group_theory.specific_groups.cyclic ⟷ Mathlib.GroupTheory.SpecificGroups.Cyclic

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

@@ -231,7 +231,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
           by_contradiction fun h =>
             Nat.find_min hex
               (Int.ofNat_lt.1 <| by
-                rw [← hk₄] <;> exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
+                rw [← hk₄] <;> exact Int.emod_lt_of_pos _ (Int.natCast_pos.2 (Nat.find_spec hex).1))
               ⟨Nat.pos_of_ne_zero h, hk₅⟩
         ⟨k / (Nat.find hex : ℤ),
           Subtype.ext_iff_val.2

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

@@ -3,7 +3,7 @@ Copyright (c) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Algebra.BigOperators.Order
+import Algebra.Order.BigOperators.Group.Finset
 import Data.Nat.Totient
 import GroupTheory.OrderOfElement
 import GroupTheory.Subgroup.Simple

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

@@ -188,7 +188,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   rw [zpow_mod_orderOf] at hk
   have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
   refine' ⟨t.to_nat, _⟩
-  rwa [← zpow_coe_nat, Int.toNat_of_nonneg this]
+  rwa [← zpow_natCast, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 -/
@@ -210,15 +210,14 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
     let ⟨x, hx₁, hx₂⟩ := hx
     let ⟨k, hk⟩ := hg x
     have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
-      ⟨k.natAbs,
-        Nat.pos_of_ne_zero fun h => hx₂ <| by rw [← hk, Int.eq_zero_of_natAbs_eq_zero h, zpow_zero],
+      ⟨k.natAbs, Nat.pos_of_ne_zero fun h => hx₂ <| by rw [← hk, Int.natAbs_eq_zero h, zpow_zero],
         match k, hk with
-        | (k : ℕ), hk => by rw [Int.natAbs_ofNat, ← zpow_coe_nat, hk] <;> exact hx₁
-        | -[k+1], hk => by rw [Int.natAbs_of_negSucc, ← Subgroup.inv_mem_iff H] <;> simp_all⟩
+        | (k : ℕ), hk => by rw [Int.natAbs_ofNat, ← zpow_natCast, hk] <;> exact hx₁
+        | -[k+1], hk => by rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H] <;> simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
         let ⟨k, hk⟩ := hg x
         have hk₁ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ zpowers (g ^ Nat.find hex) :=
-          ⟨k / Nat.find hex, by rw [← zpow_coe_nat, zpow_mul]⟩
+          ⟨k / Nat.find hex, by rw [← zpow_natCast, zpow_mul]⟩
         have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ H := by rw [zpow_mul];
           apply H.zpow_mem; exact_mod_cast (Nat.find_spec hex).2
         have hk₃ : g ^ (k % Nat.find hex) ∈ H :=
@@ -226,8 +225,8 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
             rw [← zpow_add, Int.emod_add_ediv, hk] <;> exact hx
         have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
           rw [Int.natAbs_of_nonneg
-              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
-        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_coe_nat, ← hk₄]
+              (Int.emod_nonneg _ (Int.natCast_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
+        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_natCast, ← hk₄]
         have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
           by_contradiction fun h =>
             Nat.find_min hex
@@ -238,9 +237,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
           Subtype.ext_iff_val.2
             (by
               suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) = x by simpa [zpow_mul]
-              rw [Int.mul_ediv_cancel'
-                  (Int.dvd_of_emod_eq_zero (Int.eq_zero_of_natAbs_eq_zero hk₆)),
-                hk])⟩⟩⟩
+              rw [Int.mul_ediv_cancel' (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero hk₆)), hk])⟩⟩⟩
   else
     by
     have : H = (⊥ : Subgroup α) :=
@@ -273,7 +270,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
             by
             have hgmn : g ^ (m * Nat.gcd n (Fintype.card α)) = 1 := by
               rw [pow_mul, hm, ← pow_gcd_card_eq_one_iff] <;> exact (mem_filter.1 hx).2
-            rw [zpow_coe_nat, ← pow_mul, Nat.mul_div_cancel_left', hm]
+            rw [zpow_natCast, ← pow_mul, Nat.mul_div_cancel_left', hm]
             refine' dvd_of_mul_dvd_mul_right (gcd_pos_of_pos_left (Fintype.card α) hn0) _
             conv_lhs =>
               rw [Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), ←
@@ -351,7 +348,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
               mem_filter.2
                 ⟨mem_univ _, by
                   let ⟨i, hi⟩ := b.2
-                  rw [← hi, ← zpow_coe_nat, ← zpow_mul, mul_comm, zpow_mul, zpow_coe_nat,
+                  rw [← hi, ← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, zpow_natCast,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _)

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

@@ -120,7 +120,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
   use x
   simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-  rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx 
+  rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx
   exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_addOrderOf_eq_card
@@ -139,18 +139,18 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
       by
       rw [← h]
       apply card_subgroup_dvd_card
-    rw [Nat.dvd_prime hp.1] at this 
+    rw [Nat.dvd_prime hp.1] at this
     cases this
-    · rw [Fintype.card_eq_one_iff] at this 
+    · rw [Fintype.card_eq_one_iff] at this
       cases' this with t ht
       suffices g = 1 by contradiction
       have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
-      rw [← ht 1] at hgt 
-      change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
+      rw [← ht 1] at hgt
+      change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt
       simpa using hgt
     · use g
       intro x
-      rw [← h] at this 
+      rw [← h] at this
       rw [Subgroup.eq_top_of_card_eq _ this]
       exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
@@ -185,7 +185,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
   · exact ⟨k, by exact_mod_cast hk⟩
   let t : ℤ := -k % orderOf g
-  rw [zpow_mod_orderOf] at hk 
+  rw [zpow_mod_orderOf] at hk
   have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
   refine' ⟨t.to_nat, _⟩
   rwa [← zpow_coe_nat, Int.toNat_of_nonneg this]
@@ -283,7 +283,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
       have hm0 : 0 < m :=
         Nat.pos_of_ne_zero fun hm0 => by
-          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm ; exact hm.elim' 1
+          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm; exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
       rw [← Fintype.card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       rw [hm]
@@ -314,7 +314,7 @@ variable [DecidableEq α] [Fintype α]
 theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (orderOf a)) = univ :=
   by
-  simp_rw [← SetLike.mem_coe] at ha 
+  simp_rw [← SetLike.mem_coe] at ha
   simp only [image_range_orderOf, set.eq_univ_iff_forall.mpr ha, Set.toFinset_univ]
 #align is_cyclic.image_range_order_of IsCyclic.image_range_orderOf
 #align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_addOrderOf
@@ -376,7 +376,7 @@ private theorem card_order_of_eq_totient_aux₁ :
       ∑ m in d.proper_divisors, φ m :=
     by
     refine' Finset.sum_congr rfl fun m hm => _
-    simp only [mem_filter, mem_range, mem_proper_divisors] at hm 
+    simp only [mem_filter, mem_range, mem_proper_divisors] at hm
     refine' IH m hm.2 (hm.1.trans hd) (Finset.card_pos.2 ⟨a ^ (d / m), _⟩)
     simp only [mem_filter, mem_univ, orderOf_pow a, ha, true_and_iff,
       Nat.gcd_eq_right (div_dvd_of_dvd hm.1), Nat.div_div_self hm.1 hd0]
@@ -394,7 +394,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   have hc0 : 0 < c := Fintype.card_pos_iff.2 ⟨1⟩
   apply card_order_of_eq_totient_aux₁ hn hd
   by_contra h0
-  simp only [not_lt, _root_.le_zero_iff, card_eq_zero] at h0 
+  simp only [not_lt, _root_.le_zero_iff, card_eq_zero] at h0
   apply lt_irrefl c
   calc
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card :=
@@ -411,7 +411,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     _ ≤ ∑ m in c.divisors.erase d, φ m :=
       by
       refine' sum_le_sum fun m hm => _
-      have hmc : m ∣ c := by simp only [mem_erase, mem_divisors] at hm ; tauto
+      have hmc : m ∣ c := by simp only [mem_erase, mem_divisors] at hm; tauto
       rcases(Filter (fun a : α => orderOf a = m) univ).card.eq_zero_or_pos with (h1 | h1)
       · simp [h1]; · simp [card_order_of_eq_totient_aux₁ hn hmc h1]
     _ < ∑ m in c.divisors, φ m :=
@@ -467,12 +467,12 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
     (h : Fintype.card α = p) : IsSimpleGroup α :=
   ⟨by
     have h' := Nat.Prime.one_lt (Fact.out p.prime)
-    rw [← h] at h' 
+    rw [← h] at h'
     haveI := Fintype.one_lt_card_iff_nontrivial.1 h'
     apply exists_pair_ne α, fun H Hn => by
     classical
     have hcard := card_subgroup_dvd_card H
-    rw [h, dvd_prime (Fact.out p.prime)] at hcard 
+    rw [h, dvd_prime (Fact.out p.prime)] at hcard
     refine' hcard.imp (fun h1 => _) fun hp => _
     · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
       apply eq_bot_of_subsingleton
@@ -562,7 +562,7 @@ theorem prime_card [Fintype α] : (Fintype.card α).Prime :=
     have hgo := orderOf_pow g
     rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.gcd_eq_right_iff_dvd.1 hn,
       orderOf_eq_card_of_forall_mem_zpowers, eq_comm,
-      Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo 
+      Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo
     · exact (mul_left_cancel₀ (ne_of_gt h0) ((mul_one (Fintype.card α)).trans hgo)).symm
     · intro x
       rw [h]

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

@@ -100,7 +100,7 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 variable [Group α]
 
 #print MonoidHom.map_cyclic /-
-@[to_additive MonoidAddHom.map_add_cyclic]
+@[to_additive AddMonoidHom.map_addCyclic]
 theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m :=
   by
@@ -110,11 +110,11 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
   obtain ⟨n, rfl⟩ := hG g
   rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul']
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
-#align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
+#align monoid_add_hom.map_add_cyclic AddMonoidHom.map_addCyclic
 -/
 
 #print isCyclic_of_orderOf_eq_card /-
-@[to_additive isAddCyclic_of_orderOf_eq_card]
+@[to_additive isAddCyclic_of_addOrderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
   classical
@@ -123,7 +123,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx 
   exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
-#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
+#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_addOrderOf_eq_card
 -/
 
 #print isCyclic_of_prime_card /-
@@ -259,7 +259,7 @@ open scoped Classical
 
 /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:133:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
 #print IsCyclic.card_pow_eq_one_le /-
-@[to_additive IsAddCyclic.card_pow_eq_one_le]
+@[to_additive IsAddCyclic.card_nsmul_eq_zero_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
     (univ.filterₓ fun a : α => a ^ n = 1).card ≤ n :=
   let ⟨g, hg⟩ := IsCyclic.exists_generator α
@@ -291,7 +291,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
         Nat.mul_div_cancel _ hm0]
       exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)
 #align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le
-#align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_pow_eq_one_le
+#align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_nsmul_eq_zero_le
 -/
 
 end Classical
@@ -451,16 +451,14 @@ theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 -/
 
-#print IsAddCyclic.card_orderOf_eq_totient /-
-theorem IsAddCyclic.card_orderOf_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
+theorem IsAddCyclic.card_order_of_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => addOrderOf a = d).card = totient d :=
   by
   obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›
   exact @IsCyclic.card_orderOf_eq_totient (Multiplicative α) _ ⟨⟨g, hg⟩⟩ _ _ hd
-#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totient
--/
+#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_order_of_eq_totient
 
-attribute [to_additive IsCyclic.card_orderOf_eq_totient] IsAddCyclic.card_orderOf_eq_totient
+attribute [to_additive IsCyclic.card_orderOf_eq_totient] IsAddCyclic.card_order_of_eq_totient
 
 #print isSimpleGroup_of_prime_card /-
 /-- A finite group of prime order is simple. -/
@@ -494,7 +492,7 @@ variable {G : Type _} {H : Type _} [Group G] [Group H]
 #print commutative_of_cyclic_center_quotient /-
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `comm_group_of_cycle_center_quotient` for the `comm_group` instance. -/
-@[to_additive commutative_of_add_cyclic_center_quotient
+@[to_additive commutative_of_addCyclic_center_quotient
       "A group is commutative if the quotient by\n  the center is cyclic. Also see `add_comm_group_of_cycle_center_quotient`\n  for the `add_comm_group` instance."]
 theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G)
     (a b : G) : a * b = b * a :=
@@ -514,7 +512,7 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
     _ = y ^ m * y ^ n * y ^ (-m) * (y ^ (-n) * b * a) := by rw [mem_center_iff.1 hb]
     _ = b * a := by group
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
-#align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
+#align commutative_of_add_cyclic_center_quotient commutative_of_addCyclic_center_quotient
 -/
 
 #print commGroupOfCycleCenterQuotient /-

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

@@ -188,7 +188,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   rw [zpow_mod_orderOf] at hk 
   have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
   refine' ⟨t.to_nat, _⟩
-  rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
+  rwa [← zpow_coe_nat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 -/
@@ -213,12 +213,12 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
       ⟨k.natAbs,
         Nat.pos_of_ne_zero fun h => hx₂ <| by rw [← hk, Int.eq_zero_of_natAbs_eq_zero h, zpow_zero],
         match k, hk with
-        | (k : ℕ), hk => by rw [Int.natAbs_ofNat, ← zpow_ofNat, hk] <;> exact hx₁
+        | (k : ℕ), hk => by rw [Int.natAbs_ofNat, ← zpow_coe_nat, hk] <;> exact hx₁
         | -[k+1], hk => by rw [Int.natAbs_of_negSucc, ← Subgroup.inv_mem_iff H] <;> simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
         let ⟨k, hk⟩ := hg x
         have hk₁ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ zpowers (g ^ Nat.find hex) :=
-          ⟨k / Nat.find hex, by rw [← zpow_ofNat, zpow_mul]⟩
+          ⟨k / Nat.find hex, by rw [← zpow_coe_nat, zpow_mul]⟩
         have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ H := by rw [zpow_mul];
           apply H.zpow_mem; exact_mod_cast (Nat.find_spec hex).2
         have hk₃ : g ^ (k % Nat.find hex) ∈ H :=
@@ -227,7 +227,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
           rw [Int.natAbs_of_nonneg
               (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
-        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
+        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_coe_nat, ← hk₄]
         have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
           by_contradiction fun h =>
             Nat.find_min hex
@@ -273,7 +273,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
             by
             have hgmn : g ^ (m * Nat.gcd n (Fintype.card α)) = 1 := by
               rw [pow_mul, hm, ← pow_gcd_card_eq_one_iff] <;> exact (mem_filter.1 hx).2
-            rw [zpow_ofNat, ← pow_mul, Nat.mul_div_cancel_left', hm]
+            rw [zpow_coe_nat, ← pow_mul, Nat.mul_div_cancel_left', hm]
             refine' dvd_of_mul_dvd_mul_right (gcd_pos_of_pos_left (Fintype.card α) hn0) _
             conv_lhs =>
               rw [Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), ←
@@ -351,7 +351,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
               mem_filter.2
                 ⟨mem_univ _, by
                   let ⟨i, hi⟩ := b.2
-                  rw [← hi, ← zpow_ofNat, ← zpow_mul, mul_comm, zpow_mul, zpow_ofNat,
+                  rw [← hi, ← zpow_coe_nat, ← zpow_mul, mul_comm, zpow_mul, zpow_coe_nat,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _)

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

@@ -56,7 +56,7 @@ attribute [local instance] setFintype
 open Subgroup
 
 #print IsAddCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
@@ -64,7 +64,7 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 -/
 
 #print IsCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where

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

@@ -116,7 +116,12 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 #print isCyclic_of_orderOf_eq_card /-
 @[to_additive isAddCyclic_of_orderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
-    IsCyclic α := by classical
+    IsCyclic α := by
+  classical
+  use x
+  simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
+  rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx 
+  exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 -/
@@ -128,16 +133,38 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
     (h : Fintype.card α = p) : IsCyclic α :=
   ⟨by
     obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
-    classical⟩
+    classical
+    -- for fintype (subgroup.zpowers g)
+    have : Fintype.card (Subgroup.zpowers g) ∣ p :=
+      by
+      rw [← h]
+      apply card_subgroup_dvd_card
+    rw [Nat.dvd_prime hp.1] at this 
+    cases this
+    · rw [Fintype.card_eq_one_iff] at this 
+      cases' this with t ht
+      suffices g = 1 by contradiction
+      have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
+      rw [← ht 1] at hgt 
+      change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
+      simpa using hgt
+    · use g
+      intro x
+      rw [← h] at this 
+      rw [Subgroup.eq_top_of_card_eq _ this]
+      exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
 
 #print orderOf_eq_card_of_forall_mem_zpowers /-
--- for fintype (subgroup.zpowers g)
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
-    orderOf g = Fintype.card α := by classical
+    orderOf g = Fintype.card α := by
+  classical
+  rw [Fintype.card_zpowers]
+  apply Fintype.card_of_finset'
+  simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 -/
@@ -145,7 +172,23 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 #print Infinite.orderOf_eq_zero_of_forall_mem_zpowers /-
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
-    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by classical
+    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
+  classical
+  rw [orderOf_eq_zero_iff']
+  refine' fun n hn hgn => _
+  have ho := IsOfFinOrder.orderOf_pos ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
+  obtain ⟨x, hx⟩ :=
+    Infinite.exists_not_mem_finset (Finset.image (pow g) <| Finset.range <| orderOf g)
+  apply hx
+  rw [← IsOfFinOrder.mem_powers_iff_mem_range_orderOf g x ho, Submonoid.mem_powers_iff]
+  obtain ⟨k, hk⟩ := h x
+  obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
+  · exact ⟨k, by exact_mod_cast hk⟩
+  let t : ℤ := -k % orderOf g
+  rw [zpow_mod_orderOf] at hk 
+  have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
+  refine' ⟨t.to_nat, _⟩
+  rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 -/
@@ -404,7 +447,7 @@ end Totient
 #print IsCyclic.card_orderOf_eq_totient /-
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => orderOf a = d).card = totient d := by
-  classical
+  classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 -/
 
@@ -428,7 +471,14 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
     have h' := Nat.Prime.one_lt (Fact.out p.prime)
     rw [← h] at h' 
     haveI := Fintype.one_lt_card_iff_nontrivial.1 h'
-    apply exists_pair_ne α, fun H Hn => by classical⟩
+    apply exists_pair_ne α, fun H Hn => by
+    classical
+    have hcard := card_subgroup_dvd_card H
+    rw [h, dvd_prime (Fact.out p.prime)] at hcard 
+    refine' hcard.imp (fun h1 => _) fun hp => _
+    · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
+      apply eq_bot_of_subsingleton
+    · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
 #align is_simple_group_of_prime_card isSimpleGroup_of_prime_card
 #align is_simple_add_group_of_prime_card isSimpleAddGroup_of_prime_card
 -/

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

@@ -116,12 +116,7 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 #print isCyclic_of_orderOf_eq_card /-
 @[to_additive isAddCyclic_of_orderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
-    IsCyclic α := by
-  classical
-  use x
-  simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-  rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx 
-  exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
+    IsCyclic α := by classical
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 -/
@@ -133,38 +128,16 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
     (h : Fintype.card α = p) : IsCyclic α :=
   ⟨by
     obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
-    classical
-    -- for fintype (subgroup.zpowers g)
-    have : Fintype.card (Subgroup.zpowers g) ∣ p :=
-      by
-      rw [← h]
-      apply card_subgroup_dvd_card
-    rw [Nat.dvd_prime hp.1] at this 
-    cases this
-    · rw [Fintype.card_eq_one_iff] at this 
-      cases' this with t ht
-      suffices g = 1 by contradiction
-      have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
-      rw [← ht 1] at hgt 
-      change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
-      simpa using hgt
-    · use g
-      intro x
-      rw [← h] at this 
-      rw [Subgroup.eq_top_of_card_eq _ this]
-      exact Subgroup.mem_top _⟩
+    classical⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
 
 #print orderOf_eq_card_of_forall_mem_zpowers /-
+-- for fintype (subgroup.zpowers g)
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
-    orderOf g = Fintype.card α := by
-  classical
-  rw [Fintype.card_zpowers]
-  apply Fintype.card_of_finset'
-  simpa using hx
+    orderOf g = Fintype.card α := by classical
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 -/
@@ -172,23 +145,7 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 #print Infinite.orderOf_eq_zero_of_forall_mem_zpowers /-
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
-    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
-  classical
-  rw [orderOf_eq_zero_iff']
-  refine' fun n hn hgn => _
-  have ho := IsOfFinOrder.orderOf_pos ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
-  obtain ⟨x, hx⟩ :=
-    Infinite.exists_not_mem_finset (Finset.image (pow g) <| Finset.range <| orderOf g)
-  apply hx
-  rw [← IsOfFinOrder.mem_powers_iff_mem_range_orderOf g x ho, Submonoid.mem_powers_iff]
-  obtain ⟨k, hk⟩ := h x
-  obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
-  · exact ⟨k, by exact_mod_cast hk⟩
-  let t : ℤ := -k % orderOf g
-  rw [zpow_mod_orderOf] at hk 
-  have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
-  refine' ⟨t.to_nat, _⟩
-  rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
+    (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by classical
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 -/
@@ -447,7 +404,7 @@ end Totient
 #print IsCyclic.card_orderOf_eq_totient /-
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => orderOf a = d).card = totient d := by
-  classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
+  classical
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 -/
 
@@ -471,14 +428,7 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
     have h' := Nat.Prime.one_lt (Fact.out p.prime)
     rw [← h] at h' 
     haveI := Fintype.one_lt_card_iff_nontrivial.1 h'
-    apply exists_pair_ne α, fun H Hn => by
-    classical
-    have hcard := card_subgroup_dvd_card H
-    rw [h, dvd_prime (Fact.out p.prime)] at hcard 
-    refine' hcard.imp (fun h1 => _) fun hp => _
-    · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
-      apply eq_bot_of_subsingleton
-    · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
+    apply exists_pair_ne α, fun H Hn => by classical⟩
 #align is_simple_group_of_prime_card isSimpleGroup_of_prime_card
 #align is_simple_add_group_of_prime_card isSimpleAddGroup_of_prime_card
 -/

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

@@ -266,7 +266,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
   calc
     (univ.filterₓ fun a : α => a ^ n = 1).card ≤
         (zpowers (g ^ (Fintype.card α / Nat.gcd n (Fintype.card α))) : Set α).toFinset.card :=
-      card_le_of_subset fun x hx =>
+      card_le_card fun x hx =>
         let ⟨m, hm⟩ := show x ∈ Submonoid.powers g from mem_powers_iff_mem_zpowers.2 <| hg x
         Set.mem_toFinset.2
           ⟨(m / (Fintype.card α / Nat.gcd n (Fintype.card α)) : ℕ),

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

@@ -56,7 +56,7 @@ attribute [local instance] setFintype
 open Subgroup
 
 #print IsAddCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
@@ -64,7 +64,7 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 -/
 
 #print IsCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where

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

@@ -120,7 +120,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
   use x
   simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-  rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx 
+  rw [← Fintype.card_congr (Equiv.Set.univ α), Fintype.card_zpowers] at hx 
   exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
@@ -162,7 +162,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
-  rw [orderOf_eq_card_zpowers]
+  rw [Fintype.card_zpowers]
   apply Fintype.card_of_finset'
   simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
@@ -285,7 +285,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
         Nat.pos_of_ne_zero fun hm0 => by
           rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm ; exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
-      rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
+      rw [← Fintype.card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       rw [hm]
       rw [Nat.mul_div_cancel_left _ (gcd_pos_of_pos_left _ hn0), gcd_mul_left_left, hm,
         Nat.mul_div_cancel _ hm0]
@@ -340,7 +340,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
     (Finset.univ.filterₓ fun b : α => b ^ orderOf a = 1).card = orderOf a :=
   le_antisymm (hn _ (orderOf_pos a))
     (calc
-      orderOf a = @Fintype.card (zpowers a) (id _) := orderOf_eq_card_zpowers
+      orderOf a = @Fintype.card (zpowers a) (id _) := Fintype.card_zpowers
       _ ≤
           @Fintype.card (↑(univ.filterₓ fun b : α => b ^ orderOf a = 1) : Set α)
             (Fintype.ofFinset _ fun _ => Iff.rfl) :=
@@ -610,7 +610,7 @@ theorem IsCyclic.exponent_eq_card [Group α] [IsCyclic α] [Fintype α] :
   obtain ⟨g, hg⟩ := IsCyclic.exists_generator α
   apply Nat.dvd_antisymm
   · rw [← lcm_order_eq_exponent, Finset.lcm_dvd_iff]
-    exact fun b _ => orderOf_dvd_card_univ
+    exact fun b _ => orderOf_dvd_card
   rw [← orderOf_eq_card_of_forall_mem_zpowers hg]
   exact order_dvd_exponent _
 #align is_cyclic.exponent_eq_card IsCyclic.exponent_eq_card

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

@@ -176,16 +176,16 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   classical
   rw [orderOf_eq_zero_iff']
   refine' fun n hn hgn => _
-  have ho := orderOf_pos' ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
+  have ho := IsOfFinOrder.orderOf_pos ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
   obtain ⟨x, hx⟩ :=
     Infinite.exists_not_mem_finset (Finset.image (pow g) <| Finset.range <| orderOf g)
   apply hx
-  rw [← mem_powers_iff_mem_range_order_of' g x ho, Submonoid.mem_powers_iff]
+  rw [← IsOfFinOrder.mem_powers_iff_mem_range_orderOf g x ho, Submonoid.mem_powers_iff]
   obtain ⟨k, hk⟩ := h x
   obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
   · exact ⟨k, by exact_mod_cast hk⟩
   let t : ℤ := -k % orderOf g
-  rw [zpow_eq_mod_orderOf] at hk 
+  rw [zpow_mod_orderOf] at hk 
   have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
   refine' ⟨t.to_nat, _⟩
   rwa [← zpow_ofNat, Int.toNat_of_nonneg this]

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

@@ -3,12 +3,12 @@ Copyright (c) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Mathbin.Algebra.BigOperators.Order
-import Mathbin.Data.Nat.Totient
-import Mathbin.GroupTheory.OrderOfElement
-import Mathbin.GroupTheory.Subgroup.Simple
-import Mathbin.Tactic.Group
-import Mathbin.GroupTheory.Exponent
+import Algebra.BigOperators.Order
+import Data.Nat.Totient
+import GroupTheory.OrderOfElement
+import GroupTheory.Subgroup.Simple
+import Tactic.Group
+import GroupTheory.Exponent
 
 #align_import group_theory.specific_groups.cyclic from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
 
@@ -56,7 +56,7 @@ attribute [local instance] setFintype
 open Subgroup
 
 #print IsAddCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
@@ -64,7 +64,7 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 -/
 
 #print IsCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where
@@ -257,7 +257,7 @@ section Classical
 
 open scoped Classical
 
-/- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
+/- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:133:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
 #print IsCyclic.card_pow_eq_one_le /-
 @[to_additive IsAddCyclic.card_pow_eq_one_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :

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

@@ -2,11 +2,6 @@
 Copyright (c) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
-
-! This file was ported from Lean 3 source module group_theory.specific_groups.cyclic
-! 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.Algebra.BigOperators.Order
 import Mathbin.Data.Nat.Totient
@@ -15,6 +10,8 @@ import Mathbin.GroupTheory.Subgroup.Simple
 import Mathbin.Tactic.Group
 import Mathbin.GroupTheory.Exponent
 
+#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-!
 # Cyclic groups
 

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

@@ -59,7 +59,7 @@ attribute [local instance] setFintype
 open Subgroup
 
 #print IsAddCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
@@ -67,7 +67,7 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 -/
 
 #print IsCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where
@@ -102,6 +102,7 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 
 variable [Group α]
 
+#print MonoidHom.map_cyclic /-
 @[to_additive MonoidAddHom.map_add_cyclic]
 theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m :=
@@ -113,6 +114,7 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
   rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul']
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
 #align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
+-/
 
 #print isCyclic_of_orderOf_eq_card /-
 @[to_additive isAddCyclic_of_orderOf_eq_card]
@@ -158,6 +160,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
 
+#print orderOf_eq_card_of_forall_mem_zpowers /-
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
@@ -167,7 +170,9 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
   simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
+-/
 
+#print Infinite.orderOf_eq_zero_of_forall_mem_zpowers /-
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
@@ -189,13 +194,17 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
+-/
 
+#print Bot.isCyclic /-
 @[to_additive Bot.isAddCyclic]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
 #align bot.is_add_cyclic Bot.isAddCyclic
+-/
 
+#print Subgroup.isCyclic /-
 @[to_additive AddSubgroup.isAddCyclic]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
@@ -243,6 +252,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
     clear _let_match <;> subst this <;> infer_instance
 #align subgroup.is_cyclic Subgroup.isCyclic
 #align add_subgroup.is_add_cyclic AddSubgroup.isAddCyclic
+-/
 
 open Finset Nat
 
@@ -251,6 +261,7 @@ section Classical
 open scoped Classical
 
 /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
+#print IsCyclic.card_pow_eq_one_le /-
 @[to_additive IsAddCyclic.card_pow_eq_one_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
     (univ.filterₓ fun a : α => a ^ n = 1).card ≤ n :=
@@ -284,20 +295,24 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)
 #align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le
 #align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_pow_eq_one_le
+-/
 
 end Classical
 
+#print IsCyclic.exists_monoid_generator /-
 @[to_additive]
 theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
     ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x := by simp_rw [mem_powers_iff_mem_zpowers];
   exact IsCyclic.exists_generator α
 #align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generator
 #align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_addMonoid_generator
+-/
 
 section
 
 variable [DecidableEq α] [Fintype α]
 
+#print IsCyclic.image_range_orderOf /-
 @[to_additive]
 theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (orderOf a)) = univ :=
@@ -306,13 +321,16 @@ theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
   simp only [image_range_orderOf, set.eq_univ_iff_forall.mpr ha, Set.toFinset_univ]
 #align is_cyclic.image_range_order_of IsCyclic.image_range_orderOf
 #align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_addOrderOf
+-/
 
+#print IsCyclic.image_range_card /-
 @[to_additive]
 theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (Fintype.card α)) = univ := by
   rw [← orderOf_eq_card_of_forall_mem_zpowers ha, IsCyclic.image_range_orderOf ha]
 #align is_cyclic.image_range_card IsCyclic.image_range_card
 #align is_add_cyclic.image_range_card IsAddCyclic.image_range_card
+-/
 
 end
 
@@ -321,8 +339,6 @@ section Totient
 variable [DecidableEq α] [Fintype α]
   (hn : ∀ n : ℕ, 0 < n → (univ.filterₓ fun a : α => a ^ n = 1).card ≤ n)
 
-include hn
-
 private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
     (Finset.univ.filterₓ fun b : α => b ^ orderOf a = 1).card = orderOf a :=
   le_antisymm (hn _ (orderOf_pos a))
@@ -373,6 +389,7 @@ private theorem card_order_of_eq_totient_aux₁ :
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_order_of_aux hn a]
   simpa [← cons_self_proper_divisors hd0, ← h1] using h2
 
+#print card_orderOf_eq_totient_aux₂ /-
 theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filterₓ fun a : α => orderOf a = d).card = φ d :=
   by
@@ -404,7 +421,9 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
       (sum_erase_lt_of_pos (mem_divisors.2 ⟨hd, hc0.ne'⟩) (totient_pos (pos_of_dvd_of_pos hd hc0)))
     _ = c := sum_totient _
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂
+-/
 
+#print isCyclic_of_card_pow_eq_one_le /-
 theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   have : (univ.filterₓ fun a : α => orderOf a = Fintype.card α).Nonempty :=
     card_pos.1 <| by
@@ -413,29 +432,36 @@ theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   let ⟨x, hx⟩ := this
   isCyclic_of_orderOf_eq_card x (Finset.mem_filter.1 hx).2
 #align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_le
+-/
 
+#print isAddCyclic_of_card_pow_eq_one_le /-
 theorem isAddCyclic_of_card_pow_eq_one_le {α} [AddGroup α] [DecidableEq α] [Fintype α]
     (hn : ∀ n : ℕ, 0 < n → (univ.filterₓ fun a : α => n • a = 0).card ≤ n) : IsAddCyclic α :=
   by
   obtain ⟨g, hg⟩ := @isCyclic_of_card_pow_eq_one_le (Multiplicative α) _ _ _ hn
   exact ⟨⟨g, hg⟩⟩
 #align is_add_cyclic_of_card_pow_eq_one_le isAddCyclic_of_card_pow_eq_one_le
+-/
 
 attribute [to_additive isCyclic_of_card_pow_eq_one_le] isAddCyclic_of_card_pow_eq_one_le
 
 end Totient
 
+#print IsCyclic.card_orderOf_eq_totient /-
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => orderOf a = d).card = totient d := by
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
+-/
 
+#print IsAddCyclic.card_orderOf_eq_totient /-
 theorem IsAddCyclic.card_orderOf_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => addOrderOf a = d).card = totient d :=
   by
   obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›
   exact @IsCyclic.card_orderOf_eq_totient (Multiplicative α) _ ⟨⟨g, hg⟩⟩ _ _ hd
 #align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totient
+-/
 
 attribute [to_additive IsCyclic.card_orderOf_eq_totient] IsAddCyclic.card_orderOf_eq_totient
 
@@ -468,6 +494,7 @@ open Subgroup
 
 variable {G : Type _} {H : Type _} [Group G] [Group H]
 
+#print commutative_of_cyclic_center_quotient /-
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `comm_group_of_cycle_center_quotient` for the `comm_group` instance. -/
 @[to_additive commutative_of_add_cyclic_center_quotient
@@ -491,7 +518,9 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
     _ = b * a := by group
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
 #align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
+-/
 
+#print commGroupOfCycleCenterQuotient /-
 /-- A group is commutative if the quotient by the center is cyclic. -/
 @[to_additive commutativeOfAddCycleCenterQuotient
       "A group is commutative if the quotient by\n  the center is cyclic."]
@@ -500,6 +529,7 @@ def commGroupOfCycleCenterQuotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ c
   { show Group G by infer_instance with mul_comm := commutative_of_cyclic_center_quotient f hf }
 #align comm_group_of_cycle_center_quotient commGroupOfCycleCenterQuotient
 #align commutative_of_add_cycle_center_quotient commutativeOfAddCycleCenterQuotient
+-/
 
 end QuotientCenter
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -359,7 +359,7 @@ private theorem card_order_of_eq_totient_aux₁ :
   rcases card_pos.1 hpos with ⟨a, ha'⟩
   have ha : orderOf a = d := (mem_filter.1 ha').2
   have h1 :
-    (∑ m in d.proper_divisors, (univ.filter fun a : α => orderOf a = m).card) =
+    ∑ m in d.proper_divisors, (univ.filter fun a : α => orderOf a = m).card =
       ∑ m in d.proper_divisors, φ m :=
     by
     refine' Finset.sum_congr rfl fun m hm => _
@@ -368,8 +368,7 @@ private theorem card_order_of_eq_totient_aux₁ :
     simp only [mem_filter, mem_univ, orderOf_pow a, ha, true_and_iff,
       Nat.gcd_eq_right (div_dvd_of_dvd hm.1), Nat.div_div_self hm.1 hd0]
   have h2 :
-    (∑ m in d.divisors, (univ.filter fun a : α => orderOf a = m).card) = ∑ m in d.divisors, φ m :=
-    by
+    ∑ m in d.divisors, (univ.filter fun a : α => orderOf a = m).card = ∑ m in d.divisors, φ m := by
     rw [← filter_dvd_eq_divisors hd0, sum_card_orderOf_eq_card_pow_eq_one hd0,
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_order_of_aux hn a]
   simpa [← cons_self_proper_divisors hd0, ← h1] using h2

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

@@ -282,7 +282,6 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       rw [Nat.mul_div_cancel_left _ (gcd_pos_of_pos_left _ hn0), gcd_mul_left_left, hm,
         Nat.mul_div_cancel _ hm0]
       exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)
-    
 #align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le
 #align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_pow_eq_one_le
 
@@ -342,8 +341,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
                   rw [← hi, ← zpow_ofNat, ← zpow_mul, mul_comm, zpow_mul, zpow_ofNat,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
-      _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _
-      )
+      _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _)
 
 open scoped Nat
 
@@ -406,7 +404,6 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     _ < ∑ m in c.divisors, φ m :=
       (sum_erase_lt_of_pos (mem_divisors.2 ⟨hd, hc0.ne'⟩) (totient_pos (pos_of_dvd_of_pos hd hc0)))
     _ = c := sum_totient _
-    
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂
 
 theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
@@ -493,7 +490,6 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
     _ = y ^ m * y ^ n * y ^ (-m) * (a * (y ^ (-n) * b)) := by simp [mul_assoc]
     _ = y ^ m * y ^ n * y ^ (-m) * (y ^ (-n) * b * a) := by rw [mem_center_iff.1 hb]
     _ = b * a := by group
-    
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
 #align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
 

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

@@ -59,7 +59,7 @@ attribute [local instance] setFintype
 open Subgroup
 
 #print IsAddCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
@@ -67,7 +67,7 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 -/
 
 #print IsCyclic /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where
@@ -119,10 +119,10 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
   classical
-    use x
-    simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-    rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx 
-    exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
+  use x
+  simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
+  rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx 
+  exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 -/
@@ -135,25 +135,25 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
   ⟨by
     obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
     classical
-      -- for fintype (subgroup.zpowers g)
-      have : Fintype.card (Subgroup.zpowers g) ∣ p :=
-        by
-        rw [← h]
-        apply card_subgroup_dvd_card
-      rw [Nat.dvd_prime hp.1] at this 
-      cases this
-      · rw [Fintype.card_eq_one_iff] at this 
-        cases' this with t ht
-        suffices g = 1 by contradiction
-        have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
-        rw [← ht 1] at hgt 
-        change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
-        simpa using hgt
-      · use g
-        intro x
-        rw [← h] at this 
-        rw [Subgroup.eq_top_of_card_eq _ this]
-        exact Subgroup.mem_top _⟩
+    -- for fintype (subgroup.zpowers g)
+    have : Fintype.card (Subgroup.zpowers g) ∣ p :=
+      by
+      rw [← h]
+      apply card_subgroup_dvd_card
+    rw [Nat.dvd_prime hp.1] at this 
+    cases this
+    · rw [Fintype.card_eq_one_iff] at this 
+      cases' this with t ht
+      suffices g = 1 by contradiction
+      have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
+      rw [← ht 1] at hgt 
+      change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
+      simpa using hgt
+    · use g
+      intro x
+      rw [← h] at this 
+      rw [Subgroup.eq_top_of_card_eq _ this]
+      exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
@@ -162,9 +162,9 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
-    rw [orderOf_eq_card_zpowers]
-    apply Fintype.card_of_finset'
-    simpa using hx
+  rw [orderOf_eq_card_zpowers]
+  apply Fintype.card_of_finset'
+  simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 
@@ -172,21 +172,21 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
   classical
-    rw [orderOf_eq_zero_iff']
-    refine' fun n hn hgn => _
-    have ho := orderOf_pos' ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
-    obtain ⟨x, hx⟩ :=
-      Infinite.exists_not_mem_finset (Finset.image (pow g) <| Finset.range <| orderOf g)
-    apply hx
-    rw [← mem_powers_iff_mem_range_order_of' g x ho, Submonoid.mem_powers_iff]
-    obtain ⟨k, hk⟩ := h x
-    obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
-    · exact ⟨k, by exact_mod_cast hk⟩
-    let t : ℤ := -k % orderOf g
-    rw [zpow_eq_mod_orderOf] at hk 
-    have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
-    refine' ⟨t.to_nat, _⟩
-    rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
+  rw [orderOf_eq_zero_iff']
+  refine' fun n hn hgn => _
+  have ho := orderOf_pos' ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
+  obtain ⟨x, hx⟩ :=
+    Infinite.exists_not_mem_finset (Finset.image (pow g) <| Finset.range <| orderOf g)
+  apply hx
+  rw [← mem_powers_iff_mem_range_order_of' g x ho, Submonoid.mem_powers_iff]
+  obtain ⟨k, hk⟩ := h x
+  obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
+  · exact ⟨k, by exact_mod_cast hk⟩
+  let t : ℤ := -k % orderOf g
+  rw [zpow_eq_mod_orderOf] at hk 
+  have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
+  refine' ⟨t.to_nat, _⟩
+  rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
@@ -454,12 +454,12 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
     haveI := Fintype.one_lt_card_iff_nontrivial.1 h'
     apply exists_pair_ne α, fun H Hn => by
     classical
-      have hcard := card_subgroup_dvd_card H
-      rw [h, dvd_prime (Fact.out p.prime)] at hcard 
-      refine' hcard.imp (fun h1 => _) fun hp => _
-      · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
-        apply eq_bot_of_subsingleton
-      · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
+    have hcard := card_subgroup_dvd_card H
+    rw [h, dvd_prime (Fact.out p.prime)] at hcard 
+    refine' hcard.imp (fun h1 => _) fun hp => _
+    · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
+      apply eq_bot_of_subsingleton
+    · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
 #align is_simple_group_of_prime_card isSimpleGroup_of_prime_card
 #align is_simple_add_group_of_prime_card isSimpleAddGroup_of_prime_card
 -/

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

@@ -121,7 +121,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
     use x
     simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-    rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx
+    rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx 
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
@@ -140,18 +140,18 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
         by
         rw [← h]
         apply card_subgroup_dvd_card
-      rw [Nat.dvd_prime hp.1] at this
+      rw [Nat.dvd_prime hp.1] at this 
       cases this
-      · rw [Fintype.card_eq_one_iff] at this
+      · rw [Fintype.card_eq_one_iff] at this 
         cases' this with t ht
         suffices g = 1 by contradiction
         have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
-        rw [← ht 1] at hgt
-        change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt
+        rw [← ht 1] at hgt 
+        change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt 
         simpa using hgt
       · use g
         intro x
-        rw [← h] at this
+        rw [← h] at this 
         rw [Subgroup.eq_top_of_card_eq _ this]
         exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
@@ -183,7 +183,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     obtain ⟨k, rfl | rfl⟩ := k.eq_coe_or_neg
     · exact ⟨k, by exact_mod_cast hk⟩
     let t : ℤ := -k % orderOf g
-    rw [zpow_eq_mod_orderOf] at hk
+    rw [zpow_eq_mod_orderOf] at hk 
     have : 0 ≤ t := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
     refine' ⟨t.to_nat, _⟩
     rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
@@ -275,7 +275,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
       have hm0 : 0 < m :=
         Nat.pos_of_ne_zero fun hm0 => by
-          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm; exact hm.elim' 1
+          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm ; exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
       rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       rw [hm]
@@ -303,7 +303,7 @@ variable [DecidableEq α] [Fintype α]
 theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (orderOf a)) = univ :=
   by
-  simp_rw [← SetLike.mem_coe] at ha
+  simp_rw [← SetLike.mem_coe] at ha 
   simp only [image_range_orderOf, set.eq_univ_iff_forall.mpr ha, Set.toFinset_univ]
 #align is_cyclic.image_range_order_of IsCyclic.image_range_orderOf
 #align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_addOrderOf
@@ -365,7 +365,7 @@ private theorem card_order_of_eq_totient_aux₁ :
       ∑ m in d.proper_divisors, φ m :=
     by
     refine' Finset.sum_congr rfl fun m hm => _
-    simp only [mem_filter, mem_range, mem_proper_divisors] at hm
+    simp only [mem_filter, mem_range, mem_proper_divisors] at hm 
     refine' IH m hm.2 (hm.1.trans hd) (Finset.card_pos.2 ⟨a ^ (d / m), _⟩)
     simp only [mem_filter, mem_univ, orderOf_pow a, ha, true_and_iff,
       Nat.gcd_eq_right (div_dvd_of_dvd hm.1), Nat.div_div_self hm.1 hd0]
@@ -383,7 +383,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   have hc0 : 0 < c := Fintype.card_pos_iff.2 ⟨1⟩
   apply card_order_of_eq_totient_aux₁ hn hd
   by_contra h0
-  simp only [not_lt, _root_.le_zero_iff, card_eq_zero] at h0
+  simp only [not_lt, _root_.le_zero_iff, card_eq_zero] at h0 
   apply lt_irrefl c
   calc
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card :=
@@ -400,7 +400,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     _ ≤ ∑ m in c.divisors.erase d, φ m :=
       by
       refine' sum_le_sum fun m hm => _
-      have hmc : m ∣ c := by simp only [mem_erase, mem_divisors] at hm; tauto
+      have hmc : m ∣ c := by simp only [mem_erase, mem_divisors] at hm ; tauto
       rcases(Filter (fun a : α => orderOf a = m) univ).card.eq_zero_or_pos with (h1 | h1)
       · simp [h1]; · simp [card_order_of_eq_totient_aux₁ hn hmc h1]
     _ < ∑ m in c.divisors, φ m :=
@@ -450,12 +450,12 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
     (h : Fintype.card α = p) : IsSimpleGroup α :=
   ⟨by
     have h' := Nat.Prime.one_lt (Fact.out p.prime)
-    rw [← h] at h'
+    rw [← h] at h' 
     haveI := Fintype.one_lt_card_iff_nontrivial.1 h'
     apply exists_pair_ne α, fun H Hn => by
     classical
       have hcard := card_subgroup_dvd_card H
-      rw [h, dvd_prime (Fact.out p.prime)] at hcard
+      rw [h, dvd_prime (Fact.out p.prime)] at hcard 
       refine' hcard.imp (fun h1 => _) fun hp => _
       · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
         apply eq_bot_of_subsingleton
@@ -542,7 +542,7 @@ theorem prime_card [Fintype α] : (Fintype.card α).Prime :=
     have hgo := orderOf_pow g
     rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.gcd_eq_right_iff_dvd.1 hn,
       orderOf_eq_card_of_forall_mem_zpowers, eq_comm,
-      Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo
+      Nat.div_eq_iff_eq_mul_left (Nat.pos_of_dvd_of_pos hn h0) hn] at hgo 
     · exact (mul_left_cancel₀ (ne_of_gt h0) ((mul_one (Fintype.card α)).trans hgo)).symm
     · intro x
       rw [h]

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

@@ -52,7 +52,7 @@ variable {α : Type u} {a : α}
 
 section Cyclic
 
-open BigOperators
+open scoped BigOperators
 
 attribute [local instance] setFintype
 
@@ -248,7 +248,7 @@ open Finset Nat
 
 section Classical
 
-open Classical
+open scoped Classical
 
 /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
 @[to_additive IsAddCyclic.card_pow_eq_one_le]
@@ -345,7 +345,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
       _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _
       )
 
-open Nat
+open scoped Nat
 
 -- use φ for nat.totient
 private theorem card_order_of_eq_totient_aux₁ :

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

@@ -102,12 +102,6 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 
 variable [Group α]
 
-/- warning: monoid_hom.map_cyclic -> MonoidHom.map_cyclic is a dubious translation:
-lean 3 declaration is
-  forall {G : Type.{u1}} [_inst_2 : Group.{u1} G] [h : IsCyclic.{u1} G _inst_2] (σ : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))), Exists.{1} Int (fun (m : Int) => forall (g : G), Eq.{succ u1} G (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (fun (_x : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) => G -> G) (MonoidHom.hasCoeToFun.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) σ g) (HPow.hPow.{u1, 0, u1} G Int G (instHPow.{u1, 0} G Int (DivInvMonoid.Pow.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) g m))
-but is expected to have type
-  forall {G : Type.{u1}} [_inst_2 : Group.{u1} G] [h : IsCyclic.{u1} G _inst_2] (σ : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))), Exists.{1} Int (fun (m : Int) => forall (g : G), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : G) => G) g) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : G) => G) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (MonoidHom.monoidHomClass.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))))) σ g) (HPow.hPow.{u1, 0, u1} G Int G (instHPow.{u1, 0} G Int (DivInvMonoid.Pow.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) g m))
-Case conversion may be inaccurate. Consider using '#align monoid_hom.map_cyclic MonoidHom.map_cyclicₓ'. -/
 @[to_additive MonoidAddHom.map_add_cyclic]
 theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m :=
@@ -164,12 +158,6 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
 
-/- warning: order_of_eq_card_of_forall_mem_zpowers -> orderOf_eq_card_of_forall_mem_zpowers is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Fintype.{u1} α] {g : α}, (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (Fintype.card.{u1} α _inst_2))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Fintype.{u1} α] {g : α}, (forall (x : α), Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (Fintype.card.{u1} α _inst_2))
-Case conversion may be inaccurate. Consider using '#align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowersₓ'. -/
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
@@ -180,12 +168,6 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 
-/- warning: infinite.order_of_eq_zero_of_forall_mem_zpowers -> Infinite.orderOf_eq_zero_of_forall_mem_zpowers is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Infinite.{succ u1} α] {g : α}, (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))
-but is expected to have type
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 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
@@ -208,24 +190,12 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
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 @[to_additive Bot.isAddCyclic]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
 #align bot.is_add_cyclic Bot.isAddCyclic
 
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 @[to_additive AddSubgroup.isAddCyclic]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
@@ -280,12 +250,6 @@ section Classical
 
 open Classical
 
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 /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
 @[to_additive IsAddCyclic.card_pow_eq_one_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
@@ -324,12 +288,6 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
 
 end Classical
 
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 @[to_additive]
 theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
     ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x := by simp_rw [mem_powers_iff_mem_zpowers];
@@ -341,12 +299,6 @@ section
 
 variable [DecidableEq α] [Fintype α]
 
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 @[to_additive]
 theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (orderOf a)) = univ :=
@@ -356,12 +308,6 @@ theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
 #align is_cyclic.image_range_order_of IsCyclic.image_range_orderOf
 #align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_addOrderOf
 
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 @[to_additive]
 theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (Fintype.card α)) = univ := by
@@ -430,12 +376,6 @@ private theorem card_order_of_eq_totient_aux₁ :
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_order_of_aux hn a]
   simpa [← cons_self_proper_divisors hd0, ← h1] using h2
 
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 theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filterₓ fun a : α => orderOf a = d).card = φ d :=
   by
@@ -469,12 +409,6 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂
 
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-Case conversion may be inaccurate. Consider using '#align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_leₓ'. -/
 theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   have : (univ.filterₓ fun a : α => orderOf a = Fintype.card α).Nonempty :=
     card_pos.1 <| by
@@ -484,12 +418,6 @@ theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   isCyclic_of_orderOf_eq_card x (Finset.mem_filter.1 hx).2
 #align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_le
 
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-Case conversion may be inaccurate. Consider using '#align is_add_cyclic_of_card_pow_eq_one_le isAddCyclic_of_card_pow_eq_one_leₓ'. -/
 theorem isAddCyclic_of_card_pow_eq_one_le {α} [AddGroup α] [DecidableEq α] [Fintype α]
     (hn : ∀ n : ℕ, 0 < n → (univ.filterₓ fun a : α => n • a = 0).card ≤ n) : IsAddCyclic α :=
   by
@@ -501,23 +429,11 @@ attribute [to_additive isCyclic_of_card_pow_eq_one_le] isAddCyclic_of_card_pow_e
 
 end Totient
 
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-Case conversion may be inaccurate. Consider using '#align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totientₓ'. -/
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => orderOf a = d).card = totient d := by
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 
-/- warning: is_add_cyclic.card_order_of_eq_totient -> IsAddCyclic.card_orderOf_eq_totient is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : AddGroup.{u1} α] [_inst_3 : IsAddCyclic.{u1} α _inst_2] [_inst_4 : Fintype.{u1} α] {d : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat d (Fintype.card.{u1} α _inst_4)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (fun (a : α) => instDecidableEqNat (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (Finset.univ.{u1} α _inst_4))) (Nat.totient d))
-Case conversion may be inaccurate. Consider using '#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totientₓ'. -/
 theorem IsAddCyclic.card_orderOf_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => addOrderOf a = d).card = totient d :=
   by
@@ -556,12 +472,6 @@ open Subgroup
 
 variable {G : Type _} {H : Type _} [Group G] [Group H]
 
-/- warning: commutative_of_cyclic_center_quotient -> commutative_of_cyclic_center_quotient is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotientₓ'. -/
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `comm_group_of_cycle_center_quotient` for the `comm_group` instance. -/
 @[to_additive commutative_of_add_cyclic_center_quotient
@@ -587,12 +497,6 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
 #align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
 
-/- warning: comm_group_of_cycle_center_quotient -> commGroupOfCycleCenterQuotient is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align comm_group_of_cycle_center_quotient commGroupOfCycleCenterQuotientₓ'. -/
 /-- A group is commutative if the quotient by the center is cyclic. -/
 @[to_additive commutativeOfAddCycleCenterQuotient
       "A group is commutative if the quotient by\n  the center is cyclic."]

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

@@ -79,9 +79,7 @@ class IsCyclic (α : Type u) [Group α] : Prop where
 #print isCyclic_of_subsingleton /-
 @[to_additive isAddCyclic_of_subsingleton]
 instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α] : IsCyclic α :=
-  ⟨⟨1, fun x => by
-      rw [Subsingleton.elim x 1]
-      exact mem_zpowers 1⟩⟩
+  ⟨⟨1, fun x => by rw [Subsingleton.elim x 1]; exact mem_zpowers 1⟩⟩
 #align is_cyclic_of_subsingleton isCyclic_of_subsingleton
 #align is_add_cyclic_of_subsingleton isAddCyclic_of_subsingleton
 -/
@@ -153,11 +151,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
       · rw [Fintype.card_eq_one_iff] at this
         cases' this with t ht
         suffices g = 1 by contradiction
-        have hgt :=
-          ht
-            ⟨g, by
-              change g ∈ Subgroup.zpowers g
-              exact Subgroup.mem_zpowers g⟩
+        have hgt := ht ⟨g, by change g ∈ Subgroup.zpowers g; exact Subgroup.mem_zpowers g⟩
         rw [← ht 1] at hgt
         change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt
         simpa using hgt
@@ -249,11 +243,8 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         let ⟨k, hk⟩ := hg x
         have hk₁ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ zpowers (g ^ Nat.find hex) :=
           ⟨k / Nat.find hex, by rw [← zpow_ofNat, zpow_mul]⟩
-        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ H :=
-          by
-          rw [zpow_mul]
-          apply H.zpow_mem
-          exact_mod_cast (Nat.find_spec hex).2
+        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex)) ∈ H := by rw [zpow_mul];
+          apply H.zpow_mem; exact_mod_cast (Nat.find_spec hex).2
         have hk₃ : g ^ (k % Nat.find hex) ∈ H :=
           (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
             rw [← zpow_add, Int.emod_add_ediv, hk] <;> exact hx
@@ -319,10 +310,8 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
     _ ≤ n := by
       let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
       have hm0 : 0 < m :=
-        Nat.pos_of_ne_zero fun hm0 =>
-          by
-          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm
-          exact hm.elim' 1
+        Nat.pos_of_ne_zero fun hm0 => by
+          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm; exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
       rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       rw [hm]
@@ -343,9 +332,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generatorₓ'. -/
 @[to_additive]
 theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
-    ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x :=
-  by
-  simp_rw [mem_powers_iff_mem_zpowers]
+    ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x := by simp_rw [mem_powers_iff_mem_zpowers];
   exact IsCyclic.exists_generator α
 #align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generator
 #align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_addMonoid_generator
@@ -468,19 +455,14 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
       by
       rw [eq_comm]
       refine' sum_subset (erase_subset _ _) fun m hm₁ hm₂ => _
-      have : m = d := by
-        contrapose! hm₂
-        exact mem_erase_of_ne_of_mem hm₂ hm₁
+      have : m = d := by contrapose! hm₂; exact mem_erase_of_ne_of_mem hm₂ hm₁
       simp [this, h0]
     _ ≤ ∑ m in c.divisors.erase d, φ m :=
       by
       refine' sum_le_sum fun m hm => _
-      have hmc : m ∣ c := by
-        simp only [mem_erase, mem_divisors] at hm
-        tauto
+      have hmc : m ∣ c := by simp only [mem_erase, mem_divisors] at hm; tauto
       rcases(Filter (fun a : α => orderOf a = m) univ).card.eq_zero_or_pos with (h1 | h1)
-      · simp [h1]
-      · simp [card_order_of_eq_totient_aux₁ hn hmc h1]
+      · simp [h1]; · simp [card_order_of_eq_totient_aux₁ hn hmc h1]
     _ < ∑ m in c.divisors, φ m :=
       (sum_erase_lt_of_pos (mem_divisors.2 ⟨hd, hc0.ne'⟩) (totient_pos (pos_of_dvd_of_pos hd hc0)))
     _ = c := sum_totient _

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: Johannes Hölzl
 
 ! This file was ported from Lean 3 source module group_theory.specific_groups.cyclic
-! leanprover-community/mathlib commit 0f6670b8af2dff699de1c0b4b49039b31bc13c46
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.GroupTheory.Exponent
 /-!
 # Cyclic groups
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A group `G` is called cyclic if there exists an element `g : G` such that every element of `G` is of
 the form `g ^ n` for some `n : ℕ`. This file only deals with the predicate on a group to be cyclic.
 For the concrete cyclic group of order `n`, see `data.zmod.basic`.
@@ -74,13 +77,13 @@ class IsCyclic (α : Type u) [Group α] : Prop where
 -/
 
 #print isCyclic_of_subsingleton /-
-@[to_additive is_add_cyclic_of_subsingleton]
+@[to_additive isAddCyclic_of_subsingleton]
 instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α] : IsCyclic α :=
   ⟨⟨1, fun x => by
       rw [Subsingleton.elim x 1]
       exact mem_zpowers 1⟩⟩
 #align is_cyclic_of_subsingleton isCyclic_of_subsingleton
-#align is_add_cyclic_of_subsingleton is_add_cyclic_of_subsingleton
+#align is_add_cyclic_of_subsingleton isAddCyclic_of_subsingleton
 -/
 
 #print IsCyclic.commGroup /-
@@ -120,7 +123,7 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 #align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
 
 #print isCyclic_of_orderOf_eq_card /-
-@[to_additive is_add_cyclic_of_orderOf_eq_card]
+@[to_additive isAddCyclic_of_orderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
   classical
@@ -129,12 +132,12 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
     rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
-#align is_add_cyclic_of_order_of_eq_card is_add_cyclic_of_orderOf_eq_card
+#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 -/
 
 #print isCyclic_of_prime_card /-
 /-- A finite group of prime order is cyclic. -/
-@[to_additive is_add_cyclic_of_prime_card "A finite group of prime order is cyclic."]
+@[to_additive isAddCyclic_of_prime_card "A finite group of prime order is cyclic."]
 theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card α = p) : IsCyclic α :=
   ⟨by
@@ -164,7 +167,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
         rw [Subgroup.eq_top_of_card_eq _ this]
         exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
-#align is_add_cyclic_of_prime_card is_add_cyclic_of_prime_card
+#align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 -/
 
 /- warning: order_of_eq_card_of_forall_mem_zpowers -> orderOf_eq_card_of_forall_mem_zpowers is a dubious translation:
@@ -217,11 +220,11 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_2 : Group.{u1} α], IsCyclic.{u1} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_2) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_2)) x (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.instBotSubgroup.{u1} α _inst_2)))) (Subgroup.toGroup.{u1} α _inst_2 (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.instBotSubgroup.{u1} α _inst_2)))
 Case conversion may be inaccurate. Consider using '#align bot.is_cyclic Bot.isCyclicₓ'. -/
-@[to_additive Bot.is_add_cyclic]
+@[to_additive Bot.isAddCyclic]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
-#align bot.is_add_cyclic Bot.is_add_cyclic
+#align bot.is_add_cyclic Bot.isAddCyclic
 
 /- warning: subgroup.is_cyclic -> Subgroup.isCyclic is a dubious translation:
 lean 3 declaration is
@@ -229,7 +232,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_2 : Group.{u1} α] [_inst_3 : IsCyclic.{u1} α _inst_2] (H : Subgroup.{u1} α _inst_2), IsCyclic.{u1} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_2) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_2)) x H)) (Subgroup.toGroup.{u1} α _inst_2 H)
 Case conversion may be inaccurate. Consider using '#align subgroup.is_cyclic Subgroup.isCyclicₓ'. -/
-@[to_additive AddSubgroup.is_add_cyclic]
+@[to_additive AddSubgroup.isAddCyclic]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
   let ⟨g, hg⟩ := IsCyclic.exists_generator α
@@ -278,7 +281,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         ⟨fun h => by simp at * <;> tauto, fun h => by rw [Subgroup.mem_bot.1 h] <;> exact H.one_mem⟩
     clear _let_match <;> subst this <;> infer_instance
 #align subgroup.is_cyclic Subgroup.isCyclic
-#align add_subgroup.is_add_cyclic AddSubgroup.is_add_cyclic
+#align add_subgroup.is_add_cyclic AddSubgroup.isAddCyclic
 
 open Finset Nat
 
@@ -408,7 +411,6 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _
       )
-#align card_pow_eq_one_eq_order_of_aux card_pow_eq_one_eq_order_of_aux
 
 open Nat
 
@@ -440,7 +442,6 @@ private theorem card_order_of_eq_totient_aux₁ :
     rw [← filter_dvd_eq_divisors hd0, sum_card_orderOf_eq_card_pow_eq_one hd0,
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_order_of_aux hn a]
   simpa [← cons_self_proper_divisors hd0, ← h1] using h2
-#align card_order_of_eq_totient_aux₁ card_order_of_eq_totient_aux₁
 
 /- warning: card_order_of_eq_totient_aux₂ -> card_orderOf_eq_totient_aux₂ is a dubious translation:
 lean 3 declaration is
@@ -627,7 +628,7 @@ section CommGroup
 
 variable [CommGroup α] [IsSimpleGroup α]
 
-@[to_additive IsSimpleAddGroup.is_add_cyclic]
+@[to_additive IsSimpleAddGroup.isAddCyclic]
 instance (priority := 100) : IsCyclic α :=
   by
   cases' subsingleton_or_nontrivial α with hi hi <;> haveI := hi
@@ -675,7 +676,7 @@ end CommGroup
 end IsSimpleGroup
 
 #print CommGroup.is_simple_iff_isCyclic_and_prime_card /-
-@[to_additive AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card]
+@[to_additive AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card]
 theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup α] :
     IsSimpleGroup α ↔ IsCyclic α ∧ (Fintype.card α).Prime :=
   by
@@ -686,7 +687,7 @@ theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup
     haveI : Fact (Fintype.card α).Prime := ⟨hp⟩
     exact isSimpleGroup_of_prime_card rfl
 #align comm_group.is_simple_iff_is_cyclic_and_prime_card CommGroup.is_simple_iff_isCyclic_and_prime_card
-#align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card
+#align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card
 -/
 
 section Exponent

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

@@ -55,12 +55,15 @@ attribute [local instance] setFintype
 
 open Subgroup
 
+#print IsAddCyclic /-
 /- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
 #align is_add_cyclic IsAddCyclic
+-/
 
+#print IsCyclic /-
 /- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
@@ -68,7 +71,9 @@ class IsCyclic (α : Type u) [Group α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ zpowers g
 #align is_cyclic IsCyclic
 #align is_add_cyclic IsAddCyclic
+-/
 
+#print isCyclic_of_subsingleton /-
 @[to_additive is_add_cyclic_of_subsingleton]
 instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α] : IsCyclic α :=
   ⟨⟨1, fun x => by
@@ -76,7 +81,9 @@ instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α]
       exact mem_zpowers 1⟩⟩
 #align is_cyclic_of_subsingleton isCyclic_of_subsingleton
 #align is_add_cyclic_of_subsingleton is_add_cyclic_of_subsingleton
+-/
 
+#print IsCyclic.commGroup /-
 /-- A cyclic group is always commutative. This is not an `instance` because often we have a better
 proof of `comm_group`. -/
 @[to_additive
@@ -90,9 +97,16 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
       hm ▸ hn ▸ zpow_mul_comm _ _ _ }
 #align is_cyclic.comm_group IsCyclic.commGroup
 #align is_add_cyclic.add_comm_group IsAddCyclic.addCommGroup
+-/
 
 variable [Group α]
 
+/- warning: monoid_hom.map_cyclic -> MonoidHom.map_cyclic is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} [_inst_2 : Group.{u1} G] [h : IsCyclic.{u1} G _inst_2] (σ : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))), Exists.{1} Int (fun (m : Int) => forall (g : G), Eq.{succ u1} G (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (fun (_x : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) => G -> G) (MonoidHom.hasCoeToFun.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) σ g) (HPow.hPow.{u1, 0, u1} G Int G (instHPow.{u1, 0} G Int (DivInvMonoid.Pow.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) g m))
+but is expected to have type
+  forall {G : Type.{u1}} [_inst_2 : Group.{u1} G] [h : IsCyclic.{u1} G _inst_2] (σ : MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))), Exists.{1} Int (fun (m : Int) => forall (g : G), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : G) => G) g) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : G) => G) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))) G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (MonoidHom.monoidHomClass.{u1, u1} G G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2)))))) σ g) (HPow.hPow.{u1, 0, u1} G Int G (instHPow.{u1, 0} G Int (DivInvMonoid.Pow.{u1} G (Group.toDivInvMonoid.{u1} G _inst_2))) g m))
+Case conversion may be inaccurate. Consider using '#align monoid_hom.map_cyclic MonoidHom.map_cyclicₓ'. -/
 @[to_additive MonoidAddHom.map_add_cyclic]
 theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m :=
@@ -105,6 +119,7 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
 #align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
 
+#print isCyclic_of_orderOf_eq_card /-
 @[to_additive is_add_cyclic_of_orderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
@@ -115,7 +130,9 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card is_add_cyclic_of_orderOf_eq_card
+-/
 
+#print isCyclic_of_prime_card /-
 /-- A finite group of prime order is cyclic. -/
 @[to_additive is_add_cyclic_of_prime_card "A finite group of prime order is cyclic."]
 theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
@@ -148,8 +165,15 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
         exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card is_add_cyclic_of_prime_card
+-/
 
-@[to_additive add_orderOf_eq_card_of_forall_mem_zmultiples]
+/- warning: order_of_eq_card_of_forall_mem_zpowers -> orderOf_eq_card_of_forall_mem_zpowers is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Fintype.{u1} α] {g : α}, (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (Fintype.card.{u1} α _inst_2))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Fintype.{u1} α] {g : α}, (forall (x : α), Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (Fintype.card.{u1} α _inst_2))
+Case conversion may be inaccurate. Consider using '#align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowersₓ'. -/
+@[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
@@ -157,9 +181,15 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
     apply Fintype.card_of_finset'
     simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
-#align add_order_of_eq_card_of_forall_mem_zmultiples add_orderOf_eq_card_of_forall_mem_zmultiples
-
-@[to_additive Infinite.add_orderOf_eq_zero_of_forall_mem_zmultiples]
+#align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
+
+/- warning: infinite.order_of_eq_zero_of_forall_mem_zpowers -> Infinite.orderOf_eq_zero_of_forall_mem_zpowers is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Infinite.{succ u1} α] {g : α}, (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Infinite.{succ u1} α] {g : α}, (forall (x : α), Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 g)) -> (Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) g) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))
+Case conversion may be inaccurate. Consider using '#align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowersₓ'. -/
+@[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
   classical
@@ -179,14 +209,26 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     refine' ⟨t.to_nat, _⟩
     rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
-#align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.add_orderOf_eq_zero_of_forall_mem_zmultiples
-
+#align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
+
+/- warning: bot.is_cyclic -> Bot.isCyclic is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : Group.{u1} α], IsCyclic.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} α _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.setLike.{u1} α _inst_2)) (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.hasBot.{u1} α _inst_2))) (Subgroup.toGroup.{u1} α _inst_2 (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.hasBot.{u1} α _inst_2)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : Group.{u1} α], IsCyclic.{u1} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_2) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_2)) x (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.instBotSubgroup.{u1} α _inst_2)))) (Subgroup.toGroup.{u1} α _inst_2 (Bot.bot.{u1} (Subgroup.{u1} α _inst_2) (Subgroup.instBotSubgroup.{u1} α _inst_2)))
+Case conversion may be inaccurate. Consider using '#align bot.is_cyclic Bot.isCyclicₓ'. -/
 @[to_additive Bot.is_add_cyclic]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
 #align bot.is_add_cyclic Bot.is_add_cyclic
 
+/- warning: subgroup.is_cyclic -> Subgroup.isCyclic is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : Group.{u1} α] [_inst_3 : IsCyclic.{u1} α _inst_2] (H : Subgroup.{u1} α _inst_2), IsCyclic.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} α _inst_2) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.setLike.{u1} α _inst_2)) H) (Subgroup.toGroup.{u1} α _inst_2 H)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : Group.{u1} α] [_inst_3 : IsCyclic.{u1} α _inst_2] (H : Subgroup.{u1} α _inst_2), IsCyclic.{u1} (Subtype.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_2) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_2) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_2)) x H)) (Subgroup.toGroup.{u1} α _inst_2 H)
+Case conversion may be inaccurate. Consider using '#align subgroup.is_cyclic Subgroup.isCyclicₓ'. -/
 @[to_additive AddSubgroup.is_add_cyclic]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
@@ -244,6 +286,12 @@ section Classical
 
 open Classical
 
+/- warning: is_cyclic.card_pow_eq_one_le -> IsCyclic.card_pow_eq_one_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α] [_inst_4 : IsCyclic.{u1} α _inst_1] {n : Nat}, (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (Finset.univ.{u1} α _inst_3))) n)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α] [_inst_4 : IsCyclic.{u1} α _inst_1] {n : Nat}, (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (Finset.univ.{u1} α _inst_3))) n)
+Case conversion may be inaccurate. Consider using '#align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_leₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Lean3.lean:132:4: warning: unsupported: rw with cfg: { occs := occurrences.pos[occurrences.pos] «expr[ ,]»([2, 3]) } -/
 @[to_additive IsAddCyclic.card_pow_eq_one_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
@@ -284,6 +332,12 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
 
 end Classical
 
+/- warning: is_cyclic.exists_monoid_generator -> IsCyclic.exists_monoid_generator is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Finite.{succ u1} α] [_inst_3 : IsCyclic.{u1} α _inst_1], Exists.{succ u1} α (fun (x : α) => forall (y : α), Membership.Mem.{u1, u1} α (Submonoid.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) α (Submonoid.setLike.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) y (Submonoid.powers.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) x))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : Finite.{succ u1} α] [_inst_3 : IsCyclic.{u1} α _inst_1], Exists.{succ u1} α (fun (x : α) => forall (y : α), Membership.mem.{u1, u1} α (Submonoid.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) α (Submonoid.instSetLikeSubmonoid.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1))))) y (Submonoid.powers.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) x))
+Case conversion may be inaccurate. Consider using '#align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generatorₓ'. -/
 @[to_additive]
 theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
     ∃ x : α, ∀ y : α, y ∈ Submonoid.powers x :=
@@ -291,12 +345,18 @@ theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
   simp_rw [mem_powers_iff_mem_zpowers]
   exact IsCyclic.exists_generator α
 #align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generator
-#align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_add_monoid_generator
+#align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_addMonoid_generator
 
 section
 
 variable [DecidableEq α] [Fintype α]
 
+/- warning: is_cyclic.image_range_order_of -> IsCyclic.image_range_orderOf is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {a : α} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 a)) -> (Eq.{succ u1} (Finset.{u1} α) (Finset.image.{0, u1} Nat α (fun (a : α) (b : α) => _inst_2 a b) (fun (i : Nat) => HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a i) (Finset.range (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a))) (Finset.univ.{u1} α _inst_3))
+but is expected to have type
+  forall {α : Type.{u1}} {a : α} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (x : α), Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 a)) -> (Eq.{succ u1} (Finset.{u1} α) (Finset.image.{0, u1} Nat α (fun (a : α) (b : α) => _inst_2 a b) (fun (i : Nat) => HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a i) (Finset.range (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a))) (Finset.univ.{u1} α _inst_3))
+Case conversion may be inaccurate. Consider using '#align is_cyclic.image_range_order_of IsCyclic.image_range_orderOfₓ'. -/
 @[to_additive]
 theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (orderOf a)) = univ :=
@@ -304,8 +364,14 @@ theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
   simp_rw [← SetLike.mem_coe] at ha
   simp only [image_range_orderOf, set.eq_univ_iff_forall.mpr ha, Set.toFinset_univ]
 #align is_cyclic.image_range_order_of IsCyclic.image_range_orderOf
-#align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_orderOf
-
+#align is_add_cyclic.image_range_order_of IsAddCyclic.image_range_addOrderOf
+
+/- warning: is_cyclic.image_range_card -> IsCyclic.image_range_card is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {a : α} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (x : α), Membership.Mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.hasMem.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.setLike.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 a)) -> (Eq.{succ u1} (Finset.{u1} α) (Finset.image.{0, u1} Nat α (fun (a : α) (b : α) => _inst_2 a b) (fun (i : Nat) => HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a i) (Finset.range (Fintype.card.{u1} α _inst_3))) (Finset.univ.{u1} α _inst_3))
+but is expected to have type
+  forall {α : Type.{u1}} {a : α} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (x : α), Membership.mem.{u1, u1} α (Subgroup.{u1} α _inst_1) (SetLike.instMembership.{u1, u1} (Subgroup.{u1} α _inst_1) α (Subgroup.instSetLikeSubgroup.{u1} α _inst_1)) x (Subgroup.zpowers.{u1} α _inst_1 a)) -> (Eq.{succ u1} (Finset.{u1} α) (Finset.image.{0, u1} Nat α (fun (a : α) (b : α) => _inst_2 a b) (fun (i : Nat) => HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a i) (Finset.range (Fintype.card.{u1} α _inst_3))) (Finset.univ.{u1} α _inst_3))
+Case conversion may be inaccurate. Consider using '#align is_cyclic.image_range_card IsCyclic.image_range_cardₓ'. -/
 @[to_additive]
 theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
     Finset.image (fun i => a ^ i) (range (Fintype.card α)) = univ := by
@@ -376,6 +442,12 @@ private theorem card_order_of_eq_totient_aux₁ :
   simpa [← cons_self_proper_divisors hd0, ← h1] using h2
 #align card_order_of_eq_totient_aux₁ card_order_of_eq_totient_aux₁
 
+/- warning: card_order_of_eq_totient_aux₂ -> card_orderOf_eq_totient_aux₂ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (Finset.univ.{u1} α _inst_3))) n)) -> (forall {d : Nat}, (Dvd.Dvd.{0} Nat Nat.hasDvd d (Fintype.card.{u1} α _inst_3)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (fun (a : α) => Nat.decidableEq (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (Finset.univ.{u1} α _inst_3))) (Nat.totient d)))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (Finset.univ.{u1} α _inst_3))) n)) -> (forall {d : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat d (Fintype.card.{u1} α _inst_3)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (fun (a : α) => instDecidableEqNat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (Finset.univ.{u1} α _inst_3))) (Nat.totient d)))
+Case conversion may be inaccurate. Consider using '#align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂ₓ'. -/
 theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filterₓ fun a : α => orderOf a = d).card = φ d :=
   by
@@ -414,6 +486,12 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂
 
+/- warning: is_cyclic_of_card_pow_eq_one_le -> isCyclic_of_card_pow_eq_one_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))))))) (Finset.univ.{u1} α _inst_3))) n)) -> (IsCyclic.{u1} α _inst_1)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : DecidableEq.{succ u1} α] [_inst_3 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (fun (a : α) => _inst_2 (HPow.hPow.{u1, 0, u1} α Nat α (instHPow.{u1, 0} α Nat (Monoid.Pow.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)))) a n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (Group.toDivisionMonoid.{u1} α _inst_1))))))) (Finset.univ.{u1} α _inst_3))) n)) -> (IsCyclic.{u1} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_leₓ'. -/
 theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   have : (univ.filterₓ fun a : α => orderOf a = Fintype.card α).Nonempty :=
     card_pos.1 <| by
@@ -423,6 +501,12 @@ theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   isCyclic_of_orderOf_eq_card x (Finset.mem_filter.1 hx).2
 #align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_le
 
+/- warning: is_add_cyclic_of_card_pow_eq_one_le -> isAddCyclic_of_card_pow_eq_one_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_4 : AddGroup.{u1} α] [_inst_5 : DecidableEq.{succ u1} α] [_inst_6 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (SMul.smul.{0, u1} Nat α (AddMonoid.SMul.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4))) n a) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (AddZeroClass.toHasZero.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4)))))))) (fun (a : α) => _inst_5 (SMul.smul.{0, u1} Nat α (AddMonoid.SMul.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4))) n a) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (AddZeroClass.toHasZero.{u1} α (AddMonoid.toAddZeroClass.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4)))))))) (Finset.univ.{u1} α _inst_6))) n)) -> (IsAddCyclic.{u1} α _inst_4)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_4 : AddGroup.{u1} α] [_inst_5 : DecidableEq.{succ u1} α] [_inst_6 : Fintype.{u1} α], (forall (n : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{succ u1} α (HSMul.hSMul.{0, u1, u1} Nat α α (instHSMul.{0, u1} Nat α (AddMonoid.SMul.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4)))) n a) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (NegZeroClass.toZero.{u1} α (SubNegZeroMonoid.toNegZeroClass.{u1} α (SubtractionMonoid.toSubNegZeroMonoid.{u1} α (AddGroup.toSubtractionMonoid.{u1} α _inst_4))))))) (fun (a : α) => _inst_5 (HSMul.hSMul.{0, u1, u1} Nat α α (instHSMul.{0, u1} Nat α (AddMonoid.SMul.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_4)))) n a) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (NegZeroClass.toZero.{u1} α (SubNegZeroMonoid.toNegZeroClass.{u1} α (SubtractionMonoid.toSubNegZeroMonoid.{u1} α (AddGroup.toSubtractionMonoid.{u1} α _inst_4))))))) (Finset.univ.{u1} α _inst_6))) n)) -> (IsAddCyclic.{u1} α _inst_4)
+Case conversion may be inaccurate. Consider using '#align is_add_cyclic_of_card_pow_eq_one_le isAddCyclic_of_card_pow_eq_one_leₓ'. -/
 theorem isAddCyclic_of_card_pow_eq_one_le {α} [AddGroup α] [DecidableEq α] [Fintype α]
     (hn : ∀ n : ℕ, 0 < n → (univ.filterₓ fun a : α => n • a = 0).card ≤ n) : IsAddCyclic α :=
   by
@@ -434,20 +518,33 @@ attribute [to_additive isCyclic_of_card_pow_eq_one_le] isAddCyclic_of_card_pow_e
 
 end Totient
 
+/- warning: is_cyclic.card_order_of_eq_totient -> IsCyclic.card_orderOf_eq_totient is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : IsCyclic.{u1} α _inst_1] [_inst_3 : Fintype.{u1} α] {d : Nat}, (Dvd.Dvd.{0} Nat Nat.hasDvd d (Fintype.card.{u1} α _inst_3)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (fun (a : α) => Nat.decidableEq (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (Finset.univ.{u1} α _inst_3))) (Nat.totient d))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : Group.{u1} α] [_inst_2 : IsCyclic.{u1} α _inst_1] [_inst_3 : Fintype.{u1} α] {d : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat d (Fintype.card.{u1} α _inst_3)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (fun (a : α) => instDecidableEqNat (orderOf.{u1} α (DivInvMonoid.toMonoid.{u1} α (Group.toDivInvMonoid.{u1} α _inst_1)) a) d) (Finset.univ.{u1} α _inst_3))) (Nat.totient d))
+Case conversion may be inaccurate. Consider using '#align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totientₓ'. -/
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => orderOf a = d).card = totient d := by
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 
-theorem IsAddCyclic.card_order_of_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
+/- warning: is_add_cyclic.card_order_of_eq_totient -> IsAddCyclic.card_orderOf_eq_totient is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_2 : AddGroup.{u1} α] [_inst_3 : IsAddCyclic.{u1} α _inst_2] [_inst_4 : Fintype.{u1} α] {d : Nat}, (Dvd.Dvd.{0} Nat Nat.hasDvd d (Fintype.card.{u1} α _inst_4)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (fun (a : α) => Nat.decidableEq (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (Finset.univ.{u1} α _inst_4))) (Nat.totient d))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_2 : AddGroup.{u1} α] [_inst_3 : IsAddCyclic.{u1} α _inst_2] [_inst_4 : Fintype.{u1} α] {d : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat d (Fintype.card.{u1} α _inst_4)) -> (Eq.{1} Nat (Finset.card.{u1} α (Finset.filter.{u1} α (fun (a : α) => Eq.{1} Nat (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (fun (a : α) => instDecidableEqNat (addOrderOf.{u1} α (SubNegMonoid.toAddMonoid.{u1} α (AddGroup.toSubNegMonoid.{u1} α _inst_2)) a) d) (Finset.univ.{u1} α _inst_4))) (Nat.totient d))
+Case conversion may be inaccurate. Consider using '#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totientₓ'. -/
+theorem IsAddCyclic.card_orderOf_eq_totient {α} [AddGroup α] [IsAddCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filterₓ fun a : α => addOrderOf a = d).card = totient d :=
   by
   obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›
   exact @IsCyclic.card_orderOf_eq_totient (Multiplicative α) _ ⟨⟨g, hg⟩⟩ _ _ hd
-#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_order_of_eq_totient
+#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totient
 
-attribute [to_additive IsCyclic.card_orderOf_eq_totient] IsAddCyclic.card_order_of_eq_totient
+attribute [to_additive IsCyclic.card_orderOf_eq_totient] IsAddCyclic.card_orderOf_eq_totient
 
+#print isSimpleGroup_of_prime_card /-
 /-- A finite group of prime order is simple. -/
 @[to_additive "A finite group of prime order is simple."]
 theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
@@ -465,7 +562,8 @@ theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : 
         apply eq_bot_of_subsingleton
       · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
 #align is_simple_group_of_prime_card isSimpleGroup_of_prime_card
-#align is_simple_add_group_of_prime_card is_simple_add_group_of_prime_card
+#align is_simple_add_group_of_prime_card isSimpleAddGroup_of_prime_card
+-/
 
 end Cyclic
 
@@ -475,6 +573,12 @@ open Subgroup
 
 variable {G : Type _} {H : Type _} [Group G] [Group H]
 
+/- warning: commutative_of_cyclic_center_quotient -> commutative_of_cyclic_center_quotient is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] [_inst_3 : IsCyclic.{u2} H _inst_2] (f : MonoidHom.{u1, u2} G H (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))), (LE.le.{u1} (Subgroup.{u1} G _inst_1) (Preorder.toHasLe.{u1} (Subgroup.{u1} G _inst_1) (PartialOrder.toPreorder.{u1} (Subgroup.{u1} G _inst_1) (SetLike.partialOrder.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)))) (MonoidHom.ker.{u1, u2} G _inst_1 H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2))) f) (Subgroup.center.{u1} G _inst_1)) -> (forall (a : G) (b : G), Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a))
+but is expected to have type
+  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] [_inst_3 : IsCyclic.{u2} H _inst_2] (f : MonoidHom.{u1, u2} G H (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))), (LE.le.{u1} (Subgroup.{u1} G _inst_1) (Preorder.toLE.{u1} (Subgroup.{u1} G _inst_1) (PartialOrder.toPreorder.{u1} (Subgroup.{u1} G _inst_1) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Subgroup.{u1} G _inst_1) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Subgroup.{u1} G _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u1} G _inst_1))))) (MonoidHom.ker.{u1, u2} G _inst_1 H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2))) f) (Subgroup.center.{u1} G _inst_1)) -> (forall (a : G) (b : G), Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a))
+Case conversion may be inaccurate. Consider using '#align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotientₓ'. -/
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `comm_group_of_cycle_center_quotient` for the `comm_group` instance. -/
 @[to_additive commutative_of_add_cyclic_center_quotient
@@ -500,6 +604,12 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
 #align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
 
+/- warning: comm_group_of_cycle_center_quotient -> commGroupOfCycleCenterQuotient is a dubious translation:
+lean 3 declaration is
+  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] [_inst_3 : IsCyclic.{u2} H _inst_2] (f : MonoidHom.{u1, u2} G H (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))), (LE.le.{u1} (Subgroup.{u1} G _inst_1) (Preorder.toHasLe.{u1} (Subgroup.{u1} G _inst_1) (PartialOrder.toPreorder.{u1} (Subgroup.{u1} G _inst_1) (SetLike.partialOrder.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)))) (MonoidHom.ker.{u1, u2} G _inst_1 H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2))) f) (Subgroup.center.{u1} G _inst_1)) -> (CommGroup.{u1} G)
+but is expected to have type
+  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] [_inst_3 : IsCyclic.{u2} H _inst_2] (f : MonoidHom.{u1, u2} G H (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))), (LE.le.{u1} (Subgroup.{u1} G _inst_1) (Preorder.toLE.{u1} (Subgroup.{u1} G _inst_1) (PartialOrder.toPreorder.{u1} (Subgroup.{u1} G _inst_1) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Subgroup.{u1} G _inst_1) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Subgroup.{u1} G _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u1} G _inst_1))))) (MonoidHom.ker.{u1, u2} G _inst_1 H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2))) f) (Subgroup.center.{u1} G _inst_1)) -> (CommGroup.{u1} G)
+Case conversion may be inaccurate. Consider using '#align comm_group_of_cycle_center_quotient commGroupOfCycleCenterQuotientₓ'. -/
 /-- A group is commutative if the quotient by the center is cyclic. -/
 @[to_additive commutativeOfAddCycleCenterQuotient
       "A group is commutative if the quotient by\n  the center is cyclic."]
@@ -532,6 +642,7 @@ instance (priority := 100) : IsCyclic α :=
     · rw [ht]
       apply Subgroup.mem_top
 
+#print IsSimpleGroup.prime_card /-
 @[to_additive]
 theorem prime_card [Fintype α] : (Fintype.card α).Prime :=
   by
@@ -557,11 +668,13 @@ theorem prime_card [Fintype α] : (Fintype.card α).Prime :=
     exact Subgroup.mem_zpowers _
 #align is_simple_group.prime_card IsSimpleGroup.prime_card
 #align is_simple_add_group.prime_card IsSimpleAddGroup.prime_card
+-/
 
 end CommGroup
 
 end IsSimpleGroup
 
+#print CommGroup.is_simple_iff_isCyclic_and_prime_card /-
 @[to_additive AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card]
 theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup α] :
     IsSimpleGroup α ↔ IsCyclic α ∧ (Fintype.card α).Prime :=
@@ -574,11 +687,13 @@ theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup
     exact isSimpleGroup_of_prime_card rfl
 #align comm_group.is_simple_iff_is_cyclic_and_prime_card CommGroup.is_simple_iff_isCyclic_and_prime_card
 #align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card
+-/
 
 section Exponent
 
 open Monoid
 
+#print IsCyclic.exponent_eq_card /-
 @[to_additive]
 theorem IsCyclic.exponent_eq_card [Group α] [IsCyclic α] [Fintype α] :
     exponent α = Fintype.card α :=
@@ -591,7 +706,9 @@ theorem IsCyclic.exponent_eq_card [Group α] [IsCyclic α] [Fintype α] :
   exact order_dvd_exponent _
 #align is_cyclic.exponent_eq_card IsCyclic.exponent_eq_card
 #align is_add_cyclic.exponent_eq_card IsAddCyclic.exponent_eq_card
+-/
 
+#print IsCyclic.of_exponent_eq_card /-
 @[to_additive]
 theorem IsCyclic.of_exponent_eq_card [CommGroup α] [Fintype α] (h : exponent α = Fintype.card α) :
     IsCyclic α :=
@@ -599,14 +716,18 @@ theorem IsCyclic.of_exponent_eq_card [CommGroup α] [Fintype α] (h : exponent 
   isCyclic_of_orderOf_eq_card g <| hg.trans <| exponent_eq_max'_orderOf.symm.trans h
 #align is_cyclic.of_exponent_eq_card IsCyclic.of_exponent_eq_card
 #align is_add_cyclic.of_exponent_eq_card IsAddCyclic.of_exponent_eq_card
+-/
 
+#print IsCyclic.iff_exponent_eq_card /-
 @[to_additive]
 theorem IsCyclic.iff_exponent_eq_card [CommGroup α] [Fintype α] :
     IsCyclic α ↔ exponent α = Fintype.card α :=
   ⟨fun h => IsCyclic.exponent_eq_card, IsCyclic.of_exponent_eq_card⟩
 #align is_cyclic.iff_exponent_eq_card IsCyclic.iff_exponent_eq_card
 #align is_add_cyclic.iff_exponent_eq_card IsAddCyclic.iff_exponent_eq_card
+-/
 
+#print IsCyclic.exponent_eq_zero_of_infinite /-
 @[to_additive]
 theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite α] :
     exponent α = 0 :=
@@ -614,6 +735,7 @@ theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite
   exponent_eq_zero_of_order_zero <| Infinite.orderOf_eq_zero_of_forall_mem_zpowers hg
 #align is_cyclic.exponent_eq_zero_of_infinite IsCyclic.exponent_eq_zero_of_infinite
 #align is_add_cyclic.exponent_eq_zero_of_infinite IsAddCyclic.exponent_eq_zero_of_infinite
+-/
 
 end Exponent
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -55,13 +55,13 @@ attribute [local instance] setFintype
 
 open Subgroup
 
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ AddSubgroup.zmultiples g
 #align is_add_cyclic IsAddCyclic
 
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`exists_generator] [] -/
 /-- A group is called *cyclic* if it is generated by a single element. -/
 @[to_additive IsAddCyclic]
 class IsCyclic (α : Type u) [Group α] : Prop where

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

@@ -270,7 +270,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       have hm0 : 0 < m :=
         Nat.pos_of_ne_zero fun hm0 =>
           by
-          rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
+          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm
           exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
       rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]

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

@@ -330,7 +330,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
       _ ≤
           @Fintype.card (↑(univ.filterₓ fun b : α => b ^ orderOf a = 1) : Set α)
             (Fintype.ofFinset _ fun _ => Iff.rfl) :=
-        @Fintype.card_le_of_injective (zpowers a)
+        (@Fintype.card_le_of_injective (zpowers a)
           (↑(univ.filterₓ fun b : α => b ^ orderOf a = 1) : Set α) (id _) (id _)
           (fun b =>
             ⟨b.1,
@@ -339,7 +339,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
                   let ⟨i, hi⟩ := b.2
                   rw [← hi, ← zpow_ofNat, ← zpow_mul, mul_comm, zpow_mul, zpow_ofNat,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
-          fun _ _ h => Subtype.eq (Subtype.mk.inj h)
+          fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filterₓ fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _
       )
 #align card_pow_eq_one_eq_order_of_aux card_pow_eq_one_eq_order_of_aux
@@ -409,7 +409,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
       · simp [h1]
       · simp [card_order_of_eq_totient_aux₁ hn hmc h1]
     _ < ∑ m in c.divisors, φ m :=
-      sum_erase_lt_of_pos (mem_divisors.2 ⟨hd, hc0.ne'⟩) (totient_pos (pos_of_dvd_of_pos hd hc0))
+      (sum_erase_lt_of_pos (mem_divisors.2 ⟨hd, hc0.ne'⟩) (totient_pos (pos_of_dvd_of_pos hd hc0)))
     _ = c := sum_totient _
     
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂

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

@@ -111,7 +111,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
     use x
     simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-    rw [← Fintype.card_congr (Equiv.Set.univ α), order_eq_card_zpowers] at hx
+    rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card is_add_cyclic_of_orderOf_eq_card
@@ -153,7 +153,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
-    rw [order_eq_card_zpowers]
+    rw [orderOf_eq_card_zpowers]
     apply Fintype.card_of_finset'
     simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
@@ -273,7 +273,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
           rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
           exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
-      rw [← order_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
+      rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       rw [hm]
       rw [Nat.mul_div_cancel_left _ (gcd_pos_of_pos_left _ hn0), gcd_mul_left_left, hm,
         Nat.mul_div_cancel _ hm0]
@@ -326,7 +326,7 @@ private theorem card_pow_eq_one_eq_order_of_aux (a : α) :
     (Finset.univ.filterₓ fun b : α => b ^ orderOf a = 1).card = orderOf a :=
   le_antisymm (hn _ (orderOf_pos a))
     (calc
-      orderOf a = @Fintype.card (zpowers a) (id _) := order_eq_card_zpowers
+      orderOf a = @Fintype.card (zpowers a) (id _) := orderOf_eq_card_zpowers
       _ ≤
           @Fintype.card (↑(univ.filterₓ fun b : α => b ^ orderOf a = 1) : Set α)
             (Fintype.ofFinset _ fun _ => Iff.rfl) :=

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

Try to have the date on the same line as the attribute: this makes it easier to rewrite these into machine-readable form. One date is was added: the deprecation was in the same PR as nearby lemmas.

@@ -118,7 +118,7 @@ theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →*
   rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul']
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
 #align monoid_add_hom.map_add_cyclic AddMonoidHom.map_addCyclic
-/- 2024-02-21 -/ @[deprecated] alias MonoidAddHom.map_add_cyclic := AddMonoidHom.map_addCyclic
+@[deprecated] alias MonoidAddHom.map_add_cyclic := AddMonoidHom.map_addCyclic -- 2024-02-21
 
 @[to_additive]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
@@ -130,8 +130,8 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_addOrderOf_eq_card
-/- 2024-02-21 -/ @[deprecated] alias isAddCyclic_of_orderOf_eq_card :=
-  isAddCyclic_of_addOrderOf_eq_card
+@[deprecated] -- 2024-02-21
+alias isAddCyclic_of_orderOf_eq_card := isAddCyclic_of_addOrderOf_eq_card
 
 /-- Any non-identity element of a finite group of prime order generates the group. -/
 @[to_additive "Any non-identity element of a finite group of prime order generates the group."]
@@ -319,8 +319,8 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)
 #align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le
 #align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_nsmul_eq_zero_le
-/- 2024-02-21 -/
-@[deprecated] alias IsAddCyclic.card_pow_eq_one_le := IsAddCyclic.card_nsmul_eq_zero_le
+@[deprecated] -- 2024-02-21
+alias IsAddCyclic.card_pow_eq_one_le := IsAddCyclic.card_nsmul_eq_zero_le
 
 end Classical
 
@@ -496,8 +496,8 @@ theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   isCyclic_of_orderOf_eq_card x (Finset.mem_filter.1 hx).2
 #align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_le
 
-/- 2024-02-21 -/
-@[deprecated] alias isAddCyclic_of_card_pow_eq_one_le := isAddCyclic_of_card_nsmul_eq_zero_le
+@[deprecated] -- 2024-02-21
+alias isAddCyclic_of_card_pow_eq_one_le := isAddCyclic_of_card_nsmul_eq_zero_le
 
 end Totient
 
@@ -507,8 +507,8 @@ theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 
-/- 2024-02-21 -/
-@[deprecated] alias IsAddCyclic.card_orderOf_eq_totient := IsAddCyclic.card_addOrderOf_eq_totient
+@[deprecated] -- 2024-02-21
+alias IsAddCyclic.card_orderOf_eq_totient := IsAddCyclic.card_addOrderOf_eq_totient
 
 /-- A finite group of prime order is simple. -/
 @[to_additive "A finite group of prime order is simple."]
@@ -563,7 +563,7 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
 #align commutative_of_add_cyclic_center_quotient commutative_of_addCyclic_center_quotient
 
-/- 2024-02-21 -/ @[deprecated]
+@[deprecated] -- 2024-02-21
 alias commutative_of_add_cyclic_center_quotient := commutative_of_addCyclic_center_quotient
 
 /-- A group is commutative if the quotient by the center is cyclic. -/

@@ -340,13 +340,18 @@ lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf
   exact Nat.card_congr (Equiv.Set.univ α)
 
 @[to_additive]
-lemma IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype [Fintype α] :
+lemma isCyclic_iff_exists_ofOrder_eq_natCard [Finite α] :
     IsCyclic α ↔ ∃ g : α, orderOf g = Nat.card α := by
   refine ⟨fun h ↦ h.exists_ofOrder_eq_natCard, fun h ↦ ?_⟩
   obtain ⟨g, hg⟩ := h
+  cases nonempty_fintype α
   refine isCyclic_of_orderOf_eq_card g ?_
   simp [hg]
 
+@[to_additive (attr := deprecated)] -- 2024-04-20
+protected alias IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype :=
+  isCyclic_iff_exists_ofOrder_eq_natCard
+
 section
 
 variable [DecidableEq α] [Fintype α]

Now that I am defining NNRat.cast, I want a definitive answer to this naming issue. Plenty of lemmas in mathlib already use natCast/intCast/ratCast over nat_cast/int_cast/rat_cast, and this matches with the general expectation that underscore-separated name parts correspond to a single declaration.

@@ -374,7 +374,7 @@ theorem IsCyclic.unique_zpow_zmod (ha : ∀ x : α, x ∈ zpowers a) (x : α) :
   · rw [← zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
       Int.modEq_comm, Int.modEq_iff_add_fac, ← ZMod.int_coe_zmod_eq_iff]
   · rw [← zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
-      ← ZMod.int_cast_eq_int_cast_iff] at hy
+      ← ZMod.intCast_eq_intCast_iff] at hy
     simp [hy]
 
 @[to_additive]
@@ -384,8 +384,8 @@ lemma IsCyclic.ext {G : Type*} [Group G] [Fintype G] [IsCyclic G] {d : ℕ} {a b
   specialize h g
   subst hGcard
   rw [pow_eq_pow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers hg,
-    ← ZMod.nat_cast_eq_nat_cast_iff] at h
-  simpa [ZMod.nat_cast_val, ZMod.cast_id'] using h
+    ← ZMod.natCast_eq_natCast_iff] at h
+  simpa [ZMod.natCast_val, ZMod.cast_id'] using h
 
 end
 
@@ -640,7 +640,7 @@ section SpecificInstances
 instance : IsAddCyclic ℤ := ⟨1, fun n ↦ ⟨n, by simp only [smul_eq_mul, mul_one]⟩⟩
 
 instance ZMod.instIsAddCyclic (n : ℕ) : IsAddCyclic (ZMod n) :=
-  isAddCyclic_of_surjective (Int.castRingHom _) ZMod.int_cast_surjective
+  isAddCyclic_of_surjective (Int.castRingHom _) ZMod.intCast_surjective
 
 instance ZMod.instIsSimpleAddGroup {p : ℕ} [Fact p.Prime] : IsSimpleAddGroup (ZMod p) :=
   AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card.2

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

@@ -337,7 +337,7 @@ lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf
   obtain ⟨g, hg⟩ := h.exists_generator
   use g
   rw [← card_zpowers g, (eq_top_iff' (zpowers g)).mpr hg]
-  exact (Nat.card_congr (Equiv.Set.univ α))
+  exact Nat.card_congr (Equiv.Set.univ α)
 
 @[to_additive]
 lemma IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype [Fintype α] :
@@ -752,7 +752,7 @@ noncomputable def zmodAddCyclicAddEquiv [AddGroup G] (h : IsAddCyclic G) :
     congr
     rw [← Nat.card_zmultiples]
     exact Nat.card_congr (Equiv.subtypeUnivEquiv surj)
-  exact (Int.quotientZMultiplesNatEquivZMod n)
+  exact Int.quotientZMultiplesNatEquivZMod n
     |>.symm.trans <| QuotientAddGroup.quotientAddEquivOfEq kereq
     |>.symm.trans <| QuotientAddGroup.quotientKerEquivOfSurjective (zmultiplesHom G g) surj
 

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -261,7 +261,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
           by_contradiction fun h =>
             Nat.find_min hex
               (Int.ofNat_lt.1 <| by
-                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
+                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.natCast_pos.2 (Nat.find_spec hex).1))
               ⟨Nat.pos_of_ne_zero h, hk₅⟩
         ⟨k / (Nat.find hex : ℤ),
           Subtype.ext_iff_val.2

Take the content of

• some of Algebra.BigOperators.List.Basic
• some of Algebra.BigOperators.List.Lemmas
• some of Algebra.BigOperators.Multiset.Basic
• some of Algebra.BigOperators.Multiset.Lemmas
• Algebra.BigOperators.Multiset.Order
• Algebra.BigOperators.Order

and sort it into six files:

• Algebra.Order.BigOperators.Group.List. I credit Yakov for https://github.com/leanprover-community/mathlib/pull/8543.
• Algebra.Order.BigOperators.Group.Multiset. Copyright inherited from Algebra.BigOperators.Multiset.Order.
• Algebra.Order.BigOperators.Group.Finset. Copyright inherited from Algebra.BigOperators.Order.
• Algebra.Order.BigOperators.Ring.List. I credit Stuart for https://github.com/leanprover-community/mathlib/pull/10184.
• Algebra.Order.BigOperators.Ring.Multiset. I credit Ruben for https://github.com/leanprover-community/mathlib/pull/8787.
• Algebra.Order.BigOperators.Ring.Finset. I credit Floris for https://github.com/leanprover-community/mathlib/pull/1294.

Here are the design decisions at play:

• Pure algebra and big operators algebra shouldn't import (algebraic) order theory. This PR makes that better, but not perfect because we still import Data.Nat.Order.Basic in a few List files.
• It's Algebra.Order.BigOperators instead of Algebra.BigOperators.Order because algebraic order theory is more of a theory than big operators algebra. Another reason is that algebraic order theory is the only way to mix pure order and pure algebra, while there are more ways to mix pure finiteness and pure algebra than just big operators.
• There are separate files for group/monoid lemmas vs ring lemmas. Groups/monoids are the natural setup for big operators, so their lemmas shouldn't be mixed with ring lemmas that involves both addition and multiplication. As a result, everything under Algebra.Order.BigOperators.Group should be additivisable (except a few Nat- or Int-specific lemmas). In contrast, things under Algebra.Order.BigOperators.Ring are more prone to having heavy imports.
• Lemmas are separated according to List vs Multiset vs Finset. This is not strictly necessary, and can be relaxed in cases where there aren't that many lemmas to be had. As an example, I could split out the AbsoluteValue lemmas from Algebra.Order.BigOperators.Ring.Finset to a file Algebra.Order.BigOperators.Ring.AbsoluteValue and it could stay this way until too many lemmas are in this file (or a split is needed for import reasons), in which case we would need files Algebra.Order.BigOperators.Ring.AbsoluteValue.Finset, Algebra.Order.BigOperators.Ring.AbsoluteValue.Multiset, etc...
• Finsupp big operator and finprod/finsum order lemmas also belong in Algebra.Order.BigOperators. I haven't done so in this PR because the diff is big enough like that.

@@ -3,7 +3,7 @@ Copyright (c) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Mathlib.Algebra.BigOperators.Order
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
 import Mathlib.Data.Nat.Totient
 import Mathlib.GroupTheory.OrderOfElement
 import Mathlib.GroupTheory.Subgroup.Simple

Reduce the diff of #11499

Renames

All in the Int namespace:

• ofNat_eq_cast → ofNat_eq_natCast
• cast_eq_cast_iff_Nat → natCast_inj
• natCast_eq_ofNat → ofNat_eq_natCast
• coe_nat_sub → natCast_sub
• coe_nat_nonneg → natCast_nonneg
• sign_coe_add_one → sign_natCast_add_one
• nat_succ_eq_int_succ → natCast_succ
• succ_neg_nat_succ → succ_neg_natCast_succ
• coe_pred_of_pos → natCast_pred_of_pos
• coe_nat_div → natCast_div
• coe_nat_ediv → natCast_ediv
• sign_coe_nat_of_nonzero → sign_natCast_of_ne_zero
• toNat_coe_nat → toNat_natCast
• toNat_coe_nat_add_one → toNat_natCast_add_one
• coe_nat_dvd → natCast_dvd_natCast
• coe_nat_dvd_left → natCast_dvd
• coe_nat_dvd_right → dvd_natCast
• le_coe_nat_sub → le_natCast_sub
• succ_coe_nat_pos → succ_natCast_pos
• coe_nat_modEq_iff → natCast_modEq_iff
• coe_natAbs → natCast_natAbs
• coe_nat_eq_zero → natCast_eq_zero
• coe_nat_ne_zero → natCast_ne_zero
• coe_nat_ne_zero_iff_pos → natCast_ne_zero_iff_pos
• abs_coe_nat → abs_natCast
• coe_nat_nonpos_iff → natCast_nonpos_iff

Also rename Nat.coe_nat_dvd to Nat.cast_dvd_cast

@@ -255,7 +255,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
             rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
         have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
           rw [Int.natAbs_of_nonneg
-              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
+              (Int.emod_nonneg _ (Int.natCast_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
         have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_natCast, ← hk₄]
         have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
           by_contradiction fun h =>

Reconstitute the file Algebra.Order.Monoid.WithZero from three files:

• Algebra.Order.Monoid.WithZero.Defs
• Algebra.Order.Monoid.WithZero.Basic
• Algebra.Order.WithZero

Avoid importing it in many files. Most uses were just to get le_zero_iff to work on Nat.

Before

After

@@ -454,7 +454,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   by_contra h0
   -- Must qualify `Finset.card_eq_zero` because of leanprover/lean4#2849
   -- Must specify the argument `α` to avoid mathlib4#10830
-  simp_rw [not_lt, _root_.le_zero_iff (α := ℕ), Finset.card_eq_zero] at h0
+  simp_rw [not_lt, Nat.le_zero, Finset.card_eq_zero] at h0
   apply lt_irrefl c
   calc
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card := by

We add IsPrimitiveRoot.exists_pow_or_neg_mul_pow_of_isOfFinOrder: If x is a root of unity (spelled as IsOfFinOrder x) in an n-th cyclotomic extension of ℚ, where n is odd, and ζ is a primitive n-th root of unity, then there exists r < n such that x = ζ^r or x = -ζ^r.

From flt-regular

• depends on: #11235

@@ -332,6 +332,21 @@ theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
 #align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generator
 #align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_addMonoid_generator
 
+@[to_additive]
+lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf g = Nat.card α := by
+  obtain ⟨g, hg⟩ := h.exists_generator
+  use g
+  rw [← card_zpowers g, (eq_top_iff' (zpowers g)).mpr hg]
+  exact (Nat.card_congr (Equiv.Set.univ α))
+
+@[to_additive]
+lemma IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype [Fintype α] :
+    IsCyclic α ↔ ∃ g : α, orderOf g = Nat.card α := by
+  refine ⟨fun h ↦ h.exists_ofOrder_eq_natCard, fun h ↦ ?_⟩
+  obtain ⟨g, hg⟩ := h
+  refine isCyclic_of_orderOf_eq_card g ?_
+  simp [hg]
+
 section
 
 variable [DecidableEq α] [Fintype α]

... and add a deprecated alias for the old name. This is mostly just me discovering the power of F2

@@ -217,7 +217,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     rw [← zpow_mod_orderOf] at hk
     have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (mod_cast ho.orderOf_pos.ne')
     refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩
-    rwa [← zpow_coe_nat, Int.toNat_of_nonneg this]
+    rwa [← zpow_natCast, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
@@ -240,7 +240,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         Nat.pos_of_ne_zero fun h => hx₂ <| by
           rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
             cases' k with k k
-            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_coe_nat, ← Int.ofNat_eq_coe, hk]
+            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_natCast, ← Int.ofNat_eq_coe, hk]
               exact hx₁
             · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
@@ -256,7 +256,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
           rw [Int.natAbs_of_nonneg
               (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
-        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_coe_nat, ← hk₄]
+        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_natCast, ← hk₄]
         have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
           by_contradiction fun h =>
             Nat.find_min hex
@@ -299,7 +299,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
             have hgmn : g ^ (m * Nat.gcd n (Fintype.card α)) = 1 := by
               rw [pow_mul, hm, ← pow_gcd_card_eq_one_iff]; exact (mem_filter.1 hx).2
             dsimp only
-            rw [zpow_coe_nat, ← pow_mul, Nat.mul_div_cancel_left', hm]
+            rw [zpow_natCast, ← pow_mul, Nat.mul_div_cancel_left', hm]
             refine' Nat.dvd_of_mul_dvd_mul_right (gcd_pos_of_pos_left (Fintype.card α) hn0) _
             conv_lhs =>
               rw [Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), ←
@@ -356,9 +356,9 @@ theorem IsCyclic.unique_zpow_zmod (ha : ∀ x : α, x ∈ zpowers a) (x : α) :
     ∃! n : ZMod (Fintype.card α), x = a ^ n.val := by
   obtain ⟨n, rfl⟩ := ha x
   refine ⟨n, (?_ : a ^ n = _), fun y (hy : a ^ n = _) ↦ ?_⟩
-  · rw [← zpow_coe_nat, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
+  · rw [← zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
       Int.modEq_comm, Int.modEq_iff_add_fac, ← ZMod.int_coe_zmod_eq_iff]
-  · rw [← zpow_coe_nat, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
+  · rw [← zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
       ← ZMod.int_cast_eq_int_cast_iff] at hy
     simp [hy]
 
@@ -395,7 +395,7 @@ private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
               mem_filter.2
                 ⟨mem_univ _, by
                   let ⟨i, hi⟩ := b.2
-                  rw [← hi, ← zpow_coe_nat, ← zpow_mul, mul_comm, zpow_mul, zpow_coe_nat,
+                  rw [← hi, ← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, zpow_natCast,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filter fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _

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

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

@@ -282,7 +282,7 @@ open Finset Nat
 
 section Classical
 
-open Classical
+open scoped Classical
 
 @[to_additive IsAddCyclic.card_nsmul_eq_zero_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :

Suggested in #6342 .

@@ -351,6 +351,27 @@ theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
 #align is_cyclic.image_range_card IsCyclic.image_range_card
 #align is_add_cyclic.image_range_card IsAddCyclic.image_range_card
 
+@[to_additive]
+theorem IsCyclic.unique_zpow_zmod (ha : ∀ x : α, x ∈ zpowers a) (x : α) :
+    ∃! n : ZMod (Fintype.card α), x = a ^ n.val := by
+  obtain ⟨n, rfl⟩ := ha x
+  refine ⟨n, (?_ : a ^ n = _), fun y (hy : a ^ n = _) ↦ ?_⟩
+  · rw [← zpow_coe_nat, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
+      Int.modEq_comm, Int.modEq_iff_add_fac, ← ZMod.int_coe_zmod_eq_iff]
+  · rw [← zpow_coe_nat, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
+      ← ZMod.int_cast_eq_int_cast_iff] at hy
+    simp [hy]
+
+@[to_additive]
+lemma IsCyclic.ext {G : Type*} [Group G] [Fintype G] [IsCyclic G] {d : ℕ} {a b : ZMod d}
+    (hGcard : Fintype.card G = d) (h : ∀ t : G, t ^ a.val = t ^ b.val) : a = b := by
+  obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := G)
+  specialize h g
+  subst hGcard
+  rw [pow_eq_pow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers hg,
+    ← ZMod.nat_cast_eq_nat_cast_iff] at h
+  simpa [ZMod.nat_cast_val, ZMod.cast_id'] using h
+
 end
 
 section Totient

• Remove manual translations that are now guessed correctly
• Fix some names that were incorrectly guessed by humans (and in one case fix the multiplicative name). Add deprecations for all name changes.
• Remove a couple manually additivized lemmas.

@@ -108,7 +108,7 @@ theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α := by
   contrapose! nc
   exact @isCyclic_of_subsingleton _ _ (not_nontrivial_iff_subsingleton.mp nc)
 
-@[to_additive MonoidAddHom.map_add_cyclic]
+@[to_additive]
 theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m := by
   obtain ⟨h, hG⟩ := IsCyclic.exists_generator (α := G)
@@ -117,9 +117,10 @@ theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →*
   obtain ⟨n, rfl⟩ := hG g
   rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul']
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
-#align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
+#align monoid_add_hom.map_add_cyclic AddMonoidHom.map_addCyclic
+/- 2024-02-21 -/ @[deprecated] alias MonoidAddHom.map_add_cyclic := AddMonoidHom.map_addCyclic
 
-@[to_additive isAddCyclic_of_orderOf_eq_card]
+@[to_additive]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
   classical
@@ -128,11 +129,12 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
     rw [← Fintype.card_congr (Equiv.Set.univ α), ← Fintype.card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
-#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
+#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_addOrderOf_eq_card
+/- 2024-02-21 -/ @[deprecated] alias isAddCyclic_of_orderOf_eq_card :=
+  isAddCyclic_of_addOrderOf_eq_card
 
 /-- Any non-identity element of a finite group of prime order generates the group. -/
-@[to_additive zmultiples_eq_top_of_prime_card "Any non-identity element of a finite group of prime
-  order generates the group."]
+@[to_additive "Any non-identity element of a finite group of prime order generates the group."]
 theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
     [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : zpowers g = ⊤ := by
   have := card_subgroup_dvd_card (zpowers g)
@@ -140,18 +142,18 @@ theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p :
     card_eq_iff_eq_top] at this
   exact this
 
-@[to_additive mem_zmultiples_of_prime_card]
+@[to_additive]
 theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g := by
   simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]
 
-@[to_additive mem_multiples_of_prime_card]
+@[to_additive]
 theorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g := by
   rw [mem_powers_iff_mem_zpowers]
   exact mem_zpowers_of_prime_card h hg
 
-@[to_additive multiples_eq_top_of_prime_card]
+@[to_additive]
 theorem powers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
     [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : Submonoid.powers g = ⊤ := by
   ext x
@@ -176,7 +178,7 @@ theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCycli
   obtain ⟨n, rfl⟩ := hx a
   exact ⟨n, (map_zpow _ _ _).symm⟩
 
-@[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
+@[to_additive]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
@@ -196,7 +198,7 @@ theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cycli
   rw [← Fintype.card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
   exact fun x _ ↦ ha x
 
-@[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
+@[to_additive]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by
   classical
@@ -282,7 +284,7 @@ section Classical
 
 open Classical
 
-@[to_additive IsAddCyclic.card_pow_eq_one_le]
+@[to_additive IsAddCyclic.card_nsmul_eq_zero_le]
 theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α] {n : ℕ} (hn0 : 0 < n) :
     (univ.filter fun a : α => a ^ n = 1).card ≤ n :=
   let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
@@ -316,7 +318,9 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
         Nat.mul_div_cancel _ hm0]
       exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)
 #align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le
-#align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_pow_eq_one_le
+#align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_nsmul_eq_zero_le
+/- 2024-02-21 -/
+@[deprecated] alias IsAddCyclic.card_pow_eq_one_le := IsAddCyclic.card_nsmul_eq_zero_le
 
 end Classical
 
@@ -354,6 +358,7 @@ section Totient
 variable [DecidableEq α] [Fintype α]
   (hn : ∀ n : ℕ, 0 < n → (univ.filter fun a : α => a ^ n = 1).card ≤ n)
 
+@[to_additive]
 private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
     (Finset.univ.filter fun b : α => b ^ orderOf a = 1).card = orderOf a :=
   le_antisymm (hn _ (orderOf_pos a))
@@ -377,6 +382,7 @@ private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
 
 -- Use φ for `Nat.totient`
 open Nat
+@[to_additive]
 private theorem card_orderOf_eq_totient_aux₁ :
     ∀ {d : ℕ},
       d ∣ Fintype.card α →
@@ -403,6 +409,7 @@ private theorem card_orderOf_eq_totient_aux₁ :
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_orderOf_aux hn a]
   simpa [← cons_self_properDivisors hd0, ← h1] using h2
 
+@[to_additive]
 theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filter fun a : α => orderOf a = d).card = φ d := by
   let c := Fintype.card α
@@ -410,7 +417,8 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   apply card_orderOf_eq_totient_aux₁ hn hd
   by_contra h0
   -- Must qualify `Finset.card_eq_zero` because of leanprover/lean4#2849
-  simp only [not_lt, _root_.le_zero_iff, Finset.card_eq_zero] at h0
+  -- Must specify the argument `α` to avoid mathlib4#10830
+  simp_rw [not_lt, _root_.le_zero_iff (α := ℕ), Finset.card_eq_zero] at h0
   apply lt_irrefl c
   calc
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card := by
@@ -437,6 +445,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     _ = c := sum_totient _
 #align card_order_of_eq_totient_aux₂ card_orderOf_eq_totient_aux₂
 
+@[to_additive isAddCyclic_of_card_nsmul_eq_zero_le]
 theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   have : (univ.filter fun a : α => orderOf a = Fintype.card α).Nonempty :=
     card_pos.1 <| by
@@ -446,30 +455,19 @@ theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
   isCyclic_of_orderOf_eq_card x (Finset.mem_filter.1 hx).2
 #align is_cyclic_of_card_pow_eq_one_le isCyclic_of_card_pow_eq_one_le
 
-theorem isAddCyclic_of_card_pow_eq_one_le {α} [AddGroup α] [DecidableEq α] [Fintype α]
-    (hn : ∀ n : ℕ, 0 < n → (univ.filter fun a : α => n • a = 0).card ≤ n) : IsAddCyclic α := by
-  obtain ⟨g, hg⟩ := (@isCyclic_of_card_pow_eq_one_le (Multiplicative α) _ _ (_) hn)
-  exact ⟨⟨g, hg⟩⟩
-#align is_add_cyclic_of_card_pow_eq_one_le isAddCyclic_of_card_pow_eq_one_le
-
-attribute [to_additive existing isCyclic_of_card_pow_eq_one_le] isAddCyclic_of_card_pow_eq_one_le
+/- 2024-02-21 -/
+@[deprecated] alias isAddCyclic_of_card_pow_eq_one_le := isAddCyclic_of_card_nsmul_eq_zero_le
 
 end Totient
 
+@[to_additive]
 theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
     (hd : d ∣ Fintype.card α) : (univ.filter fun a : α => orderOf a = d).card = totient d := by
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 
-theorem IsAddCyclic.card_orderOf_eq_totient {α : Type*} [AddGroup α] [IsAddCyclic α]
-    [Fintype α] {d : ℕ} (hd : d ∣ Fintype.card α) :
-    (univ.filter fun a : α => addOrderOf a = d).card = totient d := by
-  obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›
-  apply @IsCyclic.card_orderOf_eq_totient (Multiplicative α) _ ⟨⟨g, hg⟩⟩ (_) _ hd
-#align is_add_cyclic.card_order_of_eq_totient IsAddCyclic.card_orderOf_eq_totient
-
-attribute [to_additive existing IsCyclic.card_orderOf_eq_totient]
-  IsAddCyclic.card_orderOf_eq_totient
+/- 2024-02-21 -/
+@[deprecated] alias IsAddCyclic.card_orderOf_eq_totient := IsAddCyclic.card_addOrderOf_eq_totient
 
 /-- A finite group of prime order is simple. -/
 @[to_additive "A finite group of prime order is simple."]
@@ -500,7 +498,7 @@ variable {G : Type*} {H : Type*} [Group G] [Group H]
 
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `commGroup_of_cycle_center_quotient` for the `CommGroup` instance. -/
-@[to_additive commutative_of_add_cyclic_center_quotient
+@[to_additive
       "A group is commutative if the quotient by the center is cyclic.
       Also see `addCommGroup_of_cycle_center_quotient` for the `AddCommGroup` instance."]
 theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G)
@@ -522,7 +520,10 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
     _ = y ^ m * y ^ n * y ^ (-m) * (y ^ (-n) * b * a) := by rw [mem_center_iff.1 hb]
     _ = b * a := by group
 #align commutative_of_cyclic_center_quotient commutative_of_cyclic_center_quotient
-#align commutative_of_add_cyclic_center_quotient commutative_of_add_cyclic_center_quotient
+#align commutative_of_add_cyclic_center_quotient commutative_of_addCyclic_center_quotient
+
+/- 2024-02-21 -/ @[deprecated]
+alias commutative_of_add_cyclic_center_quotient := commutative_of_addCyclic_center_quotient
 
 /-- A group is commutative if the quotient by the center is cyclic. -/
 @[to_additive commutativeOfAddCycleCenterQuotient

Previously these were syntactically identical to the corresponding zpow_coe_nat and coe_nat_zsmul lemmas, now they are about OfNat.ofNat.

Unfortunately, almost every call site uses the ofNat name to refer to Nat.cast, so the downstream proofs had to be adjusted too.

@@ -215,7 +215,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     rw [← zpow_mod_orderOf] at hk
     have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (mod_cast ho.orderOf_pos.ne')
     refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩
-    rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
+    rwa [← zpow_coe_nat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
@@ -238,7 +238,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         Nat.pos_of_ne_zero fun h => hx₂ <| by
           rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
             cases' k with k k
-            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
+            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_coe_nat, ← Int.ofNat_eq_coe, hk]
               exact hx₁
             · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
@@ -254,7 +254,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
           rw [Int.natAbs_of_nonneg
               (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
-        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
+        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_coe_nat, ← hk₄]
         have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
           by_contradiction fun h =>
             Nat.find_min hex
@@ -297,7 +297,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
             have hgmn : g ^ (m * Nat.gcd n (Fintype.card α)) = 1 := by
               rw [pow_mul, hm, ← pow_gcd_card_eq_one_iff]; exact (mem_filter.1 hx).2
             dsimp only
-            rw [zpow_ofNat, ← pow_mul, Nat.mul_div_cancel_left', hm]
+            rw [zpow_coe_nat, ← pow_mul, Nat.mul_div_cancel_left', hm]
             refine' Nat.dvd_of_mul_dvd_mul_right (gcd_pos_of_pos_left (Fintype.card α) hn0) _
             conv_lhs =>
               rw [Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), ←
@@ -369,7 +369,7 @@ private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
               mem_filter.2
                 ⟨mem_univ _, by
                   let ⟨i, hi⟩ := b.2
-                  rw [← hi, ← zpow_ofNat, ← zpow_mul, mul_comm, zpow_mul, zpow_ofNat,
+                  rw [← hi, ← zpow_coe_nat, ← zpow_mul, mul_comm, zpow_mul, zpow_coe_nat,
                     pow_orderOf_eq_one, one_zpow]⟩⟩)
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filter fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _

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

@@ -416,7 +416,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card := by
       simp only [← filter_dvd_eq_divisors hc0.ne', sum_card_orderOf_eq_card_pow_eq_one hc0.ne']
       apply congr_arg card
-      simp
+      simp [c]
     _ = ∑ m in c.divisors.erase d, (univ.filter fun a : α => orderOf a = m).card := by
       rw [eq_comm]
       refine' sum_subset (erase_subset _ _) fun m hm₁ hm₂ => _

• Add a lemma stating that non-identity element of a finite group of prime order generates the group (stated in 4 ways)
• Use this to golf isCyclic_of_prime_card
• Add a version of isUnit_iff_exists_inv for non-commutative monoids.

@@ -130,35 +130,39 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 
+/-- Any non-identity element of a finite group of prime order generates the group. -/
+@[to_additive zmultiples_eq_top_of_prime_card "Any non-identity element of a finite group of prime
+  order generates the group."]
+theorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
+    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : zpowers g = ⊤ := by
+  have := card_subgroup_dvd_card (zpowers g)
+  simp_rw [h, Nat.dvd_prime hp.1, ← eq_bot_iff_card, zpowers_eq_bot, hg, false_or, ← h,
+    card_eq_iff_eq_top] at this
+  exact this
+
+@[to_additive mem_zmultiples_of_prime_card]
+theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
+    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g := by
+  simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]
+
+@[to_additive mem_multiples_of_prime_card]
+theorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
+    (h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g := by
+  rw [mem_powers_iff_mem_zpowers]
+  exact mem_zpowers_of_prime_card h hg
+
+@[to_additive multiples_eq_top_of_prime_card]
+theorem powers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ}
+    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g ≠ 1) : Submonoid.powers g = ⊤ := by
+  ext x
+  simp [mem_powers_of_prime_card h hg]
+
 /-- A finite group of prime order is cyclic. -/
 @[to_additive "A finite group of prime order is cyclic."]
 theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
-    (h : Fintype.card α = p) : IsCyclic α :=
-  ⟨by
-    obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
-    classical
-      -- for Fintype (Subgroup.zpowers g)
-      have : Fintype.card (Subgroup.zpowers g) ∣ p := by
-        rw [← h]
-        apply card_subgroup_dvd_card
-      rw [Nat.dvd_prime hp.1] at this
-      cases' this with that that
-      · rw [Fintype.card_eq_one_iff] at that
-        cases' that with t ht
-        suffices g = 1 by contradiction
-        have hgt :=
-          ht
-            ⟨g, by
-              change g ∈ Subgroup.zpowers g
-              exact Subgroup.mem_zpowers g⟩
-        rw [← ht 1] at hgt
-        change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt
-        simpa using hgt
-      · use g
-        intro x
-        rw [← h] at that
-        rw [Subgroup.eq_top_of_card_eq _ that]
-        exact Subgroup.mem_top _⟩
+    (h : Fintype.card α = p) : IsCyclic α := by
+  obtain ⟨g, hg⟩ : ∃ g, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
+  exact ⟨g, fun g' ↦ mem_zpowers_of_prime_card h hg⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 

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

@@ -616,7 +616,7 @@ theorem IsCyclic.exponent_eq_card [Group α] [IsCyclic α] [Fintype α] :
     exponent α = Fintype.card α := by
   obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
   apply Nat.dvd_antisymm
-  · rw [← lcm_order_eq_exponent, Finset.lcm_dvd_iff]
+  · rw [← lcm_orderOf_eq_exponent, Finset.lcm_dvd_iff]
     exact fun b _ => orderOf_dvd_card
   rw [← orderOf_eq_card_of_forall_mem_zpowers hg]
   exact order_dvd_exponent _

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Johan Commelin <johan@commelin.net>

@@ -681,3 +681,65 @@ lemma not_isCyclic_iff_exponent_eq_prime [Group α] {p : ℕ} (hp : p.Prime)
   · exact False.elim <| h_cyc <| isCyclic_of_orderOf_eq_card g <| by aesop
 
 end Exponent
+
+section ZMod
+
+open Subgroup AddSubgroup
+
+variable {G H : Type*}
+
+/-- The kernel of `zmultiplesHom G g` is equal to the additive subgroup generated by
+    `addOrderOf g`. -/
+theorem zmultiplesHom_ker_eq [AddGroup G] (g : G) :
+    (zmultiplesHom G g).ker = zmultiples ↑(addOrderOf g) := by
+  ext
+  simp_rw [AddMonoidHom.mem_ker, mem_zmultiples_iff, zmultiplesHom_apply,
+    ← addOrderOf_dvd_iff_zsmul_eq_zero, zsmul_eq_mul', Int.cast_id, dvd_def, eq_comm]
+
+/-- The kernel of `zpowersHom G g` is equal to the subgroup generated by `orderOf g`. -/
+theorem zpowersHom_ker_eq [Group G] (g : G) :
+    (zpowersHom G g).ker = zpowers (Multiplicative.ofAdd ↑(orderOf g)) :=
+  congr_arg AddSubgroup.toSubgroup <| zmultiplesHom_ker_eq (Additive.ofMul g)
+
+/-- The isomorphism from `ZMod n` to any cyclic additive group of `Nat.card` equal to `n`. -/
+noncomputable def zmodAddCyclicAddEquiv [AddGroup G] (h : IsAddCyclic G) :
+    ZMod (Nat.card G) ≃+ G := by
+  let n := Nat.card G
+  let ⟨g, surj⟩ := Classical.indefiniteDescription _ h.exists_generator
+  have kereq : ((zmultiplesHom G) g).ker = zmultiples ↑(Nat.card G) := by
+    rw [zmultiplesHom_ker_eq]
+    congr
+    rw [← Nat.card_zmultiples]
+    exact Nat.card_congr (Equiv.subtypeUnivEquiv surj)
+  exact (Int.quotientZMultiplesNatEquivZMod n)
+    |>.symm.trans <| QuotientAddGroup.quotientAddEquivOfEq kereq
+    |>.symm.trans <| QuotientAddGroup.quotientKerEquivOfSurjective (zmultiplesHom G g) surj
+
+/-- The isomorphism from `Multiplicative (ZMod n)` to any cyclic group of `Nat.card` equal to `n`.
+-/
+noncomputable def zmodCyclicMulEquiv [Group G] (h : IsCyclic G) :
+    Multiplicative (ZMod (Nat.card G)) ≃* G :=
+  AddEquiv.toMultiplicative <| zmodAddCyclicAddEquiv <| isAddCyclic_additive_iff.2 h
+
+/-- Two cyclic additive groups of the same cardinality are isomorphic. -/
+noncomputable def addEquivOfAddCyclicCardEq [AddGroup G] [AddGroup H] [hG : IsAddCyclic G]
+    [hH : IsAddCyclic H] (hcard : Nat.card G = Nat.card H) : G ≃+ H := hcard ▸
+  zmodAddCyclicAddEquiv hG |>.symm.trans (zmodAddCyclicAddEquiv hH)
+
+/-- Two cyclic groups of the same cardinality are isomorphic. -/
+@[to_additive existing]
+noncomputable def mulEquivOfCyclicCardEq [Group G] [Group H] [hG : IsCyclic G]
+    [hH : IsCyclic H] (hcard : Nat.card G = Nat.card H) : G ≃* H := hcard ▸
+  zmodCyclicMulEquiv hG |>.symm.trans (zmodCyclicMulEquiv hH)
+
+/-- Two groups of the same prime cardinality are isomorphic. -/
+@[to_additive "Two additive groups of the same prime cardinality are isomorphic."]
+noncomputable def mulEquivOfPrimeCardEq {p : ℕ} [Fintype G] [Fintype H] [Group G] [Group H]
+    [Fact p.Prime] (hG : Fintype.card G = p) (hH : Fintype.card H = p) : G ≃* H := by
+  have hGcyc := isCyclic_of_prime_card hG
+  have hHcyc := isCyclic_of_prime_card hH
+  apply mulEquivOfCyclicCardEq
+  rw [← Nat.card_eq_fintype_card] at hG hH
+  exact hG.trans hH.symm
+
+end ZMod

The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike is EquivLike, since that has a custom coe_injective' field that is easier to implement. All other classes should take FunLike or EquivLike as a parameter.

Zulip thread

Important changes

Previously, morphism classes would be Type-valued and extend FunLike:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  extends FunLike F A B :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

After this PR, they should be Prop-valued and take FunLike as a parameter:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  [FunLike F A B] : Prop :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

(Note that A B stay marked as outParam even though they are not purely required to be so due to the FunLike parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam is slightly faster.)

Similarly, MyEquivClass should take EquivLike as a parameter.

As a result, every mention of [MyHomClass F A B] should become [FunLike F A B] [MyHomClass F A B].

Remaining issues

Slower (failing) search

While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul is more expensive. This is due to suboptimal processing of arguments. For example:

variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)

theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y

example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _

Before this PR, applying map_mul f gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Since M and N are out_params, [MulHomClass F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found.

After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Now [FunLike F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found, before trying MulHomClass F M N which fails. Since the Mul hierarchy is very big, this can be slow to fail, especially when there is no such Mul instance.

A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N] because MulHomClass fails or succeeds much faster than the others.

As a consequence, the simpNF linter is much slower since by design it tries and fails to apply many map_ lemmas. The same issue occurs a few times in existing calls to simp [map_mul], where map_mul is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.

simp not firing sometimes

This affects map_smulₛₗ and related definitions. For simp lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw can find every argument to map_smulₛₗ successfully but simp can't: leanprover/lean4#3701.

Missing instances due to unification failing

Especially in the category theory library, we might sometimes have a type A which is also accessible as a synonym (Bundled A hA).1. Instance synthesis doesn't always work if we have f : A →* B but x * y : (Bundled A hA).1 or vice versa. This seems to be mostly fixed by keeping A B as outParams in MulHomClass F A B. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1 instead of using the syntax in the discrimination tree.)

Workaround for issues

The timeouts can be worked around for now by specifying which map_mul we mean, either as map_mul f for some explicit f, or as e.g. MonoidHomClass.map_mul.

map_smulₛₗ not firing as simp lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulₛₗ a simp lemma instead of the generic map_smulₛₗ. Writing simp [map_smulₛₗ _] also works.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -164,7 +164,8 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 
 @[to_additive]
 theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
-    [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) : IsCyclic G := by
+    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) :
+    IsCyclic G := by
   obtain ⟨x, hx⟩ := hH
   refine ⟨f x, fun a ↦ ?_⟩
   obtain ⟨a, rfl⟩ := hf a

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -670,8 +670,8 @@ lemma not_isCyclic_iff_exponent_eq_prime [Group α] {p : ℕ} (hp : p.Prime)
   orders of `g` are `1`, `p`, or `p ^ 2`. It can't be the former because `g ≠ 1`, and it can't
   the latter because the group isn't cyclic. -/
   have := (Nat.mem_divisors (m := p ^ 2)).mpr ⟨hα' ▸ orderOf_dvd_card (x := g), by aesop⟩
-  simp? [Nat.divisors_prime_pow hp 2] at this says
-    simp only [Nat.divisors_prime_pow hp 2, Finset.mem_map, Finset.mem_range,
+  simp? [Nat.divisors_prime_pow hp 2]  at this says
+    simp only [Nat.divisors_prime_pow hp 2, Nat.reduceAdd, Finset.mem_map, Finset.mem_range,
       Function.Embedding.coeFn_mk] at this
   obtain ⟨a, ha, ha'⟩ := this
   interval_cases a

Change a few lemma names that have historically bothered me.

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

@@ -284,7 +284,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
   calc
     (univ.filter fun a : α => a ^ n = 1).card ≤
         (zpowers (g ^ (Fintype.card α / Nat.gcd n (Fintype.card α))) : Set α).toFinset.card :=
-      card_le_of_subset fun x hx =>
+      card_le_card fun x hx =>
         let ⟨m, hm⟩ := show x ∈ Submonoid.powers g from mem_powers_iff_mem_zpowers.2 <| hg x
         Set.mem_toFinset.2
           ⟨(m / (Fintype.card α / Nat.gcd n (Fintype.card α)) : ℕ), by

This PR shows a few facts related to Monoid.exponent, especially when it is prime:

1. A nontrivial finite cancellative monoid has exponent greater than 1.
2. A nontrivial monoid has prime exponent p if and only if every non-identity element has order p.
3. A group of order p ^ 2 with p prime is not cyclic if and only if it has exponent p.

@@ -651,4 +651,32 @@ protected theorem ZMod.exponent (n : ℕ) : AddMonoid.exponent (ZMod n) = n := b
   · rw [IsAddCyclic.exponent_eq_zero_of_infinite]
   · rw [IsAddCyclic.exponent_eq_card, card]
 
+/-- A group of order `p ^ 2` is not cyclic if and only if its exponent is `p`. -/
+@[to_additive]
+lemma not_isCyclic_iff_exponent_eq_prime [Group α] {p : ℕ} (hp : p.Prime)
+    (hα : Nat.card α = p ^ 2) : ¬ IsCyclic α ↔ Monoid.exponent α = p := by
+  -- G is a nontrivial fintype of cardinality `p ^ 2`
+  let _inst : Fintype α := @Fintype.ofFinite α <| Nat.finite_of_card_ne_zero <| by aesop
+  have hα' : Fintype.card α = p ^ 2 := by simpa using hα
+  have := (Fintype.one_lt_card_iff_nontrivial (α := α)).mp <|
+    hα' ▸ one_lt_pow hp.one_lt two_ne_zero
+  /- in the forward direction, we apply `exponent_eq_prime_iff`, and the reverse direction follows
+  immediately because if `α` has exponent `p`, it has no element of order `p ^ 2`. -/
+  refine ⟨fun h_cyc ↦ (Monoid.exponent_eq_prime_iff hp).mpr fun g hg ↦ ?_, fun h_exp h_cyc ↦ by
+    obtain (rfl|rfl) := eq_zero_or_one_of_sq_eq_self <| hα' ▸ h_exp ▸ (h_cyc.exponent_eq_card).symm
+    · exact Nat.not_prime_zero hp
+    · exact Nat.not_prime_one hp⟩
+  /- we must show every non-identity element has order `p`. By Lagrange's theorem, the only possible
+  orders of `g` are `1`, `p`, or `p ^ 2`. It can't be the former because `g ≠ 1`, and it can't
+  the latter because the group isn't cyclic. -/
+  have := (Nat.mem_divisors (m := p ^ 2)).mpr ⟨hα' ▸ orderOf_dvd_card (x := g), by aesop⟩
+  simp? [Nat.divisors_prime_pow hp 2] at this says
+    simp only [Nat.divisors_prime_pow hp 2, Finset.mem_map, Finset.mem_range,
+      Function.Embedding.coeFn_mk] at this
+  obtain ⟨a, ha, ha'⟩ := this
+  interval_cases a
+  · exact False.elim <| hg <| orderOf_eq_one_iff.mp <| by aesop
+  · aesop
+  · exact False.elim <| h_cyc <| isCyclic_of_orderOf_eq_card g <| by aesop
+
 end Exponent

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

@@ -125,7 +125,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
     use x
     simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-    rw [← Fintype.card_congr (Equiv.Set.univ α), ←Fintype.card_zpowers] at hx
+    rw [← Fintype.card_congr (Equiv.Set.univ α), ← Fintype.card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
@@ -175,7 +175,7 @@ theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCycli
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
-    rw [←Fintype.card_zpowers]
+    rw [← Fintype.card_zpowers]
     apply Fintype.card_of_finset'
     simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
@@ -188,7 +188,7 @@ theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cycli
   use a
   contrapose! k_lt_card_G
   convert orderOf_le_of_pow_eq_one k_pos.bot_lt k_lt_card_G
-  rw [←Fintype.card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
+  rw [← Fintype.card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
   exact fun x _ ↦ ha x
 
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
@@ -202,12 +202,12 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
       Infinite.exists_not_mem_finset
         (Finset.image (fun x => g ^ x) <| Finset.range <| orderOf g)
     apply hx
-    rw [←ho.mem_powers_iff_mem_range_orderOf, Submonoid.mem_powers_iff]
+    rw [← ho.mem_powers_iff_mem_range_orderOf, Submonoid.mem_powers_iff]
     obtain ⟨k, hk⟩ := h x
     dsimp at hk
     obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg
     · exact ⟨k, mod_cast hk⟩
-    rw [←zpow_mod_orderOf] at hk
+    rw [← zpow_mod_orderOf] at hk
     have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (mod_cast ho.orderOf_pos.ne')
     refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩
     rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
@@ -233,7 +233,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         Nat.pos_of_ne_zero fun h => hx₂ <| by
           rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
             cases' k with k k
-            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ←Int.ofNat_eq_coe, hk]
+            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
               exact hx₁
             · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>

Teach the conversion Cyclic ↦ addCyclic to to_additive.

Affected files:

GroupTheory/SpecificGroups/Cyclic
Tactic/ToAdditive

@@ -58,12 +58,12 @@ class IsAddCyclic (α : Type u) [AddGroup α] : Prop where
 #align is_add_cyclic IsAddCyclic
 
 /-- A group is called *cyclic* if it is generated by a single element. -/
-@[to_additive IsAddCyclic]
+@[to_additive]
 class IsCyclic (α : Type u) [Group α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ zpowers g
 #align is_cyclic IsCyclic
 
-@[to_additive isAddCyclic_of_subsingleton]
+@[to_additive]
 instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α] : IsCyclic α :=
   ⟨⟨1, fun x => by
       rw [Subsingleton.elim x 1]
@@ -102,10 +102,8 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 
 variable [Group α]
 
-/-- A non-cyclic multiplicative group is non-trivial.
-The additive version is `Nontrivial.of_not_isAddCyclic`. -/
-@[to_additive Nontrivial.of_not_isAddCyclic "A non-cyclic additive group is non-trivial.
-The multiplicative version is `Nontrivial.of_not_isCyclic`."]
+/-- A non-cyclic multiplicative group is non-trivial. -/
+@[to_additive "A non-cyclic additive group is non-trivial."]
 theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α := by
   contrapose! nc
   exact @isCyclic_of_subsingleton _ _ (not_nontrivial_iff_subsingleton.mp nc)
@@ -133,7 +131,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 
 /-- A finite group of prime order is cyclic. -/
-@[to_additive isAddCyclic_of_prime_card "A finite group of prime order is cyclic."]
+@[to_additive "A finite group of prime order is cyclic."]
 theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card α = p) : IsCyclic α :=
   ⟨by
@@ -164,7 +162,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 
-@[to_additive isAddCyclic_of_surjective]
+@[to_additive]
 theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
     [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) : IsCyclic G := by
   obtain ⟨x, hx⟩ := hH
@@ -183,7 +181,7 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 
-@[to_additive exists_nsmul_ne_zero_of_isAddCyclic]
+@[to_additive]
 theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]
     {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 := by
   rcases G_cyclic with ⟨a, ha⟩
@@ -216,13 +214,13 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
-@[to_additive Bot.isAddCyclic]
+@[to_additive]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
 #align bot.is_add_cyclic Bot.isAddCyclic
 
-@[to_additive AddSubgroup.isAddCyclic]
+@[to_additive]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
   let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
@@ -538,7 +536,7 @@ section CommGroup
 
 variable [CommGroup α] [IsSimpleGroup α]
 
-@[to_additive IsSimpleAddGroup.isAddCyclic]
+@[to_additive]
 instance (priority := 100) isCyclic : IsCyclic α := by
   cases' subsingleton_or_nontrivial α with hi hi <;> haveI := hi
   · apply isCyclic_of_subsingleton
@@ -583,7 +581,7 @@ end CommGroup
 
 end IsSimpleGroup
 
-@[to_additive AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card]
+@[to_additive]
 theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup α] :
     IsSimpleGroup α ↔ IsCyclic α ∧ (Fintype.card α).Prime := by
   constructor

A non-cyclic group is also non-trivial, whether it is multiplicative or additive.

@@ -102,6 +102,14 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 
 variable [Group α]
 
+/-- A non-cyclic multiplicative group is non-trivial.
+The additive version is `Nontrivial.of_not_isAddCyclic`. -/
+@[to_additive Nontrivial.of_not_isAddCyclic "A non-cyclic additive group is non-trivial.
+The multiplicative version is `Nontrivial.of_not_isCyclic`."]
+theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α := by
+  contrapose! nc
+  exact @isCyclic_of_subsingleton _ _ (not_nontrivial_iff_subsingleton.mp nc)
+
 @[to_additive MonoidAddHom.map_add_cyclic]
 theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m := by

From flt-regular.

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

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

@@ -71,6 +71,20 @@ instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α]
 #align is_cyclic_of_subsingleton isCyclic_of_subsingleton
 #align is_add_cyclic_of_subsingleton isAddCyclic_of_subsingleton
 
+@[simp]
+theorem isCyclic_multiplicative_iff [AddGroup α] : IsCyclic (Multiplicative α) ↔ IsAddCyclic α :=
+  ⟨fun H ↦ ⟨H.1⟩, fun H ↦ ⟨H.1⟩⟩
+
+instance isCyclic_multiplicative [AddGroup α] [IsAddCyclic α] : IsCyclic (Multiplicative α) :=
+  isCyclic_multiplicative_iff.mpr inferInstance
+
+@[simp]
+theorem isAddCyclic_additive_iff [Group α] : IsAddCyclic (Additive α) ↔ IsCyclic α :=
+  ⟨fun H ↦ ⟨H.1⟩, fun H ↦ ⟨H.1⟩⟩
+
+instance isAddCyclic_additive [Group α] [IsCyclic α] : IsAddCyclic (Additive α) :=
+  isAddCyclic_additive_iff.mpr inferInstance
+
 /-- A cyclic group is always commutative. This is not an `instance` because often we have a better
 proof of `CommGroup`. -/
 @[to_additive
@@ -142,6 +156,15 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
 #align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 
+@[to_additive isAddCyclic_of_surjective]
+theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
+    [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) : IsCyclic G := by
+  obtain ⟨x, hx⟩ := hH
+  refine ⟨f x, fun a ↦ ?_⟩
+  obtain ⟨a, rfl⟩ := hf a
+  obtain ⟨n, rfl⟩ := hx a
+  exact ⟨n, (map_zpow _ _ _).symm⟩
+
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
@@ -564,6 +587,19 @@ theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup
 #align comm_group.is_simple_iff_is_cyclic_and_prime_card CommGroup.is_simple_iff_isCyclic_and_prime_card
 #align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card
 
+section SpecificInstances
+
+instance : IsAddCyclic ℤ := ⟨1, fun n ↦ ⟨n, by simp only [smul_eq_mul, mul_one]⟩⟩
+
+instance ZMod.instIsAddCyclic (n : ℕ) : IsAddCyclic (ZMod n) :=
+  isAddCyclic_of_surjective (Int.castRingHom _) ZMod.int_cast_surjective
+
+instance ZMod.instIsSimpleAddGroup {p : ℕ} [Fact p.Prime] : IsSimpleAddGroup (ZMod p) :=
+  AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card.2
+    ⟨inferInstance, by simpa using (Fact.out : p.Prime)⟩
+
+end SpecificInstances
+
 section Exponent
 
 open Monoid
@@ -603,13 +639,6 @@ theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite
 #align is_cyclic.exponent_eq_zero_of_infinite IsCyclic.exponent_eq_zero_of_infinite
 #align is_add_cyclic.exponent_eq_zero_of_infinite IsAddCyclic.exponent_eq_zero_of_infinite
 
-instance ZMod.instIsAddCyclic (n : ℕ) : IsAddCyclic (ZMod n) where
-  exists_generator := ⟨1, fun n ↦ ⟨n, by simp⟩⟩
-
-instance ZMod.instIsSimpleAddGroup {p : ℕ} [Fact p.Prime] : IsSimpleAddGroup (ZMod p) :=
-  AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card.2
-    ⟨by infer_instance, by simpa using (Fact.out : p.Prime)⟩
-
 @[simp]
 protected theorem ZMod.exponent (n : ℕ) : AddMonoid.exponent (ZMod n) = n := by
   cases n

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

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

@@ -177,9 +177,9 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     obtain ⟨k, hk⟩ := h x
     dsimp at hk
     obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg
-    · exact ⟨k, by exact_mod_cast hk⟩
+    · exact ⟨k, mod_cast hk⟩
     rw [←zpow_mod_orderOf] at hk
-    have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (by exact_mod_cast ho.orderOf_pos.ne')
+    have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (mod_cast ho.orderOf_pos.ne')
     refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩
     rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
@@ -213,7 +213,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
           rw [zpow_mul]
           apply H.zpow_mem
-          exact_mod_cast (Nat.find_spec hex).2
+          exact mod_cast (Nat.find_spec hex).2
         have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
           (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
             rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx

The cardinality of a subgroup is greater than the order of any of its elements.

Rename

• order_eq_card_zpowers → Fintype.card_zpowers
• order_eq_card_zpowers' → Nat.card_zpowers (and turn it around to match Nat.card_subgroupPowers)
• Submonoid.powers_subset → Submonoid.powers_le
• orderOf_dvd_card_univ → orderOf_dvd_card
• orderOf_subgroup → Subgroup.orderOf
• Subgroup.nonempty → Subgroup.coe_nonempty

@@ -105,7 +105,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   classical
     use x
     simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
-    rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx
+    rw [← Fintype.card_congr (Equiv.Set.univ α), ←Fintype.card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
 #align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
@@ -146,7 +146,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
     orderOf g = Fintype.card α := by
   classical
-    rw [orderOf_eq_card_zpowers]
+    rw [←Fintype.card_zpowers]
     apply Fintype.card_of_finset'
     simpa using hx
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
@@ -159,7 +159,7 @@ theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cycli
   use a
   contrapose! k_lt_card_G
   convert orderOf_le_of_pow_eq_one k_pos.bot_lt k_lt_card_G
-  rw [orderOf_eq_card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
+  rw [←Fintype.card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
   exact fun x _ ↦ ha x
 
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
@@ -276,7 +276,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
           rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
           exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
-      rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
+      rw [Fintype.card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]
       nth_rw 2 [hm]; nth_rw 3 [hm]
       rw [Nat.mul_div_cancel_left _ (gcd_pos_of_pos_left _ hn0), gcd_mul_left_left, hm,
         Nat.mul_div_cancel _ hm0]
@@ -324,7 +324,7 @@ private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
     (Finset.univ.filter fun b : α => b ^ orderOf a = 1).card = orderOf a :=
   le_antisymm (hn _ (orderOf_pos a))
     (calc
-      orderOf a = @Fintype.card (zpowers a) (id _) := orderOf_eq_card_zpowers
+      orderOf a = @Fintype.card (zpowers a) (id _) := Fintype.card_zpowers.symm
       _ ≤
           @Fintype.card (↑(univ.filter fun b : α => b ^ orderOf a = 1) : Set α)
             (Fintype.ofFinset _ fun _ => Iff.rfl) :=
@@ -574,7 +574,7 @@ theorem IsCyclic.exponent_eq_card [Group α] [IsCyclic α] [Fintype α] :
   obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
   apply Nat.dvd_antisymm
   · rw [← lcm_order_eq_exponent, Finset.lcm_dvd_iff]
-    exact fun b _ => orderOf_dvd_card_univ
+    exact fun b _ => orderOf_dvd_card
   rw [← orderOf_eq_card_of_forall_mem_zpowers hg]
   exact order_dvd_exponent _
 #align is_cyclic.exponent_eq_card IsCyclic.exponent_eq_card

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.

@@ -168,18 +168,18 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
   classical
     rw [orderOf_eq_zero_iff']
     refine' fun n hn hgn => _
-    have ho := orderOf_pos' ((isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, hn, hgn⟩)
+    have ho := isOfFinOrder_iff_pow_eq_one.mpr ⟨n, hn, hgn⟩
     obtain ⟨x, hx⟩ :=
       Infinite.exists_not_mem_finset
         (Finset.image (fun x => g ^ x) <| Finset.range <| orderOf g)
     apply hx
-    rw [← mem_powers_iff_mem_range_order_of' (x := g) (y := x) ho, Submonoid.mem_powers_iff]
+    rw [←ho.mem_powers_iff_mem_range_orderOf, Submonoid.mem_powers_iff]
     obtain ⟨k, hk⟩ := h x
     dsimp at hk
     obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg
     · exact ⟨k, by exact_mod_cast hk⟩
-    rw [zpow_eq_mod_orderOf] at hk
-    have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (by exact_mod_cast ho.ne')
+    rw [←zpow_mod_orderOf] at hk
+    have : 0 ≤ (-k % orderOf g : ℤ) := Int.emod_nonneg (-k) (by exact_mod_cast ho.orderOf_pos.ne')
     refine' ⟨(-k % orderOf g : ℤ).toNat, _⟩
     rwa [← zpow_ofNat, Int.toNat_of_nonneg this]
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers

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

@@ -375,7 +375,8 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   have hc0 : 0 < c := Fintype.card_pos_iff.2 ⟨1⟩
   apply card_orderOf_eq_totient_aux₁ hn hd
   by_contra h0
-  simp only [not_lt, _root_.le_zero_iff, card_eq_zero] at h0
+  -- Must qualify `Finset.card_eq_zero` because of leanprover/lean4#2849
+  simp only [not_lt, _root_.le_zero_iff, Finset.card_eq_zero] at h0
   apply lt_irrefl c
   calc
     c = ∑ m in c.divisors, (univ.filter fun a : α => orderOf a = m).card := by

@@ -605,6 +605,10 @@ theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite
 instance ZMod.instIsAddCyclic (n : ℕ) : IsAddCyclic (ZMod n) where
   exists_generator := ⟨1, fun n ↦ ⟨n, by simp⟩⟩
 
+instance ZMod.instIsSimpleAddGroup {p : ℕ} [Fact p.Prime] : IsSimpleAddGroup (ZMod p) :=
+  AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card.2
+    ⟨by infer_instance, by simpa using (Fact.out : p.Prime)⟩
+
 @[simp]
 protected theorem ZMod.exponent (n : ℕ) : AddMonoid.exponent (ZMod n) = n := by
   cases n

@@ -602,4 +602,13 @@ theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite
 #align is_cyclic.exponent_eq_zero_of_infinite IsCyclic.exponent_eq_zero_of_infinite
 #align is_add_cyclic.exponent_eq_zero_of_infinite IsAddCyclic.exponent_eq_zero_of_infinite
 
+instance ZMod.instIsAddCyclic (n : ℕ) : IsAddCyclic (ZMod n) where
+  exists_generator := ⟨1, fun n ↦ ⟨n, by simp⟩⟩
+
+@[simp]
+protected theorem ZMod.exponent (n : ℕ) : AddMonoid.exponent (ZMod n) = n := by
+  cases n
+  · rw [IsAddCyclic.exponent_eq_zero_of_infinite]
+  · rw [IsAddCyclic.exponent_eq_card, card]
+
 end Exponent

Includes an auto-fix feature with the --fix flag so that spacing suggestions will be automatically applied in review also.

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

@@ -154,7 +154,7 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 
 @[to_additive exists_nsmul_ne_zero_of_isAddCyclic]
 theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]
-  {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 := by
+    {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 := by
   rcases G_cyclic with ⟨a, ha⟩
   use a
   contrapose! k_lt_card_G

Adds a theorem saying the cardinality of a multiplicative subgroup of a field, cast to the field, is nonzero. As well as sum_subgroup_units_zero_of_ne_bot and other theorems about summing over multiplicative subgroups.

Co-authored-by: Pratyush Mishra <prat@upenn.edu> Co-authored-by: Buster <rcopley@gmail.com>

@@ -152,6 +152,16 @@ theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x,
 #align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers
 #align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples
 
+@[to_additive exists_nsmul_ne_zero_of_isAddCyclic]
+theorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]
+  {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) : ∃ a : G, a ^ k ≠ 1 := by
+  rcases G_cyclic with ⟨a, ha⟩
+  use a
+  contrapose! k_lt_card_G
+  convert orderOf_le_of_pow_eq_one k_pos.bot_lt k_lt_card_G
+  rw [orderOf_eq_card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]
+  exact fun x _ ↦ ha x
+
 @[to_additive Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples]
 theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
     (h : ∀ x, x ∈ zpowers g) : orderOf g = 0 := by

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -263,7 +263,7 @@ theorem IsCyclic.card_pow_eq_one_le [DecidableEq α] [Fintype α] [IsCyclic α]
       let ⟨m, hm⟩ := Nat.gcd_dvd_right n (Fintype.card α)
       have hm0 : 0 < m :=
         Nat.pos_of_ne_zero fun hm0 => by
-          rw [hm0, MulZeroClass.mul_zero, Fintype.card_eq_zero_iff] at hm
+          rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm
           exact hm.elim' 1
       simp only [Set.toFinset_card, SetLike.coe_sort_coe]
       rw [← orderOf_eq_card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]

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

This has nice performance benefits.

@@ -89,7 +89,7 @@ def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
 variable [Group α]
 
 @[to_additive MonoidAddHom.map_add_cyclic]
-theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →* G) :
+theorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (σ : G →* G) :
     ∃ m : ℤ, ∀ g : G, σ g = g ^ m := by
   obtain ⟨h, hG⟩ := IsCyclic.exists_generator (α := G)
   obtain ⟨m, hm⟩ := hG (σ h)
@@ -416,7 +416,7 @@ theorem IsCyclic.card_orderOf_eq_totient [IsCyclic α] [Fintype α] {d : ℕ}
   classical apply card_orderOf_eq_totient_aux₂ (fun n => IsCyclic.card_pow_eq_one_le) hd
 #align is_cyclic.card_order_of_eq_totient IsCyclic.card_orderOf_eq_totient
 
-theorem IsAddCyclic.card_orderOf_eq_totient {α : Type _} [AddGroup α] [IsAddCyclic α]
+theorem IsAddCyclic.card_orderOf_eq_totient {α : Type*} [AddGroup α] [IsAddCyclic α]
     [Fintype α] {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filter fun a : α => addOrderOf a = d).card = totient d := by
   obtain ⟨g, hg⟩ := id ‹IsAddCyclic α›
@@ -451,7 +451,7 @@ section QuotientCenter
 
 open Subgroup
 
-variable {G : Type _} {H : Type _} [Group G] [Group H]
+variable {G : Type*} {H : Type*} [Group G] [Group H]
 
 /-- A group is commutative if the quotient by the center is cyclic.
   Also see `commGroup_of_cycle_center_quotient` for the `CommGroup` instance. -/

• 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) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
-
-! This file was ported from Lean 3 source module group_theory.specific_groups.cyclic
-! leanprover-community/mathlib commit 0f6670b8af2dff699de1c0b4b49039b31bc13c46
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.BigOperators.Order
 import Mathlib.Data.Nat.Totient
@@ -15,6 +10,8 @@ import Mathlib.GroupTheory.Subgroup.Simple
 import Mathlib.Tactic.Group
 import Mathlib.GroupTheory.Exponent
 
+#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
+
 /-!
 # Cyclic 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.

@@ -199,7 +199,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
             cases' k with k k
             · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ←Int.ofNat_eq_coe, hk]
               exact hx₁
-            . rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
+            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
     ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
         let ⟨k, hk⟩ := hg x
         have hk : g ^ k = x := hk

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

@@ -166,7 +166,7 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
       Infinite.exists_not_mem_finset
         (Finset.image (fun x => g ^ x) <| Finset.range <| orderOf g)
     apply hx
-    rw [← mem_powers_iff_mem_range_order_of' (x:=g) (y:=x) ho, Submonoid.mem_powers_iff]
+    rw [← mem_powers_iff_mem_range_order_of' (x := g) (y := x) ho, Submonoid.mem_powers_iff]
     obtain ⟨k, hk⟩ := h x
     dsimp at hk
     obtain ⟨k, rfl | rfl⟩ := k.eq_nat_or_neg

@@ -29,7 +29,7 @@ For the concrete cyclic group of order `n`, see `Data.ZMod.Basic`.
 ## Main statements
 
 * `isCyclic_of_prime_card` proves that a finite group of prime order is cyclic.
-* `isSimpleGroup_of_prime_card`, `is_simple_group.is_cyclic`,
+* `isSimpleGroup_of_prime_card`, `IsSimpleGroup.isCyclic`,
   and `IsSimpleGroup.prime_card` classify finite simple abelian groups.
 * `IsCyclic.exponent_eq_card`: For a finite cyclic group `G`, the exponent is equal to
   the group's cardinality.
@@ -66,18 +66,18 @@ class IsCyclic (α : Type u) [Group α] : Prop where
   exists_generator : ∃ g : α, ∀ x, x ∈ zpowers g
 #align is_cyclic IsCyclic
 
-@[to_additive is_add_cyclic_of_subsingleton]
+@[to_additive isAddCyclic_of_subsingleton]
 instance (priority := 100) isCyclic_of_subsingleton [Group α] [Subsingleton α] : IsCyclic α :=
   ⟨⟨1, fun x => by
       rw [Subsingleton.elim x 1]
       exact mem_zpowers 1⟩⟩
 #align is_cyclic_of_subsingleton isCyclic_of_subsingleton
-#align is_add_cyclic_of_subsingleton is_add_cyclic_of_subsingleton
+#align is_add_cyclic_of_subsingleton isAddCyclic_of_subsingleton
 
 /-- A cyclic group is always commutative. This is not an `instance` because often we have a better
 proof of `CommGroup`. -/
 @[to_additive
-      "A cyclic group is always commutative. This is not an `instance` because often we have\n
+      "A cyclic group is always commutative. This is not an `instance` because often we have
       a better proof of `AddCommGroup`."]
 def IsCyclic.commGroup [hg : Group α] [IsCyclic α] : CommGroup α :=
   { hg with
@@ -102,7 +102,7 @@ theorem MonoidHom.map_cyclic {G : Type _} [Group G] [h : IsCyclic G] (σ : G →
 #align monoid_hom.map_cyclic MonoidHom.map_cyclic
 #align monoid_add_hom.map_add_cyclic MonoidAddHom.map_add_cyclic
 
-@[to_additive is_add_cyclic_of_orderOf_eq_card]
+@[to_additive isAddCyclic_of_orderOf_eq_card]
 theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
     IsCyclic α := by
   classical
@@ -111,16 +111,16 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
     rw [← Fintype.card_congr (Equiv.Set.univ α), orderOf_eq_card_zpowers] at hx
     exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 #align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card
-#align is_add_cyclic_of_order_of_eq_card is_add_cyclic_of_orderOf_eq_card
+#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_orderOf_eq_card
 
 /-- A finite group of prime order is cyclic. -/
-@[to_additive is_add_cyclic_of_prime_card "A finite group of prime order is cyclic."]
+@[to_additive isAddCyclic_of_prime_card "A finite group of prime order is cyclic."]
 theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card α = p) : IsCyclic α :=
   ⟨by
     obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
     classical
-      -- for fintype (subgroup.zpowers g)
+      -- for Fintype (Subgroup.zpowers g)
       have : Fintype.card (Subgroup.zpowers g) ∣ p := by
         rw [← h]
         apply card_subgroup_dvd_card
@@ -143,7 +143,7 @@ theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [
         rw [Subgroup.eq_top_of_card_eq _ that]
         exact Subgroup.mem_top _⟩
 #align is_cyclic_of_prime_card isCyclic_of_prime_card
-#align is_add_cyclic_of_prime_card is_add_cyclic_of_prime_card
+#align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card
 
 @[to_additive addOrderOf_eq_card_of_forall_mem_zmultiples]
 theorem orderOf_eq_card_of_forall_mem_zpowers [Fintype α] {g : α} (hx : ∀ x, x ∈ zpowers g) :
@@ -178,13 +178,13 @@ theorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite α] {g : α}
 #align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers
 #align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples
 
-@[to_additive Bot.is_add_cyclic]
+@[to_additive Bot.isAddCyclic]
 instance Bot.isCyclic {α : Type u} [Group α] : IsCyclic (⊥ : Subgroup α) :=
   ⟨⟨1, fun x => ⟨0, Subtype.eq <| (zpow_zero (1 : α)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)⟩⟩⟩
 #align bot.is_cyclic Bot.isCyclic
-#align bot.is_add_cyclic Bot.is_add_cyclic
+#align bot.is_add_cyclic Bot.isAddCyclic
 
-@[to_additive AddSubgroup.is_add_cyclic]
+@[to_additive AddSubgroup.isAddCyclic]
 instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
   haveI := Classical.propDecidable
   let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
@@ -233,7 +233,7 @@ instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup
         ⟨fun h => by simp at *; tauto, fun h => by rw [Subgroup.mem_bot.1 h]; exact H.one_mem⟩
     subst this; infer_instance
 #align subgroup.is_cyclic Subgroup.isCyclic
-#align add_subgroup.is_add_cyclic AddSubgroup.is_add_cyclic
+#align add_subgroup.is_add_cyclic AddSubgroup.isAddCyclic
 
 open Finset Nat
 
@@ -333,12 +333,9 @@ private theorem card_pow_eq_one_eq_orderOf_aux (a : α) :
           fun _ _ h => Subtype.eq (Subtype.mk.inj h))
       _ = (univ.filter fun b : α => b ^ orderOf a = 1).card := Fintype.card_ofFinset _ _
       )
--- Porting note: private so presumably is not used
--- #align card_pow_eq_one_eq_order_of_aux card_pow_eq_one_eq_order_of_aux
 
+-- Use φ for `Nat.totient`
 open Nat
-
--- use φ for nat.totient
 private theorem card_orderOf_eq_totient_aux₁ :
     ∀ {d : ℕ},
       d ∣ Fintype.card α →
@@ -364,8 +361,6 @@ private theorem card_orderOf_eq_totient_aux₁ :
     rw [← filter_dvd_eq_divisors hd0, sum_card_orderOf_eq_card_pow_eq_one hd0,
       filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_orderOf_aux hn a]
   simpa [← cons_self_properDivisors hd0, ← h1] using h2
--- Porting note: private so presumably is not used
--- #align card_order_of_eq_totient_aux₁ card_order_of_eq_totient_aux₁
 
 theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
     (univ.filter fun a : α => orderOf a = d).card = φ d := by
@@ -438,18 +433,18 @@ attribute [to_additive existing IsCyclic.card_orderOf_eq_totient]
 @[to_additive "A finite group of prime order is simple."]
 theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
     (h : Fintype.card α = p) : IsSimpleGroup α :=
-    have : Nontrivial α := by
-      have h' := Nat.Prime.one_lt (Fact.out (p := p.Prime))
-      rw [← h] at h'
-      exact Fintype.one_lt_card_iff_nontrivial.1 h'
-    ⟨ fun H _ => by
-      classical
-        have hcard := card_subgroup_dvd_card H
-        rw [h, dvd_prime (Fact.out (p := p.Prime))] at hcard
-        refine' hcard.imp (fun h1 => _) fun hp => _
-        · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
-          apply eq_bot_of_subsingleton
-        · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
+  have : Nontrivial α := by
+    have h' := Nat.Prime.one_lt (Fact.out (p := p.Prime))
+    rw [← h] at h'
+    exact Fintype.one_lt_card_iff_nontrivial.1 h'
+  ⟨fun H _ => by
+    classical
+      have hcard := card_subgroup_dvd_card H
+      rw [h, dvd_prime (Fact.out (p := p.Prime))] at hcard
+      refine' hcard.imp (fun h1 => _) fun hp => _
+      · haveI := Fintype.card_le_one_iff_subsingleton.1 (le_of_eq h1)
+        apply eq_bot_of_subsingleton
+      · exact eq_top_of_card_eq _ (hp.trans h.symm)⟩
 #align is_simple_group_of_prime_card isSimpleGroup_of_prime_card
 #align is_simple_add_group_of_prime_card isSimpleAddGroup_of_prime_card
 
@@ -462,11 +457,10 @@ open Subgroup
 variable {G : Type _} {H : Type _} [Group G] [Group H]
 
 /-- A group is commutative if the quotient by the center is cyclic.
-  Also see `commGroup_of_cycle_center_quotient` for the `comm_group` instance. -/
+  Also see `commGroup_of_cycle_center_quotient` for the `CommGroup` instance. -/
 @[to_additive commutative_of_add_cyclic_center_quotient
-      "A group is commutative if the quotient by\n
-      the center is cyclic. Also see `addCommGroup_of_cycle_center_quotient`\n
-      for the `add_comm_group` instance."]
+      "A group is commutative if the quotient by the center is cyclic.
+      Also see `addCommGroup_of_cycle_center_quotient` for the `AddCommGroup` instance."]
 theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G)
     (a b : G) : a * b = b * a :=
   let ⟨⟨x, y, (hxy : f y = x)⟩, (hx : ∀ a : f.range, a ∈ zpowers _)⟩ :=
@@ -476,8 +470,7 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
   have hm : x ^ m = f a := by simpa [Subtype.ext_iff] using hm
   have hn : x ^ n = f b := by simpa [Subtype.ext_iff] using hn
   have ha : y ^ (-m) * a ∈ center G :=
-    hf (by
-    rw [f.mem_ker, f.map_mul, f.map_zpow, hxy, zpow_neg x m, hm, inv_mul_self])
+    hf (by rw [f.mem_ker, f.map_mul, f.map_zpow, hxy, zpow_neg x m, hm, inv_mul_self])
   have hb : y ^ (-n) * b ∈ center G :=
     hf (by rw [f.mem_ker, f.map_mul, f.map_zpow, hxy, zpow_neg x n, hn, inv_mul_self])
   calc
@@ -491,7 +484,7 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
 
 /-- A group is commutative if the quotient by the center is cyclic. -/
 @[to_additive commutativeOfAddCycleCenterQuotient
-      "A group is commutative if the quotient by\n  the center is cyclic."]
+      "A group is commutative if the quotient by the center is cyclic."]
 def commGroupOfCycleCenterQuotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G) :
     CommGroup G :=
   { show Group G by infer_instance with mul_comm := commutative_of_cyclic_center_quotient f hf }
@@ -519,6 +512,8 @@ instance (priority := 100) isCyclic : IsCyclic α := by
       apply Subgroup.mem_zpowers
     · rw [ht]
       apply Subgroup.mem_top
+#align is_simple_group.is_cyclic IsSimpleGroup.isCyclic
+#align is_simple_add_group.is_add_cyclic IsSimpleAddGroup.isAddCyclic
 
 @[to_additive]
 theorem prime_card [Fintype α] : (Fintype.card α).Prime := by
@@ -549,7 +544,7 @@ end CommGroup
 
 end IsSimpleGroup
 
-@[to_additive AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card]
+@[to_additive AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card]
 theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup α] :
     IsSimpleGroup α ↔ IsCyclic α ∧ (Fintype.card α).Prime := by
   constructor
@@ -559,7 +554,7 @@ theorem CommGroup.is_simple_iff_isCyclic_and_prime_card [Fintype α] [CommGroup
     haveI : Fact (Fintype.card α).Prime := ⟨hp⟩
     exact isSimpleGroup_of_prime_card rfl
 #align comm_group.is_simple_iff_is_cyclic_and_prime_card CommGroup.is_simple_iff_isCyclic_and_prime_card
-#align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_is_add_cyclic_and_prime_card
+#align add_comm_group.is_simple_iff_is_add_cyclic_and_prime_card AddCommGroup.is_simple_iff_isAddCyclic_and_prime_card
 
 section Exponent
 
@@ -601,4 +596,3 @@ theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite
 #align is_add_cyclic.exponent_eq_zero_of_infinite IsAddCyclic.exponent_eq_zero_of_infinite
 
 end Exponent
-