundedOrder.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.instCompleteLatticeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))) (CommMonoid.primaryComponent.{u1} G _inst_1 p hp) (CommMonoid.primaryComponent.{u1} G _inst_1 p' hp'))
+Case conversion may be inaccurate. Consider using '#align comm_monoid.primary_component.disjoint CommMonoid.primaryComponent.disjointₓ'. -/
 /-- The `p`- and `q`-primary components are disjoint for `p ≠ q`. -/
 @[to_additive "The `p`- and `q`-primary components are disjoint for `p ≠ q`."]
 theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠ p') :
@@ -269,12 +339,24 @@ namespace Monoid.IsTorsion
 
 variable {G}
 
+/- warning: monoid.is_torsion.torsion_eq_top -> Monoid.IsTorsion.torsion_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], (Monoid.IsTorsion.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) -> (Eq.{succ u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (CommMonoid.torsion.{u1} G _inst_1) (Top.top.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.hasTop.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], (Monoid.IsTorsion.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) -> (Eq.{succ u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (CommMonoid.torsion.{u1} G _inst_1) (Top.top.{u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (Submonoid.instTopSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_eq_top Monoid.IsTorsion.torsion_eq_topₓ'. -/
 /-- The torsion submonoid of a torsion monoid is `⊤`. -/
 @[simp, to_additive "The additive torsion submonoid of an additive torsion monoid is `⊤`."]
 theorem torsion_eq_top (tG : IsTorsion G) : torsion G = ⊤ := by ext <;> tauto
 #align monoid.is_torsion.torsion_eq_top Monoid.IsTorsion.torsion_eq_top
 #align add_monoid.is_torsion.torsion_eq_top AddMonoid.IsTorsion.torsion_eq_top
 
+/- warning: monoid.is_torsion.torsion_mul_equiv -> Monoid.IsTorsion.torsionMulEquiv is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], (Monoid.IsTorsion.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) -> (MulEquiv.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.setLike.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) (CommMonoid.torsion.{u1} G _inst_1)) G (Submonoid.mul.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) (CommMonoid.torsion.{u1} G _inst_1)) (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommMonoid.{u1} G], (Monoid.IsTorsion.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) -> (MulEquiv.{u1, u1} (Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))) G (Submonoid.instSetLikeSubmonoid.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)))) x (CommMonoid.torsion.{u1} G _inst_1))) G (Submonoid.mul.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1)) (CommMonoid.torsion.{u1} G _inst_1)) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (CommMonoid.toMonoid.{u1} G _inst_1))))
+Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_mul_equiv Monoid.IsTorsion.torsionMulEquivₓ'. -/
 /-- A torsion monoid is isomorphic to its torsion submonoid. -/
 @[to_additive "An additive torsion monoid is isomorphic to its torsion submonoid."]
 def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
@@ -282,22 +364,40 @@ def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
 #align monoid.is_torsion.torsion_mul_equiv Monoid.IsTorsion.torsionMulEquiv
 #align add_monoid.is_torsion.torsion_add_equiv AddMonoid.IsTorsion.torsionAddEquiv
 
+/- warning: monoid.is_torsion.torsion_mul_equiv_apply -> Monoid.IsTorsion.torsionMulEquiv_apply is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_applyₓ'. -/
 @[to_additive]
 theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
     tG.torsionMulEquiv a = MulEquiv.submonoidCongr tG.torsion_eq_top a :=
   rfl
 #align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_apply
-#align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsion_add_equiv_apply
-
+#align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsionAddEquiv_apply
+
+/- warning: monoid.is_torsion.torsion_mul_equiv_symm_apply_coe -> Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align monoid.is_torsion.torsion_mul_equiv_symm_apply_coe Monoid.IsTorsion.torsionMulEquiv_symm_apply_coeₓ'. -/
 @[to_additive]
 theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
     tG.torsionMulEquiv.symm a = ⟨Submonoid.topEquiv.symm a, tG _⟩ :=
   rfl
 #align monoid.is_torsion.torsion_mul_equiv_symm_apply_coe Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe
-#align add_monoid.is_torsion.torsion_add_equiv_symm_apply_coe AddMonoid.IsTorsion.torsion_add_equiv_symm_apply_coe
+#align add_monoid.is_torsion.torsion_add_equiv_symm_apply_coe AddMonoid.IsTorsion.torsionAddEquiv_symm_apply_coe
 
 end Monoid.IsTorsion
 
+/- warning: torsion.of_torsion -> Torsion.ofTorsion is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align torsion.of_torsion Torsion.ofTorsionₓ'. -/
 /-- Torsion submonoids of a torsion submonoid are isomorphic to the submonoid. -/
 @[simp,
   to_additive AddCommMonoid.Torsion.ofTorsion
@@ -315,13 +415,16 @@ variable (G) [CommGroup G]
 
 namespace CommGroup
 
+#print CommGroup.torsion /-
 /-- The torsion subgroup of an abelian group. -/
 @[to_additive "The torsion subgroup of an additive abelian group."]
 def torsion : Subgroup G :=
   { CommMonoid.torsion G with inv_mem' := fun x => IsOfFinOrder.inv }
 #align comm_group.torsion CommGroup.torsion
 #align add_comm_group.torsion AddCommGroup.torsion
+-/
 
+#print CommGroup.torsion_eq_torsion_submonoid /-
 /-- The torsion submonoid of an abelian group equals the torsion subgroup as a submonoid. -/
 @[to_additive add_torsion_eq_add_torsion_submonoid
       "The additive torsion submonoid of an abelian group equals the torsion subgroup as a submonoid."]
@@ -329,11 +432,13 @@ theorem torsion_eq_torsion_submonoid : CommMonoid.torsion G = (torsion G).toSubm
   rfl
 #align comm_group.torsion_eq_torsion_submonoid CommGroup.torsion_eq_torsion_submonoid
 #align add_comm_group.add_torsion_eq_add_torsion_submonoid AddCommGroup.add_torsion_eq_add_torsion_submonoid
+-/
 
 variable (p : ℕ) [hp : Fact p.Prime]
 
 include hp
 
+#print CommGroup.primaryComponent /-
 /-- The `p`-primary component is the subgroup of elements with order prime-power of `p`. -/
 @[to_additive
       "The `p`-primary component is the subgroup of elements with additive order prime-power of `p`.",
@@ -343,9 +448,16 @@ def primaryComponent : Subgroup G :=
     inv_mem' := fun g ⟨n, hn⟩ => ⟨n, (orderOf_inv g).trans hn⟩ }
 #align comm_group.primary_component CommGroup.primaryComponent
 #align add_comm_group.primary_component AddCommGroup.primaryComponent
+-/
 
 variable {G} {p}
 
+/- warning: comm_group.primary_component.is_p_group -> CommGroup.primaryComponent.isPGroup is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : CommGroup.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)], IsPGroup.{u1} p (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1)) G (Subgroup.setLike.{u1} G (CommGroup.toGroup.{u1} G _inst_1))) (CommGroup.primaryComponent.{u1} G _inst_1 p hp)) (Subgroup.toGroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1) (CommGroup.primaryComponent.{u1} G _inst_1 p hp))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : CommGroup.{u1} G] {p : Nat} [hp : Fact (Nat.Prime p)], IsPGroup.{u1} p (Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Subgroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1)) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1)) G (Subgroup.instSetLikeSubgroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1))) x (CommGroup.primaryComponent.{u1} G _inst_1 p hp))) (Subgroup.toGroup.{u1} G (CommGroup.toGroup.{u1} G _inst_1) (CommGroup.primaryComponent.{u1} G _inst_1 p hp))
+Case conversion may be inaccurate. Consider using '#align comm_group.primary_component.is_p_group CommGroup.primaryComponent.isPGroupₓ'. -/
 /-- The `p`-primary component is a `p` group. -/
 theorem primaryComponent.isPGroup : IsPGroup p <| primaryComponent G p := fun g =>
   (propext exists_orderOf_eq_prime_pow_iff.symm).mpr
@@ -360,13 +472,21 @@ namespace Monoid
 
 variable (G) [Monoid G]
 
+#print Monoid.IsTorsionFree /-
 /-- A predicate on a monoid saying that only 1 is of finite order. -/
 @[to_additive "A predicate on an additive monoid saying that only 0 is of finite order."]
 def IsTorsionFree :=
   ∀ g : G, g ≠ 1 → ¬IsOfFinOrder g
 #align monoid.is_torsion_free Monoid.IsTorsionFree
 #align add_monoid.is_torsion_free AddMonoid.IsTorsionFree
+-/
 
+/- warning: monoid.not_is_torsion_free_iff -> Monoid.not_isTorsionFree_iff is a dubious translation:
+lean 3 declaration is
+  forall (G : Type.{u1}) [_inst_1 : Monoid.{u1} G], Iff (Not (Monoid.IsTorsionFree.{u1} G _inst_1)) (Exists.{succ u1} G (fun (g : G) => And (Ne.{succ u1} G g (OfNat.ofNat.{u1} G 1 (OfNat.mk.{u1} G 1 (One.one.{u1} G (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G _inst_1)))))) (IsOfFinOrder.{u1} G _inst_1 g)))
+but is expected to have type
+  forall (G : Type.{u1}) [_inst_1 : Monoid.{u1} G], Iff (Not (Monoid.IsTorsionFree.{u1} G _inst_1)) (Exists.{succ u1} G (fun (g : G) => And (Ne.{succ u1} G g (OfNat.ofNat.{u1} G 1 (One.toOfNat1.{u1} G (Monoid.toOne.{u1} G _inst_1)))) (IsOfFinOrder.{u1} G _inst_1 g)))
+Case conversion may be inaccurate. Consider using '#align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iffₓ'. -/
 /-- A nontrivial monoid is not torsion-free if any nontrivial element has finite order. -/
 @[simp,
   to_additive "An additive monoid is not torsion free if any nontrivial element has finite order."]
@@ -383,6 +503,7 @@ open Monoid
 
 variable [Group G]
 
+#print IsTorsion.not_torsion_free /-
 /-- A nontrivial torsion group is not torsion-free. -/
 @[to_additive AddMonoid.IsTorsion.not_torsion_free
       "A nontrivial additive torsion group is not torsion-free."]
@@ -393,7 +514,9 @@ theorem IsTorsion.not_torsion_free [hN : Nontrivial G] : IsTorsion G → ¬IsTor
     exact ⟨x, hx, tG x⟩
 #align is_torsion.not_torsion_free IsTorsion.not_torsion_free
 #align add_monoid.is_torsion.not_torsion_free AddMonoid.IsTorsion.not_torsion_free
+-/
 
+#print IsTorsionFree.not_torsion /-
 /-- A nontrivial torsion-free group is not torsion. -/
 @[to_additive AddMonoid.IsTorsionFree.not_torsion
       "A nontrivial torsion-free additive group is not torsion."]
@@ -404,7 +527,14 @@ theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬Is
     exact ⟨x, (tfG x) hx⟩
 #align is_torsion_free.not_torsion IsTorsionFree.not_torsion
 #align add_monoid.is_torsion_free.not_torsion AddMonoid.IsTorsionFree.not_torsion
+-/
 
+/- warning: is_torsion_free.subgroup -> IsTorsionFree.subgroup is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G], (Monoid.IsTorsionFree.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) -> (forall (H : Subgroup.{u1} G _inst_1), Monoid.IsTorsionFree.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} G _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)) H) (Subgroup.toGroup.{u1} G _inst_1 H))))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G], (Monoid.IsTorsionFree.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) -> (forall (H : Subgroup.{u1} G _inst_1), Monoid.IsTorsionFree.{u1} (Subtype.{succ u1} G (fun (x : G) => Membership.mem.{u1, u1} G (Subgroup.{u1} G _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.instSetLikeSubgroup.{u1} G _inst_1)) x H)) (Submonoid.toMonoid.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) (Subgroup.toSubmonoid.{u1} G _inst_1 H)))
+Case conversion may be inaccurate. Consider using '#align is_torsion_free.subgroup IsTorsionFree.subgroupₓ'. -/
 /-- Subgroups of torsion-free groups are torsion-free. -/
 @[to_additive "Subgroups of additive torsion-free groups are additively torsion-free."]
 theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsionFree H :=
@@ -412,8 +542,14 @@ theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsi
   (isOfFinOrder_iff_coe H.toSubmonoid h).Not.mpr <|
     tG h <| by norm_cast <;> simp [hne, not_false_iff]
 #align is_torsion_free.subgroup IsTorsionFree.subgroup
-#align is_torsion_free.add_subgroup IsTorsionFree.add_subgroup
-
+#align is_torsion_free.add_subgroup IsTorsionFree.addSubgroup
+
+/- warning: is_torsion_free.prod -> IsTorsionFree.prod is a dubious translation:
+lean 3 declaration is
+  forall {η : Type.{u1}} {Gs : η -> Type.{u2}} [_inst_2 : forall (i : η), Group.{u2} (Gs i)], (forall (i : η), Monoid.IsTorsionFree.{u2} (Gs i) (DivInvMonoid.toMonoid.{u2} (Gs i) (Group.toDivInvMonoid.{u2} (Gs i) (_inst_2 i)))) -> (Monoid.IsTorsionFree.{max u1 u2} (forall (i : η), Gs i) (Pi.monoid.{u1, u2} η (fun (i : η) => Gs i) (fun (i : η) => DivInvMonoid.toMonoid.{u2} (Gs i) (Group.toDivInvMonoid.{u2} (Gs i) (_inst_2 i)))))
+but is expected to have type
+  forall {η : Type.{u2}} {Gs : η -> Type.{u1}} [_inst_2 : forall (i : η), Group.{u1} (Gs i)], (forall (i : η), Monoid.IsTorsionFree.{u1} (Gs i) (DivInvMonoid.toMonoid.{u1} (Gs i) (Group.toDivInvMonoid.{u1} (Gs i) (_inst_2 i)))) -> (Monoid.IsTorsionFree.{max u2 u1} (forall (i : η), Gs i) (Pi.monoid.{u2, u1} η (fun (i : η) => Gs i) (fun (i : η) => DivInvMonoid.toMonoid.{u1} (Gs i) (Group.toDivInvMonoid.{u1} (Gs i) (_inst_2 i)))))
+Case conversion may be inaccurate. Consider using '#align is_torsion_free.prod IsTorsionFree.prodₓ'. -/
 /-- Direct products of torsion free groups are torsion free. -/
 @[to_additive AddMonoid.IsTorsionFree.prod
       "Direct products of additive torsion free groups are torsion free."]
@@ -434,6 +570,7 @@ open CommGroup (torsion)
 
 variable (G) [CommGroup G]
 
+#print IsTorsionFree.quotient_torsion /-
 /-- Quotienting a group by its torsion subgroup yields a torsion free group. -/
 @[to_additive AddIsTorsionFree.quotient_torsion
       "Quotienting a group by its additive torsion subgroup yields an additive torsion free group."]
@@ -447,6 +584,7 @@ theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun
         ((isOfFinOrder_iff_pow_eq_one _).mpr ⟨m * n, mul_pos mpos npos, (pow_mul g m n).symm ▸ hn⟩)
 #align is_torsion_free.quotient_torsion IsTorsionFree.quotient_torsion
 #align add_is_torsion_free.quotient_torsion AddIsTorsionFree.quotient_torsion
+-/
 
 end CommGroup
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -114,7 +114,7 @@ theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsi
     by
     obtain ⟨ngn, ngnpos, hngn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tH <| f g)
     have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
-    lift g ^ ngn to N using hN.symm ▸ hmem
+    lift g ^ ngn to N using hN.symm ▸ hmem with gn
     obtain ⟨nn, nnpos, hnn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tN gn)
     exact
       ⟨ngn * nn, mul_pos ngnpos nnpos, by

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -114,7 +114,7 @@ theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsi
     by
     obtain ⟨ngn, ngnpos, hngn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tH <| f g)
     have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
-    lift g ^ ngn to N using hN.symm ▸ hmem with gn
+    lift g ^ ngn to N using hN.symm ▸ hmem
     obtain ⟨nn, nnpos, hnn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tN gn)
     exact
       ⟨ngn * nn, mul_pos ngnpos nnpos, by

mathlib commit https://github.com/leanprover-community/mathlib/commit/1f4705ccdfe1e557fc54a0ce081a05e33d2e6240

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Julian Berman
 
 ! This file was ported from Lean 3 source module group_theory.torsion
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 1f4705ccdfe1e557fc54a0ce081a05e33d2e6240
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -276,12 +276,26 @@ theorem torsion_eq_top (tG : IsTorsion G) : torsion G = ⊤ := by ext <;> tauto
 #align add_monoid.is_torsion.torsion_eq_top AddMonoid.IsTorsion.torsion_eq_top
 
 /-- A torsion monoid is isomorphic to its torsion submonoid. -/
-@[to_additive "An additive torsion monoid is isomorphic to its torsion submonoid.", simps]
+@[to_additive "An additive torsion monoid is isomorphic to its torsion submonoid."]
 def torsionMulEquiv (tG : IsTorsion G) : torsion G ≃* G :=
   (MulEquiv.submonoidCongr tG.torsion_eq_top).trans Submonoid.topEquiv
 #align monoid.is_torsion.torsion_mul_equiv Monoid.IsTorsion.torsionMulEquiv
 #align add_monoid.is_torsion.torsion_add_equiv AddMonoid.IsTorsion.torsionAddEquiv
 
+@[to_additive]
+theorem torsionMulEquiv_apply (tG : IsTorsion G) (a : torsion G) :
+    tG.torsionMulEquiv a = MulEquiv.submonoidCongr tG.torsion_eq_top a :=
+  rfl
+#align monoid.is_torsion.torsion_mul_equiv_apply Monoid.IsTorsion.torsionMulEquiv_apply
+#align add_monoid.is_torsion.torsion_add_equiv_apply AddMonoid.IsTorsion.torsion_add_equiv_apply
+
+@[to_additive]
+theorem torsionMulEquiv_symm_apply_coe (tG : IsTorsion G) (a : G) :
+    tG.torsionMulEquiv.symm a = ⟨Submonoid.topEquiv.symm a, tG _⟩ :=
+  rfl
+#align monoid.is_torsion.torsion_mul_equiv_symm_apply_coe Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe
+#align add_monoid.is_torsion.torsion_add_equiv_symm_apply_coe AddMonoid.IsTorsion.torsion_add_equiv_symm_apply_coe
+
 end Monoid.IsTorsion
 
 /-- Torsion submonoids of a torsion submonoid are isomorphic to the submonoid. -/

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

@@ -170,9 +170,9 @@ namespace AddMonoid
 /-- A module whose scalars are additively torsion is additively torsion. -/
 theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R) : IsTorsion M :=
   fun f =>
-  (is_of_fin_add_order_iff_nsmul_eq_zero _).mpr <|
+  (isOfFinAddOrder_iff_nsmul_eq_zero _).mpr <|
     by
-    obtain ⟨n, npos, hn⟩ := (is_of_fin_add_order_iff_nsmul_eq_zero _).mp (tR 1)
+    obtain ⟨n, npos, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero _).mp (tR 1)
     exact ⟨n, npos, by simp only [nsmul_eq_smul_cast R _ f, ← nsmul_one, hn, zero_smul]⟩
 #align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsion
 

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

A mix of various changes; generated with a script and manually tweaked.

@@ -28,7 +28,7 @@ This file defines torsion groups, i.e. groups where all elements have finite ord
 ## Implementation
 
 All torsion monoids are really groups (which is proven here as `Monoid.IsTorsion.group`), but since
-the definition can be stated on monoids it is implemented on `monoid` to match other declarations in
+the definition can be stated on monoids it is implemented on `Monoid` to match other declarations in
 the group theory library.
 
 ## Tags

@@ -365,7 +365,7 @@ variable {G}
 @[to_additive (attr := simp) "An additive monoid is not torsion free if any
   nontrivial element has finite order."]
 theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsOfFinOrder g := by
-  simp_rw [IsTorsionFree, Ne.def, not_forall, Classical.not_not, exists_prop]
+  simp_rw [IsTorsionFree, Ne, not_forall, Classical.not_not, exists_prop]
 #align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iff
 #align add_monoid.not_is_torsion_free_iff AddMonoid.not_isTorsionFree_iff
 

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

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

where the latter is more natural

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

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

but it seems to no longer apply.

Remarks on the PR :

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

@@ -75,7 +75,7 @@ noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
   { ‹Monoid G› with
     inv := fun g => g ^ (orderOf g - 1)
     mul_left_inv := fun g => by
-      erw [← pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]
+      erw [← pow_succ, tsub_add_cancel_of_le, pow_orderOf_eq_one]
       exact (tG g).orderOf_pos }
 #align is_torsion.group IsTorsion.group
 #align is_torsion.add_group IsTorsion.addGroup

and some subgroup results.

From PFR

@@ -449,3 +449,21 @@ theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun
 #align add_is_torsion_free.quotient_torsion AddIsTorsionFree.quotient_torsion
 
 end CommGroup
+
+lemma isTorsionFree_iff_noZeroSMulDivisors_nat {M : Type*} [AddMonoid M] :
+    AddMonoid.IsTorsionFree M ↔ NoZeroSMulDivisors ℕ M := by
+  simp_rw [AddMonoid.IsTorsionFree, isOfFinAddOrder_iff_nsmul_eq_zero, not_exists, not_and,
+    pos_iff_ne_zero, noZeroSMulDivisors_iff, forall_swap (β := ℕ)]
+  exact forall₂_congr fun _ _ ↦ by tauto
+
+lemma isTorsionFree_iff_noZeroSMulDivisors_int [AddGroup G] :
+    AddMonoid.IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G := by
+  simp_rw [AddMonoid.IsTorsionFree, isOfFinAddOrder_iff_zsmul_eq_zero, not_exists, not_and,
+    noZeroSMulDivisors_iff, forall_swap (β := ℤ)]
+  exact forall₂_congr fun _ _ ↦ by tauto
+
+@[deprecated] -- 2024-02-29
+alias AddMonoid.IsTorsionFree_iff_noZeroSMulDivisors := isTorsionFree_iff_noZeroSMulDivisors_int
+
+lemma IsTorsionFree.of_noZeroSMulDivisors {M : Type*} [AddMonoid M] [NoZeroSMulDivisors ℕ M] :
+    AddMonoid.IsTorsionFree M := isTorsionFree_iff_noZeroSMulDivisors_nat.2 ‹_›

Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>

@@ -138,7 +138,7 @@ theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g =
       "The group exponent exists for any bounded additive torsion group."]
 theorem IsTorsion.exponentExists (tG : IsTorsion G)
     (bounded : (Set.range fun g : G => orderOf g).Finite) : ExponentExists G :=
-  exponentExists_iff_ne_zero.mpr <|
+  exponent_ne_zero.mp <|
     (exponent_ne_zero_iff_range_orderOf_finite fun g => (tG g).orderOf_pos).mpr bounded
 #align is_torsion.exponent_exists IsTorsion.exponentExists
 #align is_add_torsion.exponent_exists IsAddTorsion.exponentExists
@@ -146,7 +146,7 @@ theorem IsTorsion.exponentExists (tG : IsTorsion G)
 /-- Finite groups are torsion groups. -/
 @[to_additive is_add_torsion_of_finite "Finite additive groups are additive torsion groups."]
 theorem isTorsion_of_finite [Finite G] : IsTorsion G :=
-  ExponentExists.isTorsion <| exponentExists_iff_ne_zero.mpr exponent_ne_zero_of_finite
+  ExponentExists.isTorsion .of_finite
 #align is_torsion_of_finite isTorsion_of_finite
 #align is_add_torsion_of_finite is_add_torsion_of_finite
 

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

@@ -373,6 +373,12 @@ theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsO
 lemma isTorsionFree_of_subsingleton [Subsingleton G] : IsTorsionFree G :=
   fun _a ha _ => ha <| Subsingleton.elim _ _
 
+@[to_additive]
+lemma isTorsionFree_iff_torsion_eq_bot {G} [CommGroup G] :
+    IsTorsionFree G ↔ CommGroup.torsion G = ⊥ := by
+  rw [IsTorsionFree, eq_bot_iff, SetLike.le_def]
+  simp [not_imp_not, CommGroup.mem_torsion]
+
 end Monoid
 
 section Group

@@ -106,12 +106,12 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
 @[to_additive AddIsTorsion.extension_closed "Additive torsion groups are closed under extensions."]
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
     (tN : IsTorsion N) : IsTorsion G := fun g => by
-    obtain ⟨ngn, ngnpos, hngn⟩ := (tH <| f g).exists_pow_eq_one
-    have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
-    lift g ^ ngn to N using hN.symm ▸ hmem with gn h
-    obtain ⟨nn, nnpos, hnn⟩ := (tN gn).exists_pow_eq_one
-    exact isOfFinOrder_iff_pow_eq_one.mpr <| ⟨ngn * nn, mul_pos ngnpos nnpos, by
-        rw [pow_mul, ← h, ← Subgroup.coe_pow, hnn, Subgroup.coe_one]⟩
+  obtain ⟨ngn, ngnpos, hngn⟩ := (tH <| f g).exists_pow_eq_one
+  have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
+  lift g ^ ngn to N using hN.symm ▸ hmem with gn h
+  obtain ⟨nn, nnpos, hnn⟩ := (tN gn).exists_pow_eq_one
+  exact isOfFinOrder_iff_pow_eq_one.mpr <| ⟨ngn * nn, mul_pos ngnpos nnpos, by
+      rw [pow_mul, ← h, ← Subgroup.coe_pow, hnn, Subgroup.coe_one]⟩
 #align is_torsion.extension_closed IsTorsion.extension_closed
 #align add_is_torsion.extension_closed AddIsTorsion.extension_closed
 

Many lemmas in GroupTheory.OrderOfElement were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation.

Renames

• Function.eq_of_lt_minimalPeriod_of_iterate_eq → Function.iterate_injOn_Iio_minimalPeriod
• Function.eq_iff_lt_minimalPeriod_of_iterate_eq → Function.iterate_eq_iterate_iff_of_lt_minimalPeriod
• isOfFinOrder_iff_coe → Submonoid.isOfFinOrder_coe
• orderOf_pos' → IsOfFinOrder.orderOf_pos
• pow_eq_mod_orderOf → pow_mod_orderOf (and turned around)
• pow_injective_of_lt_orderOf → pow_injOn_Iio_orderOf
• mem_powers_iff_mem_range_order_of' → IsOfFinOrder.mem_powers_iff_mem_range_orderOf
• orderOf_pow'' → IsOfFinOrder.orderOf_pow
• orderOf_pow_coprime → Nat.Coprime.orderOf_pow
• zpow_eq_mod_orderOf → zpow_mod_orderOf (and turned around)
• exists_pow_eq_one → isOfFinOrder_of_finite
• pow_apply_eq_pow_mod_orderOf_cycleOf_apply → pow_mod_orderOf_cycleOf_apply

New lemmas

• IsOfFinOrder.powers_eq_image_range_orderOf
• IsOfFinOrder.natCard_powers_le_orderOf
• IsOfFinOrder.finite_powers
• finite_powers
• infinite_powers
• Nat.card_submonoidPowers
• IsOfFinOrder.mem_powers_iff_mem_zpowers
• IsOfFinOrder.powers_eq_zpowers
• IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf
• IsOfFinOrder.exists_pow_eq_one

Other changes

• Move decidableMemPowers/fintypePowers to GroupTheory.Submonoid.Membership and decidableMemZpowers/fintypeZpowers to GroupTheory.Subgroup.ZPowers.
• finEquivPowers, finEquivZpowers, powersEquivPowers and zpowersEquivZpowers now assume IsOfFinTorsion x instead of Finite G.
• isOfFinOrder_iff_pow_eq_one now takes one less explicit argument.
• Delete Equiv.Perm.IsCycle.exists_pow_eq_one since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use isOfFinOrder_of_finite instead.

@@ -76,7 +76,7 @@ noncomputable def IsTorsion.group [Monoid G] (tG : IsTorsion G) : Group G :=
     inv := fun g => g ^ (orderOf g - 1)
     mul_left_inv := fun g => by
       erw [← pow_succ', tsub_add_cancel_of_le, pow_orderOf_eq_one]
-      exact orderOf_pos' (tG g) }
+      exact (tG g).orderOf_pos }
 #align is_torsion.group IsTorsion.group
 #align is_torsion.add_group IsTorsion.addGroup
 
@@ -87,7 +87,7 @@ variable [Group G] {N : Subgroup G} [Group H]
 /-- Subgroups of torsion groups are torsion groups. -/
 @[to_additive "Subgroups of additive torsion groups are additive torsion groups."]
 theorem IsTorsion.subgroup (tG : IsTorsion G) (H : Subgroup G) : IsTorsion H := fun h =>
-  (isOfFinOrder_iff_coe H.toSubmonoid h).mpr <| tG h
+  Submonoid.isOfFinOrder_coe.1 <| tG h
 #align is_torsion.subgroup IsTorsion.subgroup
 #align is_torsion.add_subgroup IsTorsion.addSubgroup
 
@@ -105,14 +105,12 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
 /-- Torsion groups are closed under extensions. -/
 @[to_additive AddIsTorsion.extension_closed "Additive torsion groups are closed under extensions."]
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
-    (tN : IsTorsion N) : IsTorsion G := fun g =>
-  (isOfFinOrder_iff_pow_eq_one _).mpr <| by
-    obtain ⟨ngn, ngnpos, hngn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tH <| f g)
+    (tN : IsTorsion N) : IsTorsion G := fun g => by
+    obtain ⟨ngn, ngnpos, hngn⟩ := (tH <| f g).exists_pow_eq_one
     have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
     lift g ^ ngn to N using hN.symm ▸ hmem with gn h
-    obtain ⟨nn, nnpos, hnn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp (tN gn)
-    exact
-      ⟨ngn * nn, mul_pos ngnpos nnpos, by
+    obtain ⟨nn, nnpos, hnn⟩ := (tN gn).exists_pow_eq_one
+    exact isOfFinOrder_iff_pow_eq_one.mpr <| ⟨ngn * nn, mul_pos ngnpos nnpos, by
         rw [pow_mul, ← h, ← Subgroup.coe_pow, hnn, Subgroup.coe_one]⟩
 #align is_torsion.extension_closed IsTorsion.extension_closed
 #align add_is_torsion.extension_closed AddIsTorsion.extension_closed
@@ -131,7 +129,7 @@ theorem IsTorsion.quotient_iff {f : G →* H} (hf : Function.Surjective f) (hN :
       "If a group exponent exists, the group is additively torsion."]
 theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g => by
   obtain ⟨n, npos, hn⟩ := h
-  exact (isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, npos, hn g⟩
+  exact isOfFinOrder_iff_pow_eq_one.mpr ⟨n, npos, hn g⟩
 #align exponent_exists.is_torsion ExponentExists.isTorsion
 #align exponent_exists.is_add_torsion ExponentExists.is_add_torsion
 
@@ -141,7 +139,7 @@ theorem ExponentExists.isTorsion (h : ExponentExists G) : IsTorsion G := fun g =
 theorem IsTorsion.exponentExists (tG : IsTorsion G)
     (bounded : (Set.range fun g : G => orderOf g).Finite) : ExponentExists G :=
   exponentExists_iff_ne_zero.mpr <|
-    (exponent_ne_zero_iff_range_orderOf_finite fun g => orderOf_pos' (tG g)).mpr bounded
+    (exponent_ne_zero_iff_range_orderOf_finite fun g => (tG g).orderOf_pos).mpr bounded
 #align is_torsion.exponent_exists IsTorsion.exponentExists
 #align is_add_torsion.exponent_exists IsAddTorsion.exponentExists
 
@@ -164,8 +162,8 @@ namespace AddMonoid
 /-- A module whose scalars are additively torsion is additively torsion. -/
 theorem IsTorsion.module_of_torsion [Semiring R] [Module R M] (tR : IsTorsion R) : IsTorsion M :=
   fun f =>
-  (isOfFinAddOrder_iff_nsmul_eq_zero _).mpr <| by
-    obtain ⟨n, npos, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero _).mp (tR 1)
+  isOfFinAddOrder_iff_nsmul_eq_zero.mpr <| by
+    obtain ⟨n, npos, hn⟩ := (tR 1).exists_nsmul_eq_zero
     exact ⟨n, npos, by simp only [nsmul_eq_smul_cast R _ f, ← nsmul_one, hn, zero_smul]⟩
 #align add_monoid.is_torsion.module_of_torsion AddMonoid.IsTorsion.module_of_torsion
 
@@ -406,9 +404,7 @@ theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬Is
 /-- Subgroups of torsion-free groups are torsion-free. -/
 @[to_additive "Subgroups of additive torsion-free groups are additively torsion-free."]
 theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsionFree H :=
-  fun h hne =>
-  (isOfFinOrder_iff_coe H.toSubmonoid h).not.mpr <|
-    tG h <| by norm_cast
+  fun h hne ↦ Submonoid.isOfFinOrder_coe.not.1 <| tG h <| by norm_cast
 #align is_torsion_free.subgroup IsTorsionFree.subgroup
 #align is_torsion_free.add_subgroup IsTorsionFree.addSubgroup
 
@@ -438,11 +434,11 @@ variable (G) [CommGroup G]
 theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun g hne hfin =>
   hne <| by
     induction' g using QuotientGroup.induction_on' with g
-    obtain ⟨m, mpos, hm⟩ := (isOfFinOrder_iff_pow_eq_one _).mp hfin
-    obtain ⟨n, npos, hn⟩ := (isOfFinOrder_iff_pow_eq_one _).mp ((QuotientGroup.eq_one_iff _).mp hm)
+    obtain ⟨m, mpos, hm⟩ := hfin.exists_pow_eq_one
+    obtain ⟨n, npos, hn⟩ := ((QuotientGroup.eq_one_iff _).mp hm).exists_pow_eq_one
     exact
       (QuotientGroup.eq_one_iff g).mpr
-        ((isOfFinOrder_iff_pow_eq_one _).mpr ⟨m * n, mul_pos mpos npos, (pow_mul g m n).symm ▸ hn⟩)
+        (isOfFinOrder_iff_pow_eq_one.mpr ⟨m * n, mul_pos mpos npos, (pow_mul g m n).symm ▸ hn⟩)
 #align is_torsion_free.quotient_torsion IsTorsionFree.quotient_torsion
 #align add_is_torsion_free.quotient_torsion AddIsTorsionFree.quotient_torsion
 

@@ -361,6 +361,8 @@ def IsTorsionFree :=
 #align monoid.is_torsion_free Monoid.IsTorsionFree
 #align add_monoid.is_torsion_free AddMonoid.IsTorsionFree
 
+variable {G}
+
 /-- A nontrivial monoid is not torsion-free if any nontrivial element has finite order. -/
 @[to_additive (attr := simp) "An additive monoid is not torsion free if any
   nontrivial element has finite order."]
@@ -369,6 +371,10 @@ theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsO
 #align monoid.not_is_torsion_free_iff Monoid.not_isTorsionFree_iff
 #align add_monoid.not_is_torsion_free_iff AddMonoid.not_isTorsionFree_iff
 
+@[to_additive (attr := simp)]
+lemma isTorsionFree_of_subsingleton [Subsingleton G] : IsTorsionFree G :=
+  fun _a ha _ => ha <| Subsingleton.elim _ _
+
 end Monoid
 
 section Group
@@ -381,7 +387,7 @@ variable [Group G]
 @[to_additive AddMonoid.IsTorsion.not_torsion_free
       "A nontrivial additive torsion group is not torsion-free."]
 theorem IsTorsion.not_torsion_free [hN : Nontrivial G] : IsTorsion G → ¬IsTorsionFree G := fun tG =>
-  (not_isTorsionFree_iff _).mpr <| by
+  not_isTorsionFree_iff.mpr <| by
     obtain ⟨x, hx⟩ := (nontrivial_iff_exists_ne (1 : G)).mp hN
     exact ⟨x, hx, tG x⟩
 #align is_torsion.not_torsion_free IsTorsion.not_torsion_free

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

This has nice performance benefits.

@@ -44,7 +44,7 @@ periodic group, aperiodic group, torsion subgroup, torsion abelian group
 -/
 
 
-variable {G H : Type _}
+variable {G H : Type*}
 
 namespace Monoid
 
@@ -157,7 +157,7 @@ end Group
 section Module
 
 -- A (semi/)ring of scalars and a commutative monoid of elements
-variable (R M : Type _) [AddCommMonoid M]
+variable (R M : Type*) [AddCommMonoid M]
 
 namespace AddMonoid
 
@@ -409,7 +409,7 @@ theorem IsTorsionFree.subgroup (tG : IsTorsionFree G) (H : Subgroup G) : IsTorsi
 /-- Direct products of torsion free groups are torsion free. -/
 @[to_additive AddMonoid.IsTorsionFree.prod
       "Direct products of additive torsion free groups are torsion free."]
-theorem IsTorsionFree.prod {η : Type _} {Gs : η → Type _} [∀ i, Group (Gs i)]
+theorem IsTorsionFree.prod {η : Type*} {Gs : η → Type*} [∀ i, Group (Gs i)]
     (tfGs : ∀ i, IsTorsionFree (Gs i)) : IsTorsionFree <| ∀ i, Gs i := fun w hne h =>
   hne <|
     funext fun i => Classical.not_not.mp <| mt (tfGs i (w i)) <| Classical.not_not.mpr <| h.apply i

We define the torsion subgroup of the units of a number field and prove some results about it, mostly: it is finite, cyclic and an unit is torsion iff its value is 1 at all infinite places. Some results linking to rootsOfUnity are also proved.

This PR also includes a direct coercion from (𝓞 K)ˣ to K that is very convenient, although I am not sure it's done properly.

@@ -323,6 +323,9 @@ theorem torsion_eq_torsion_submonoid : CommMonoid.torsion G = (torsion G).toSubm
 #align comm_group.torsion_eq_torsion_submonoid CommGroup.torsion_eq_torsion_submonoid
 #align add_comm_group.add_torsion_eq_add_torsion_submonoid AddCommGroup.add_torsion_eq_add_torsion_submonoid
 
+@[to_additive]
+theorem mem_torsion (g : G) : g ∈ torsion G ↔ IsOfFinOrder g := Iff.rfl
+
 variable (p : ℕ) [hp : Fact p.Prime]
 
 /-- The `p`-primary component is the subgroup of elements with order prime-power of `p`. -/
@@ -438,4 +441,3 @@ theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun
 #align add_is_torsion_free.quotient_torsion AddIsTorsionFree.quotient_torsion
 
 end CommGroup
-

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Julian Berman. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Julian Berman
-
-! This file was ported from Lean 3 source module group_theory.torsion
-! leanprover-community/mathlib commit 1f4705ccdfe1e557fc54a0ce081a05e33d2e6240
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.GroupTheory.Exponent
 import Mathlib.GroupTheory.OrderOfElement
@@ -14,6 +9,8 @@ import Mathlib.GroupTheory.PGroup
 import Mathlib.GroupTheory.QuotientGroup
 import Mathlib.GroupTheory.Submonoid.Operations
 
+#align_import group_theory.torsion from "leanprover-community/mathlib"@"1f4705ccdfe1e557fc54a0ce081a05e33d2e6240"
+
 /-!
 # Torsion groups
 

This PR is the result of running

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

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

@@ -251,8 +251,7 @@ theorem primaryComponent.disjoint {p' : ℕ} [hp' : Fact p'.Prime] (hne : p ≠
   Submonoid.disjoint_def.mpr <| by
     rintro g ⟨_ | n, hn⟩ ⟨n', hn'⟩
     · rwa [pow_zero, orderOf_eq_one_iff] at hn
-    ·
-      exact
+    · exact
         absurd (eq_of_prime_pow_eq hp.out.prime hp'.out.prime n.succ_pos (hn.symm.trans hn')) hne
 #align comm_monoid.primary_component.disjoint CommMonoid.primaryComponent.disjoint
 #align add_comm_monoid.primary_component.disjoint AddCommMonoid.primaryComponent.disjoint