ring_theory.nilpotent ⟷ Mathlib.RingTheory.Nilpotent

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

@@ -159,7 +159,7 @@ theorem isRadical_iff_span_singleton [CommSemiring R] :
 theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x :=
   ⟨fun h x => h k x, fun h =>
-    k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ' x n).symm ▸ hd.hMul_right x) 0 hk
+    k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ x n).symm ▸ hd.hMul_right x) 0 hk
       (fun x hd => pow_one x ▸ hd) fun n _ hn x hd => h x <| hn _ <| (pow_mul x k n).subst hd⟩
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
 -/

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

@@ -217,8 +217,8 @@ variable [Ring R] (h_comm : Commute x y)
 #print Commute.isNilpotent_sub /-
 theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) :=
   by
-  rw [← neg_right_iff] at h_comm 
-  rw [← isNilpotent_neg_iff] at hy 
+  rw [← neg_right_iff] at h_comm
+  rw [← isNilpotent_neg_iff] at hy
   rw [sub_eq_add_neg]
   exact h_comm.is_nilpotent_add hx hy
 #align commute.is_nilpotent_sub Commute.isNilpotent_sub

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
-import Mathbin.Data.Nat.Choose.Sum
-import Mathbin.Algebra.Algebra.Bilinear
-import Mathbin.RingTheory.Ideal.Operations
+import Data.Nat.Choose.Sum
+import Algebra.Algebra.Bilinear
+import RingTheory.Ideal.Operations
 
 #align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
 

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

@@ -159,7 +159,7 @@ theorem isRadical_iff_span_singleton [CommSemiring R] :
 theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x :=
   ⟨fun h x => h k x, fun h =>
-    k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ' x n).symm ▸ hd.mul_right x) 0 hk
+    k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ' x n).symm ▸ hd.hMul_right x) 0 hk
       (fun x hd => pow_one x ▸ hd) fun n _ hn x hd => h x <| hn _ <| (pow_mul x k n).subst hd⟩
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
 -/

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module ring_theory.nilpotent
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Choose.Sum
 import Mathbin.Algebra.Algebra.Bilinear
 import Mathbin.RingTheory.Ideal.Operations
 
+#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
 /-!
 # Nilpotent elements
 

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

@@ -51,26 +51,34 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
 #align is_nilpotent.mk IsNilpotent.mk
 -/
 
+#print IsNilpotent.zero /-
 theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
+-/
 
+#print IsNilpotent.neg /-
 theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   by
   obtain ⟨n, hn⟩ := h
   use n
   rw [neg_pow, hn, MulZeroClass.mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
+-/
 
+#print isNilpotent_neg_iff /-
 @[simp]
 theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩
 #align is_nilpotent_neg_iff isNilpotent_neg_iff
+-/
 
+#print IsNilpotent.map /-
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
   use hr.some; rw [← map_pow, hr.some_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
+-/
 
 #print IsReduced /-
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
@@ -80,10 +88,12 @@ class IsReduced (R : Type _) [Zero R] [Pow R ℕ] : Prop where
 #align is_reduced IsReduced
 -/
 
+#print isReduced_of_noZeroDivisors /-
 instance (priority := 900) isReduced_of_noZeroDivisors [MonoidWithZero R] [NoZeroDivisors R] :
     IsReduced R :=
   ⟨fun _ ⟨_, hn⟩ => pow_eq_zero hn⟩
 #align is_reduced_of_no_zero_divisors isReduced_of_noZeroDivisors
+-/
 
 #print isReduced_of_subsingleton /-
 instance (priority := 900) isReduced_of_subsingleton [Zero R] [Pow R ℕ] [Subsingleton R] :
@@ -98,11 +108,14 @@ theorem IsNilpotent.eq_zero [Zero R] [Pow R ℕ] [IsReduced R] (h : IsNilpotent
 #align is_nilpotent.eq_zero IsNilpotent.eq_zero
 -/
 
+#print isNilpotent_iff_eq_zero /-
 @[simp]
 theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x ↔ x = 0 :=
   ⟨fun h => h.eq_zero, fun h => h.symm ▸ IsNilpotent.zero⟩
 #align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zero
+-/
 
+#print isReduced_of_injective /-
 theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _}
     [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
     IsReduced R := by
@@ -112,12 +125,15 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
   rw [map_zero]
   exact (hx.map f).eq_zero
 #align is_reduced_of_injective isReduced_of_injective
+-/
 
+#print RingHom.ker_isRadical_iff_reduced_of_surjective /-
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
     (RingHom.ker f).IsRadical ↔ IsReduced S := by
   simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker] <;> rfl
 #align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjective
+-/
 
 #print IsRadical /-
 /-- An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
@@ -127,9 +143,11 @@ def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=
 #align is_radical IsRadical
 -/
 
+#print zero_isRadical_iff /-
 theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
   simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]; exact forall_swap
 #align zero_is_radical_iff zero_isRadical_iff
+-/
 
 #print isRadical_iff_span_singleton /-
 theorem isRadical_iff_span_singleton [CommSemiring R] :
@@ -149,10 +167,12 @@ theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
 -/
 
+#print isReduced_iff_pow_one_lt /-
 theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by
   simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]
 #align is_reduced_iff_pow_one_lt isReduced_iff_pow_one_lt
+-/
 
 namespace Commute
 
@@ -160,8 +180,7 @@ section Semiring
 
 variable [Semiring R] (h_comm : Commute x y)
 
-include h_comm
-
+#print Commute.isNilpotent_add /-
 theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y) :=
   by
   obtain ⟨n, hn⟩ := hx
@@ -175,17 +194,22 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   · rw [pow_eq_zero_of_le hi hn, MulZeroClass.zero_mul]
   · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
+-/
 
+#print Commute.isNilpotent_mul_left /-
 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   by
   obtain ⟨n, hn⟩ := h
   use n
   rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
+-/
 
+#print Commute.isNilpotent_mul_right /-
 theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) := by rw [h_comm.eq];
   exact h_comm.symm.is_nilpotent_mul_left h
 #align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_right
+-/
 
 end Semiring
 
@@ -193,8 +217,7 @@ section Ring
 
 variable [Ring R] (h_comm : Commute x y)
 
-include h_comm
-
+#print Commute.isNilpotent_sub /-
 theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) :=
   by
   rw [← neg_right_iff] at h_comm 
@@ -202,6 +225,7 @@ theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   rw [sub_eq_add_neg]
   exact h_comm.is_nilpotent_add hx hy
 #align commute.is_nilpotent_sub Commute.isNilpotent_sub
+-/
 
 end Ring
 
@@ -218,27 +242,37 @@ def nilradical (R : Type _) [CommSemiring R] : Ideal R :=
 #align nilradical nilradical
 -/
 
+#print mem_nilradical /-
 theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
+-/
 
+#print nilradical_eq_sInf /-
 theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
     nilradical R = sInf {J : Ideal R | J.IsPrime} :=
   (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
 #align nilradical_eq_Inf nilradical_eq_sInf
+-/
 
+#print nilpotent_iff_mem_prime /-
 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J := by
   rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]; rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
+-/
 
+#print nilradical_le_prime /-
 theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :=
   (nilradical_eq_sInf R).symm ▸ sInf_le H
 #align nilradical_le_prime nilradical_le_prime
+-/
 
+#print nilradical_eq_zero /-
 @[simp]
 theorem nilradical_eq_zero (R : Type _) [CommSemiring R] [IsReduced R] : nilradical R = 0 :=
   Ideal.ext fun _ => isNilpotent_iff_eq_zero
 #align nilradical_eq_zero nilradical_eq_zero
+-/
 
 end CommSemiring
 
@@ -246,19 +280,23 @@ namespace LinearMap
 
 variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 
+#print LinearMap.isNilpotent_mulLeft_iff /-
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_left_eq_zero_iff, pow_mul_left] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
+-/
 
+#print LinearMap.isNilpotent_mulRight_iff /-
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_right_eq_zero_iff, pow_mul_right] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
+-/
 
 end LinearMap
 
@@ -268,12 +306,14 @@ variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
+#print Module.End.IsNilpotent.mapQ /-
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
   by
   obtain ⟨k, hk⟩ := hnp
   use k
   simp [← p.mapq_pow, hk]
 #align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQ
+-/
 
 end Module.End
 

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

@@ -223,7 +223,7 @@ theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
 #align mem_nilradical mem_nilradical
 
 theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
-    nilradical R = sInf { J : Ideal R | J.IsPrime } :=
+    nilradical R = sInf {J : Ideal R | J.IsPrime} :=
   (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
 #align nilradical_eq_Inf nilradical_eq_sInf
 

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

@@ -197,8 +197,8 @@ include h_comm
 
 theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) :=
   by
-  rw [← neg_right_iff] at h_comm
-  rw [← isNilpotent_neg_iff] at hy
+  rw [← neg_right_iff] at h_comm 
+  rw [← isNilpotent_neg_iff] at hy 
   rw [sub_eq_add_neg]
   exact h_comm.is_nilpotent_add hx hy
 #align commute.is_nilpotent_sub Commute.isNilpotent_sub
@@ -249,14 +249,14 @@ variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
-      simp only [mul_left_eq_zero_iff, pow_mul_left] at hn⊢ <;>
+      simp only [mul_left_eq_zero_iff, pow_mul_left] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
 
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
-      simp only [mul_right_eq_zero_iff, pow_mul_right] at hn⊢ <;>
+      simp only [mul_right_eq_zero_iff, pow_mul_right] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
 

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

@@ -51,22 +51,10 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
 #align is_nilpotent.mk IsNilpotent.mk
 -/
 
-/- warning: is_nilpotent.zero -> IsNilpotent.zero is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))))))
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1)))
-Case conversion may be inaccurate. Consider using '#align is_nilpotent.zero IsNilpotent.zeroₓ'. -/
 theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
 
-/- warning: is_nilpotent.neg -> IsNilpotent.neg is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x))
-but is expected to have type
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x))
-Case conversion may be inaccurate. Consider using '#align is_nilpotent.neg IsNilpotent.negₓ'. -/
 theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   by
   obtain ⟨n, hn⟩ := h
@@ -74,23 +62,11 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   rw [neg_pow, hn, MulZeroClass.mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
-/- warning: is_nilpotent_neg_iff -> isNilpotent_neg_iff is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x)
-but is expected to have type
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x)) (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x)
-Case conversion may be inaccurate. Consider using '#align is_nilpotent_neg_iff isNilpotent_neg_iffₓ'. -/
 @[simp]
 theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩
 #align is_nilpotent_neg_iff isNilpotent_neg_iff
 
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 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
   use hr.some; rw [← map_pow, hr.some_spec, map_zero]
@@ -104,12 +80,6 @@ class IsReduced (R : Type _) [Zero R] [Pow R ℕ] : Prop where
 #align is_reduced IsReduced
 -/
 
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 instance (priority := 900) isReduced_of_noZeroDivisors [MonoidWithZero R] [NoZeroDivisors R] :
     IsReduced R :=
   ⟨fun _ ⟨_, hn⟩ => pow_eq_zero hn⟩
@@ -128,23 +98,11 @@ theorem IsNilpotent.eq_zero [Zero R] [Pow R ℕ] [IsReduced R] (h : IsNilpotent
 #align is_nilpotent.eq_zero IsNilpotent.eq_zero
 -/
 
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 @[simp]
 theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x ↔ x = 0 :=
   ⟨fun h => h.eq_zero, fun h => h.symm ▸ IsNilpotent.zero⟩
 #align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zero
 
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 theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _}
     [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
     IsReduced R := by
@@ -155,12 +113,6 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
   exact (hx.map f).eq_zero
 #align is_reduced_of_injective isReduced_of_injective
 
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 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
     (RingHom.ker f).IsRadical ↔ IsReduced S := by
@@ -175,12 +127,6 @@ def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=
 #align is_radical IsRadical
 -/
 
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 theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
   simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]; exact forall_swap
 #align zero_is_radical_iff zero_isRadical_iff
@@ -203,12 +149,6 @@ theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
 -/
 
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 theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by
   simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]
@@ -222,12 +162,6 @@ variable [Semiring R] (h_comm : Commute x y)
 
 include h_comm
 
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 theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y) :=
   by
   obtain ⟨n, hn⟩ := hx
@@ -242,12 +176,6 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
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 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   by
   obtain ⟨n, hn⟩ := h
@@ -255,12 +183,6 @@ theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
 
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 theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) := by rw [h_comm.eq];
   exact h_comm.symm.is_nilpotent_mul_left h
 #align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_right
@@ -273,12 +195,6 @@ variable [Ring R] (h_comm : Commute x y)
 
 include h_comm
 
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 theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) :=
   by
   rw [← neg_right_iff] at h_comm
@@ -302,53 +218,23 @@ def nilradical (R : Type _) [CommSemiring R] : Ideal R :=
 #align nilradical nilradical
 -/
 
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 theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
 
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 theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
     nilradical R = sInf { J : Ideal R | J.IsPrime } :=
   (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
 #align nilradical_eq_Inf nilradical_eq_sInf
 
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 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J := by
   rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]; rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
 
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 theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :=
   (nilradical_eq_sInf R).symm ▸ sInf_le H
 #align nilradical_le_prime nilradical_le_prime
 
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 @[simp]
 theorem nilradical_eq_zero (R : Type _) [CommSemiring R] [IsReduced R] : nilradical R = 0 :=
   Ideal.ext fun _ => isNilpotent_iff_eq_zero
@@ -360,12 +246,6 @@ namespace LinearMap
 
 variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 
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 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
@@ -373,12 +253,6 @@ theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpot
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
 
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-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.to_smulCommClass.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
-Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iffₓ'. -/
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
@@ -394,9 +268,6 @@ variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
-/- warning: module.End.is_nilpotent.mapq -> Module.End.IsNilpotent.mapQ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQₓ'. -/
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
   by
   obtain ⟨k, hk⟩ := hnp

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

@@ -92,10 +92,8 @@ but is expected to have type
   forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {r : R} {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)], (IsNilpotent.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) (MonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) _inst_2) (Monoid.Pow.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) (MonoidWithZero.toMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) _inst_2)) (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f r))
 Case conversion may be inaccurate. Consider using '#align is_nilpotent.map IsNilpotent.mapₓ'. -/
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
-    [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) :=
-  by
-  use hr.some
-  rw [← map_pow, hr.some_spec, map_zero]
+    [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
+  use hr.some; rw [← map_pow, hr.some_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
 
 #print IsReduced /-
@@ -183,10 +181,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], Iff (IsRadical.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (MonoidWithZero.toSemigroupWithZero.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1)))) (IsReduced.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)))
 Case conversion may be inaccurate. Consider using '#align zero_is_radical_iff zero_isRadical_iffₓ'. -/
-theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R :=
-  by
-  simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
-  exact forall_swap
+theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
+  simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]; exact forall_swap
 #align zero_is_radical_iff zero_isRadical_iff
 
 #print isRadical_iff_span_singleton /-
@@ -265,9 +261,7 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y))
 Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_rightₓ'. -/
-theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) :=
-  by
-  rw [h_comm.eq]
+theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) := by rw [h_comm.eq];
   exact h_comm.symm.is_nilpotent_mul_left h
 #align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_right
 
@@ -335,10 +329,8 @@ lean 3 declaration is
 but is expected to have type
   forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
 Case conversion may be inaccurate. Consider using '#align nilpotent_iff_mem_prime nilpotent_iff_mem_primeₓ'. -/
-theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J :=
-  by
-  rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]
-  rfl
+theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J := by
+  rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]; rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
 
 /- warning: nilradical_le_prime -> nilradical_le_prime is a dubious translation:

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

@@ -403,10 +403,7 @@ variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
 /- warning: module.End.is_nilpotent.mapq -> Module.End.IsNilpotent.mapQ is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) 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_inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
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_inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) 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M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
+<too large>
 Case conversion may be inaccurate. Consider using '#align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQₓ'. -/
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -89,7 +89,7 @@ theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : MonoidWithZero.{u1} S] {r : R} {F : Type.{u2}} [_inst_3 : MonoidWithZeroHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u1} S (MulZeroClass.toHasZero.{u1} S (MulZeroOneClass.toMulZeroClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (Monoid.Pow.{u1} S (MonoidWithZero.toMonoid.{u1} S _inst_2)) (coeFn.{succ u2, succ u1} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u2, succ u1, succ u1} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u2, u1, u1} F R S (MulOneClass.toHasMul.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MulOneClass.toHasMul.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (MonoidHomClass.toMulHomClass.{u2, u1, u1} F R S (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2) _inst_3)))) f r))
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {r : R} {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)], (IsNilpotent.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) (MonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) _inst_2) (Monoid.Pow.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) (MonoidWithZero.toMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) _inst_2)) (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f r))
+  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {r : R} {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)], (IsNilpotent.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) (MonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) _inst_2) (Monoid.Pow.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) (MonoidWithZero.toMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) r) _inst_2)) (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f r))
 Case conversion may be inaccurate. Consider using '#align is_nilpotent.map IsNilpotent.mapₓ'. -/
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) :=
@@ -145,7 +145,7 @@ theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : MonoidWithZero.{u1} S] {F : Type.{u2}} [_inst_3 : MonoidWithZeroHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)] (f : F), (Function.Injective.{succ u1, succ u1} R S (coeFn.{succ u2, succ u1} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u2, succ u1, succ u1} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u2, u1, u1} F R S (MulOneClass.toHasMul.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MulOneClass.toHasMul.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (MonoidHomClass.toMulHomClass.{u2, u1, u1} F R S (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2) _inst_3)))) f)) -> (forall [_inst_4 : IsReduced.{u1} S (MulZeroClass.toHasZero.{u1} S (MulZeroOneClass.toMulZeroClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (Monoid.Pow.{u1} S (MonoidWithZero.toMonoid.{u1} S _inst_2))], IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)))
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)] (f : F), (Function.Injective.{succ u2, succ u2} R S (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f)) -> (forall [_inst_4 : IsReduced.{u2} S (MonoidWithZero.toZero.{u2} S _inst_2) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S _inst_2))], IsReduced.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)))
+  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)] (f : F), (Function.Injective.{succ u2, succ u2} R S (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f)) -> (forall [_inst_4 : IsReduced.{u2} S (MonoidWithZero.toZero.{u2} S _inst_2) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S _inst_2))], IsReduced.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)))
 Case conversion may be inaccurate. Consider using '#align is_reduced_of_injective isReduced_of_injectiveₓ'. -/
 theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _}
     [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
@@ -161,7 +161,7 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u1, succ u2} R S (coeFn.{succ u3, max (succ u1) (succ u2)} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u3, succ u1, succ u2} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u3, u1, u2} F R S (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Distrib.toHasMul.{u2} S (NonUnitalNonAssocSemiring.toDistrib.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{u3, u1, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))) _inst_3)))) f)) -> (Iff (Ideal.IsRadical.{u1} R _inst_1 (RingHom.ker.{u1, u2, u3} R S F (CommSemiring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (MulZeroClass.toHasZero.{u2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (Monoid.Pow.{u2} S (Ring.toMonoid.{u2} S (CommRing.toRing.{u2} S _inst_2)))))
 but is expected to have type
-  forall {R : Type.{u3}} {S : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u3, succ u2} R S (FunLike.coe.{succ u1, succ u3, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u3, u2} F R S (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u3, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) _inst_3))) f)) -> (Iff (Ideal.IsRadical.{u3} R _inst_1 (RingHom.ker.{u3, u2, u1} R S F (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (CommMonoidWithZero.toZero.{u2} S (CommSemiring.toCommMonoidWithZero.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))))))
+  forall {R : Type.{u3}} {S : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u3, succ u2} R S (FunLike.coe.{succ u1, succ u3, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{u1, u3, u2} F R S (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u3, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) _inst_3))) f)) -> (Iff (Ideal.IsRadical.{u3} R _inst_1 (RingHom.ker.{u3, u2, u1} R S F (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (CommMonoidWithZero.toZero.{u2} S (CommSemiring.toCommMonoidWithZero.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))))))
 Case conversion may be inaccurate. Consider using '#align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjectiveₓ'. -/
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -406,7 +406,7 @@ variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.hasZero.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.hasZero.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
 Case conversion may be inaccurate. Consider using '#align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQₓ'. -/
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
   by

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

@@ -343,7 +343,7 @@ theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime 
 
 /- warning: nilradical_le_prime -> nilradical_le_prime is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (nilradical.{u1} R _inst_1) J
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toHasLe.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (nilradical.{u1} R _inst_1) J
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))) (nilradical.{u1} R _inst_1) J
 Case conversion may be inaccurate. Consider using '#align nilradical_le_prime nilradical_le_primeₓ'. -/
@@ -404,7 +404,7 @@ variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
 /- warning: module.End.is_nilpotent.mapq -> Module.End.IsNilpotent.mapQ is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.hasZero.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.hasZero.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.hasZero.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.hasZero.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
 but is expected to have type
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
 Case conversion may be inaccurate. Consider using '#align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQₓ'. -/

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

@@ -318,16 +318,16 @@ theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
 
-/- warning: nilradical_eq_Inf -> nilradical_eq_infₛ is a dubious translation:
+/- warning: nilradical_eq_Inf -> nilradical_eq_sInf is a dubious translation:
 lean 3 declaration is
-  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.infₛ.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.hasInf.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.sInf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.hasInf.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
 but is expected to have type
-  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.infₛ.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
-Case conversion may be inaccurate. Consider using '#align nilradical_eq_Inf nilradical_eq_infₛₓ'. -/
-theorem nilradical_eq_infₛ (R : Type _) [CommSemiring R] :
-    nilradical R = infₛ { J : Ideal R | J.IsPrime } :=
-  (Ideal.radical_eq_infₛ ⊥).trans <| by simp_rw [and_iff_right bot_le]
-#align nilradical_eq_Inf nilradical_eq_infₛ
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.sInf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
+Case conversion may be inaccurate. Consider using '#align nilradical_eq_Inf nilradical_eq_sInfₓ'. -/
+theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
+    nilradical R = sInf { J : Ideal R | J.IsPrime } :=
+  (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
+#align nilradical_eq_Inf nilradical_eq_sInf
 
 /- warning: nilpotent_iff_mem_prime -> nilpotent_iff_mem_prime is a dubious translation:
 lean 3 declaration is
@@ -337,7 +337,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align nilpotent_iff_mem_prime nilpotent_iff_mem_primeₓ'. -/
 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J :=
   by
-  rw [← mem_nilradical, nilradical_eq_infₛ, Submodule.mem_infₛ]
+  rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]
   rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
 
@@ -348,7 +348,7 @@ but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))) (nilradical.{u1} R _inst_1) J
 Case conversion may be inaccurate. Consider using '#align nilradical_le_prime nilradical_le_primeₓ'. -/
 theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :=
-  (nilradical_eq_infₛ R).symm ▸ infₛ_le H
+  (nilradical_eq_sInf R).symm ▸ sInf_le H
 #align nilradical_le_prime nilradical_le_prime
 
 /- warning: nilradical_eq_zero -> nilradical_eq_zero is a dubious translation:

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

@@ -161,7 +161,7 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u1, succ u2} R S (coeFn.{succ u3, max (succ u1) (succ u2)} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u3, succ u1, succ u2} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u3, u1, u2} F R S (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Distrib.toHasMul.{u2} S (NonUnitalNonAssocSemiring.toDistrib.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{u3, u1, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))) _inst_3)))) f)) -> (Iff (Ideal.IsRadical.{u1} R _inst_1 (RingHom.ker.{u1, u2, u3} R S F (CommSemiring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (MulZeroClass.toHasZero.{u2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (Monoid.Pow.{u2} S (Ring.toMonoid.{u2} S (CommRing.toRing.{u2} S _inst_2)))))
 but is expected to have type
-  forall {R : Type.{u3}} {S : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u3, succ u2} R S (FunLike.coe.{succ u1, succ u3, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u3, u2} F R S (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u3, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))) _inst_3))) f)) -> (Iff (Ideal.IsRadical.{u3} R _inst_1 (RingHom.ker.{u3, u2, u1} R S F (CommSemiring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (CommMonoidWithZero.toZero.{u2} S (CommSemiring.toCommMonoidWithZero.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))
+  forall {R : Type.{u3}} {S : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u3, succ u2} R S (FunLike.coe.{succ u1, succ u3, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u3, u2} F R S (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u3, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) _inst_3))) f)) -> (Iff (Ideal.IsRadical.{u3} R _inst_1 (RingHom.ker.{u3, u2, u1} R S F (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (CommMonoidWithZero.toZero.{u2} S (CommSemiring.toCommMonoidWithZero.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))))))
 Case conversion may be inaccurate. Consider using '#align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjectiveₓ'. -/
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :

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

@@ -368,23 +368,31 @@ namespace LinearMap
 
 variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 
-#print LinearMap.isNilpotent_mulLeft_iff /-
+/- warning: linear_map.is_nilpotent_mul_left_iff -> LinearMap.isNilpotent_mulLeft_iff is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.hasZero.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulLeft.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.to_smulCommClass.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulLeft.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.to_smulCommClass.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iffₓ'. -/
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_left_eq_zero_iff, pow_mul_left] at hn⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
--/
 
-#print LinearMap.isNilpotent_mulRight_iff /-
+/- warning: linear_map.is_nilpotent_mul_right_iff -> LinearMap.isNilpotent_mulRight_iff is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.hasZero.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.to_smulCommClass.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iffₓ'. -/
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_right_eq_zero_iff, pow_mul_right] at hn⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
--/
 
 end LinearMap
 

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

@@ -368,31 +368,23 @@ namespace LinearMap
 
 variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 
-/- warning: linear_map.is_nilpotent_mul_left_iff -> LinearMap.isNilpotent_mulLeft_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iffₓ'. -/
+#print LinearMap.isNilpotent_mulLeft_iff /-
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_left_eq_zero_iff, pow_mul_left] at hn⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
+-/
 
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-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.hasZero.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
-but is expected to have type
-  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
-Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iffₓ'. -/
+#print LinearMap.isNilpotent_mulRight_iff /-
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
       simp only [mul_right_eq_zero_iff, pow_mul_right] at hn⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
+-/
 
 end LinearMap
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1

@@ -63,7 +63,7 @@ theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
 
 /- warning: is_nilpotent.neg -> IsNilpotent.neg is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x))
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x))
 but is expected to have type
   forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x))
 Case conversion may be inaccurate. Consider using '#align is_nilpotent.neg IsNilpotent.negₓ'. -/
@@ -76,7 +76,7 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
 
 /- warning: is_nilpotent_neg_iff -> isNilpotent_neg_iff is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x)
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) x)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x)
 but is expected to have type
   forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x)) (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x)
 Case conversion may be inaccurate. Consider using '#align is_nilpotent_neg_iff isNilpotent_neg_iffₓ'. -/
@@ -281,7 +281,7 @@ include h_comm
 
 /- warning: commute.is_nilpotent_sub -> Commute.isNilpotent_sub is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Ring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y))
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Ring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) x y))
 but is expected to have type
   forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Ring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R _inst_1)) x y))
 Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_sub Commute.isNilpotent_subₓ'. -/

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

@@ -312,7 +312,7 @@ def nilradical (R : Type _) [CommSemiring R] : Ideal R :=
 lean 3 declaration is
   forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (Membership.Mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x (nilradical.{u1} R _inst_1)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x)
 but is expected to have type
-  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.instSetLikeSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x (nilradical.{u1} R _inst_1)) (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x)
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x (nilradical.{u1} R _inst_1)) (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x)
 Case conversion may be inaccurate. Consider using '#align mem_nilradical mem_nilradicalₓ'. -/
 theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
@@ -333,7 +333,7 @@ theorem nilradical_eq_infₛ (R : Type _) [CommSemiring R] :
 lean 3 declaration is
   forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.Mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
 but is expected to have type
-  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.instSetLikeSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
 Case conversion may be inaccurate. Consider using '#align nilpotent_iff_mem_prime nilpotent_iff_mem_primeₓ'. -/
 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module ring_theory.nilpotent
-! leanprover-community/mathlib commit da420a8c6dd5bdfb85c4ced85c34388f633bc6ff
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.Ideal.Operations
 /-!
 # Nilpotent elements
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main definitions
 
   * `is_nilpotent`

mathlib commit https://github.com/leanprover-community/mathlib/commit/f24cc2891c0e328f0ee8c57387103aa462c44b5e

@@ -31,6 +31,7 @@ universe u v
 
 variable {R S : Type u} {x y : R}
 
+#print IsNilpotent /-
 /-- An element is said to be nilpotent if some natural-number-power of it equals zero.
 
 Note that we require only the bare minimum assumptions for the definition to make sense. Even
@@ -39,15 +40,30 @@ power-associative. -/
 def IsNilpotent [Zero R] [Pow R ℕ] (x : R) : Prop :=
   ∃ n : ℕ, x ^ n = 0
 #align is_nilpotent IsNilpotent
+-/
 
+#print IsNilpotent.mk /-
 theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) : IsNilpotent x :=
   ⟨n, e⟩
 #align is_nilpotent.mk IsNilpotent.mk
+-/
 
+/- warning: is_nilpotent.zero -> IsNilpotent.zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align is_nilpotent.zero IsNilpotent.zeroₓ'. -/
 theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
 
+/- warning: is_nilpotent.neg -> IsNilpotent.neg is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x))
+Case conversion may be inaccurate. Consider using '#align is_nilpotent.neg IsNilpotent.negₓ'. -/
 theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   by
   obtain ⟨n, hn⟩ := h
@@ -55,11 +71,23 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   rw [neg_pow, hn, MulZeroClass.mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
+/- warning: is_nilpotent_neg_iff -> isNilpotent_neg_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) x)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x)
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} [_inst_1 : Ring.{u1} R], Iff (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) x)) (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x)
+Case conversion may be inaccurate. Consider using '#align is_nilpotent_neg_iff isNilpotent_neg_iffₓ'. -/
 @[simp]
 theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩
 #align is_nilpotent_neg_iff isNilpotent_neg_iff
 
+/- warning: is_nilpotent.map -> IsNilpotent.map is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : MonoidWithZero.{u1} S] {r : R} {F : Type.{u2}} [_inst_3 : MonoidWithZeroHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)], (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u1} S (MulZeroClass.toHasZero.{u1} S (MulZeroOneClass.toMulZeroClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (Monoid.Pow.{u1} S (MonoidWithZero.toMonoid.{u1} S _inst_2)) (coeFn.{succ u2, succ u1} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u2, succ u1, succ u1} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u2, u1, u1} F R S (MulOneClass.toHasMul.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MulOneClass.toHasMul.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (MonoidHomClass.toMulHomClass.{u2, u1, u1} F R S (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2) _inst_3)))) f r))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {r : R} {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)], (IsNilpotent.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)) r) -> (forall (f : F), IsNilpotent.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) (MonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) _inst_2) (Monoid.Pow.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) (MonoidWithZero.toMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) r) _inst_2)) (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f r))
+Case conversion may be inaccurate. Consider using '#align is_nilpotent.map IsNilpotent.mapₓ'. -/
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) :=
   by
@@ -67,31 +95,55 @@ theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type
   rw [← map_pow, hr.some_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
 
+#print IsReduced /-
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
 @[mk_iff]
 class IsReduced (R : Type _) [Zero R] [Pow R ℕ] : Prop where
   eq_zero : ∀ x : R, IsNilpotent x → x = 0
 #align is_reduced IsReduced
+-/
 
+/- warning: is_reduced_of_no_zero_divisors -> isReduced_of_noZeroDivisors is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)))], IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MonoidWithZero.toZero.{u1} R _inst_1)], IsReduced.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))
+Case conversion may be inaccurate. Consider using '#align is_reduced_of_no_zero_divisors isReduced_of_noZeroDivisorsₓ'. -/
 instance (priority := 900) isReduced_of_noZeroDivisors [MonoidWithZero R] [NoZeroDivisors R] :
     IsReduced R :=
   ⟨fun _ ⟨_, hn⟩ => pow_eq_zero hn⟩
 #align is_reduced_of_no_zero_divisors isReduced_of_noZeroDivisors
 
+#print isReduced_of_subsingleton /-
 instance (priority := 900) isReduced_of_subsingleton [Zero R] [Pow R ℕ] [Subsingleton R] :
     IsReduced R :=
   ⟨fun _ _ => Subsingleton.elim _ _⟩
 #align is_reduced_of_subsingleton isReduced_of_subsingleton
+-/
 
+#print IsNilpotent.eq_zero /-
 theorem IsNilpotent.eq_zero [Zero R] [Pow R ℕ] [IsReduced R] (h : IsNilpotent x) : x = 0 :=
   IsReduced.eq_zero x h
 #align is_nilpotent.eq_zero IsNilpotent.eq_zero
+-/
 
+/- warning: is_nilpotent_iff_eq_zero -> isNilpotent_iff_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) x) (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)))))))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : IsReduced.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))], Iff (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) x) (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zeroₓ'. -/
 @[simp]
 theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x ↔ x = 0 :=
   ⟨fun h => h.eq_zero, fun h => h.symm ▸ IsNilpotent.zero⟩
 #align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zero
 
+/- warning: is_reduced_of_injective -> isReduced_of_injective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] [_inst_2 : MonoidWithZero.{u1} S] {F : Type.{u2}} [_inst_3 : MonoidWithZeroHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)] (f : F), (Function.Injective.{succ u1, succ u1} R S (coeFn.{succ u2, succ u1} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u2, succ u1, succ u1} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u2, u1, u1} F R S (MulOneClass.toHasMul.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (MulOneClass.toHasMul.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (MonoidHomClass.toMulHomClass.{u2, u1, u1} F R S (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u2, u1, u1} F R S (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2) _inst_3)))) f)) -> (forall [_inst_4 : IsReduced.{u1} S (MulZeroClass.toHasZero.{u1} S (MulZeroOneClass.toMulZeroClass.{u1} S (MonoidWithZero.toMulZeroOneClass.{u1} S _inst_2))) (Monoid.Pow.{u1} S (MonoidWithZero.toMonoid.{u1} S _inst_2))], IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u2}} [_inst_1 : MonoidWithZero.{u2} R] [_inst_2 : MonoidWithZero.{u2} S] {F : Type.{u1}} [_inst_3 : MonoidWithZeroHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)] (f : F), (Function.Injective.{succ u2, succ u2} R S (FunLike.coe.{succ u1, succ u2, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u2, u2} F R S (MulOneClass.toMul.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1))) (MulOneClass.toMul.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2))) (MonoidHomClass.toMulHomClass.{u1, u2, u2} F R S (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1)) (MulZeroOneClass.toMulOneClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2)) (MonoidWithZeroHomClass.toMonoidHomClass.{u1, u2, u2} F R S (MonoidWithZero.toMulZeroOneClass.{u2} R _inst_1) (MonoidWithZero.toMulZeroOneClass.{u2} S _inst_2) _inst_3))) f)) -> (forall [_inst_4 : IsReduced.{u2} S (MonoidWithZero.toZero.{u2} S _inst_2) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S _inst_2))], IsReduced.{u2} R (MonoidWithZero.toZero.{u2} R _inst_1) (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align is_reduced_of_injective isReduced_of_injectiveₓ'. -/
 theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _}
     [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
     IsReduced R := by
@@ -102,38 +154,62 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
   exact (hx.map f).eq_zero
 #align is_reduced_of_injective isReduced_of_injective
 
+/- warning: ring_hom.ker_is_radical_iff_reduced_of_surjective -> RingHom.ker_isRadical_iff_reduced_of_surjective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u1, succ u2} R S (coeFn.{succ u3, max (succ u1) (succ u2)} F (fun (_x : F) => R -> S) (FunLike.hasCoeToFun.{succ u3, succ u1, succ u2} F R (fun (_x : R) => S) (MulHomClass.toFunLike.{u3, u1, u2} F R S (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Distrib.toHasMul.{u2} S (NonUnitalNonAssocSemiring.toDistrib.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (NonUnitalRingHomClass.toMulHomClass.{u3, u1, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u3, u1, u2} F R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))) _inst_3)))) f)) -> (Iff (Ideal.IsRadical.{u1} R _inst_1 (RingHom.ker.{u1, u2, u3} R S F (CommSemiring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (MulZeroClass.toHasZero.{u2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (Monoid.Pow.{u2} S (Ring.toMonoid.{u2} S (CommRing.toRing.{u2} S _inst_2)))))
+but is expected to have type
+  forall {R : Type.{u3}} {S : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : RingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))] {f : F}, (Function.Surjective.{succ u3, succ u2} R S (FunLike.coe.{succ u1, succ u3, succ u2} F R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{u1, u3, u2} F R S (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u3, u2} F R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u3, u2} F R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))) _inst_3))) f)) -> (Iff (Ideal.IsRadical.{u3} R _inst_1 (RingHom.ker.{u3, u2, u1} R S F (CommSemiring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3 f)) (IsReduced.{u2} S (CommMonoidWithZero.toZero.{u2} S (CommSemiring.toCommMonoidWithZero.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (Monoid.Pow.{u2} S (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))
+Case conversion may be inaccurate. Consider using '#align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjectiveₓ'. -/
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
     (RingHom.ker f).IsRadical ↔ IsReduced S := by
   simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker] <;> rfl
 #align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjective
 
+#print IsRadical /-
 /-- An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
   divides a power of `x`. -/
 def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=
   ∀ (n : ℕ) (x), y ∣ x ^ n → y ∣ x
 #align is_radical IsRadical
+-/
 
+/- warning: zero_is_radical_iff -> zero_isRadical_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], Iff (IsRadical.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (MonoidWithZero.toSemigroupWithZero.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))))))) (IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R], Iff (IsRadical.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (MonoidWithZero.toSemigroupWithZero.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1)))) (IsReduced.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align zero_is_radical_iff zero_isRadical_iffₓ'. -/
 theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R :=
   by
   simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
   exact forall_swap
 #align zero_is_radical_iff zero_isRadical_iff
 
+#print isRadical_iff_span_singleton /-
 theorem isRadical_iff_span_singleton [CommSemiring R] :
     IsRadical y ↔ (Ideal.span ({y} : Set R)).IsRadical :=
   by
   simp_rw [IsRadical, ← Ideal.mem_span_singleton]
   exact forall_swap.trans (forall_congr' fun r => exists_imp_distrib.symm)
 #align is_radical_iff_span_singleton isRadical_iff_span_singleton
+-/
 
+#print isRadical_iff_pow_one_lt /-
 theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x :=
   ⟨fun h x => h k x, fun h =>
     k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ' x n).symm ▸ hd.mul_right x) 0 hk
       (fun x hd => pow_one x ▸ hd) fun n _ hn x hd => h x <| hn _ <| (pow_mul x k n).subst hd⟩
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
+-/
 
+/- warning: is_reduced_iff_pow_one_lt -> isReduced_iff_pow_one_lt is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] (k : Nat), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) k) -> (Iff (IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))) (forall (x : R), (Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))) x k) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1))))))) -> (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R _inst_1)))))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : MonoidWithZero.{u1} R] (k : Nat), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) k) -> (Iff (IsReduced.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))) (forall (x : R), (Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R _inst_1))) x k) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1)))) -> (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R _inst_1))))))
+Case conversion may be inaccurate. Consider using '#align is_reduced_iff_pow_one_lt isReduced_iff_pow_one_ltₓ'. -/
 theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsReduced R ↔ ∀ x : R, x ^ k = 0 → x = 0 := by
   simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]
@@ -147,6 +223,12 @@ variable [Semiring R] (h_comm : Commute x y)
 
 include h_comm
 
+/- warning: commute.is_nilpotent_add -> Commute.isNilpotent_add is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y))
+Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_add Commute.isNilpotent_addₓ'. -/
 theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y) :=
   by
   obtain ⟨n, hn⟩ := hx
@@ -161,6 +243,12 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
+/- warning: commute.is_nilpotent_mul_left -> Commute.isNilpotent_mul_left is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y))
+Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_leftₓ'. -/
 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   by
   obtain ⟨n, hn⟩ := h
@@ -168,6 +256,12 @@ theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
 
+/- warning: commute.is_nilpotent_mul_right -> Commute.isNilpotent_mul_right is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Semiring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y))
+Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_rightₓ'. -/
 theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) :=
   by
   rw [h_comm.eq]
@@ -182,6 +276,12 @@ variable [Ring R] (h_comm : Commute x y)
 
 include h_comm
 
+/- warning: commute.is_nilpotent_sub -> Commute.isNilpotent_sub is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Ring.{u1} R], (Commute.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) x y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) x) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) y) -> (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} {y : R} [_inst_1 : Ring.{u1} R], (Commute.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) x y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) x) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) y) -> (IsNilpotent.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R _inst_1)) x y))
+Case conversion may be inaccurate. Consider using '#align commute.is_nilpotent_sub Commute.isNilpotent_subₓ'. -/
 theorem isNilpotent_sub (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x - y) :=
   by
   rw [← neg_right_iff] at h_comm
@@ -198,30 +298,62 @@ section CommSemiring
 
 variable [CommSemiring R]
 
+#print nilradical /-
 /-- The nilradical of a commutative semiring is the ideal of nilpotent elements. -/
 def nilradical (R : Type _) [CommSemiring R] : Ideal R :=
   (0 : Ideal R).radical
 #align nilradical nilradical
+-/
 
+/- warning: mem_nilradical -> mem_nilradical is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (Membership.Mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x (nilradical.{u1} R _inst_1)) (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x)
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.instSetLikeSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x (nilradical.{u1} R _inst_1)) (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x)
+Case conversion may be inaccurate. Consider using '#align mem_nilradical mem_nilradicalₓ'. -/
 theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
 
+/- warning: nilradical_eq_Inf -> nilradical_eq_infₛ is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.infₛ.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.hasInf.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (InfSet.infₛ.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))) (setOf.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (fun (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) => Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2) J)))
+Case conversion may be inaccurate. Consider using '#align nilradical_eq_Inf nilradical_eq_infₛₓ'. -/
 theorem nilradical_eq_infₛ (R : Type _) [CommSemiring R] :
     nilradical R = infₛ { J : Ideal R | J.IsPrime } :=
   (Ideal.radical_eq_infₛ ⊥).trans <| by simp_rw [and_iff_right bot_le]
 #align nilradical_eq_Inf nilradical_eq_infₛ
 
+/- warning: nilpotent_iff_mem_prime -> nilpotent_iff_mem_prime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.Mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.hasMem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
+but is expected to have type
+  forall {R : Type.{u1}} {x : R} [_inst_1 : CommSemiring.{u1} R], Iff (IsNilpotent.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J) -> (Membership.mem.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.instSetLikeSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) x J))
+Case conversion may be inaccurate. Consider using '#align nilpotent_iff_mem_prime nilpotent_iff_mem_primeₓ'. -/
 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J :=
   by
   rw [← mem_nilradical, nilradical_eq_infₛ, Submodule.mem_infₛ]
   rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
 
+/- warning: nilradical_le_prime -> nilradical_le_prime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (nilradical.{u1} R _inst_1) J
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) [H : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) J], LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))) (nilradical.{u1} R _inst_1) J
+Case conversion may be inaccurate. Consider using '#align nilradical_le_prime nilradical_le_primeₓ'. -/
 theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :=
   (nilradical_eq_infₛ R).symm ▸ infₛ_le H
 #align nilradical_le_prime nilradical_le_prime
 
+/- warning: nilradical_eq_zero -> nilradical_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R] [_inst_3 : IsReduced.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (OfNat.ofNat.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) 0 (OfNat.mk.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) 0 (Zero.zero.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (MulZeroClass.toHasZero.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (IdemSemiring.toSemiring.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Submodule.idemSemiring.{u1, u1} R _inst_2 R (CommSemiring.toSemiring.{u1} R _inst_2) (Algebra.id.{u1} R _inst_2))))))))))
+but is expected to have type
+  forall (R : Type.{u1}) [_inst_2 : CommSemiring.{u1} R] [_inst_3 : IsReduced.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_2)) (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))], Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (nilradical.{u1} R _inst_2) (OfNat.ofNat.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) 0 (Zero.toOfNat0.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (CommMonoidWithZero.toZero.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (CommSemiring.toCommMonoidWithZero.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (IdemCommSemiring.toCommSemiring.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)) (Ideal.instIdemCommSemiringIdealToSemiring.{u1} R _inst_2))))))
+Case conversion may be inaccurate. Consider using '#align nilradical_eq_zero nilradical_eq_zeroₓ'. -/
 @[simp]
 theorem nilradical_eq_zero (R : Type _) [CommSemiring R] [IsReduced R] : nilradical R = 0 :=
   Ideal.ext fun _ => isNilpotent_iff_eq_zero
@@ -233,6 +365,12 @@ namespace LinearMap
 
 variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 
+/- warning: linear_map.is_nilpotent_mul_left_iff -> LinearMap.isNilpotent_mulLeft_iff is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.hasZero.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulLeft.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulLeft.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iffₓ'. -/
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
@@ -240,6 +378,12 @@ theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpot
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
 
+/- warning: linear_map.is_nilpotent_mul_right_iff -> LinearMap.isNilpotent_mulRight_iff is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.hasZero.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)))) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+but is expected to have type
+  forall (R : Type.{u1}) {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (a : A), Iff (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R A A (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Module.End.monoid.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (LinearMap.mulRight.{u1, u2} R A _inst_1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2)) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.to_smulCommClass.{u1, u2, u2} R _inst_1 A _inst_2 _inst_3 A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Semiring.toModule.{u2} A _inst_2) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (IsScalarTower.right.{u1, u2} R A _inst_1 _inst_2 _inst_3) a)) (IsNilpotent.{u2} A (MonoidWithZero.toZero.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2)) (Monoid.Pow.{u2} A (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A _inst_2))) a)
+Case conversion may be inaccurate. Consider using '#align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iffₓ'. -/
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
@@ -255,6 +399,12 @@ variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
+/- warning: module.End.is_nilpotent.mapq -> Module.End.IsNilpotent.mapQ is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.hasZero.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.hasZero.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) 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(Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {f : Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} (hp : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p (Submodule.comap.{u1, u1, u2, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f p)), (IsNilpotent.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R M M (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Module.End.monoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) f) -> (IsNilpotent.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (LinearMap.instZeroLinearMap.{u1, u1, u2, u2} R R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Monoid.Pow.{u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Module.End.monoid.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) p) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 p)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 p))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_3 p R M _inst_1 _inst_2 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) p f hp))
+Case conversion may be inaccurate. Consider using '#align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQₓ'. -/
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
   by
   obtain ⟨k, hk⟩ := hnp

mathlib commit https://github.com/leanprover-community/mathlib/commit/f24cc2891c0e328f0ee8c57387103aa462c44b5e

@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module ring_theory.nilpotent
-! leanprover-community/mathlib commit e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74
+! leanprover-community/mathlib commit da420a8c6dd5bdfb85c4ced85c34388f633bc6ff
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Choose.Sum
 import Mathbin.Algebra.Algebra.Bilinear
-import Mathbin.RingTheory.Ideal.QuotientOperations
+import Mathbin.RingTheory.Ideal.Operations
 
 /-!
 # Nilpotent elements
@@ -108,13 +108,6 @@ theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [
   simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker] <;> rfl
 #align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjective
 
-theorem Ideal.isRadical_iff_quotient_reduced [CommRing R] (I : Ideal R) :
-    I.IsRadical ↔ IsReduced (R ⧸ I) :=
-  by
-  conv_lhs => rw [← @Ideal.mk_ker R _ I]
-  exact RingHom.ker_isRadical_iff_reduced_of_surjective (@Ideal.Quotient.mk_surjective R _ I)
-#align ideal.is_radical_iff_quotient_reduced Ideal.isRadical_iff_quotient_reduced
-
 /-- An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
   divides a power of `x`. -/
 def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=
@@ -271,74 +264,3 @@ theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
 
 end Module.End
 
-section Ideal
-
-variable [CommSemiring R] [CommRing S] [Algebra R S] (I : Ideal S)
-
-/-- Let `P` be a property on ideals. If `P` holds for square-zero ideals, and if
-  `P I → P (J ⧸ I) → P J`, then `P` holds for all nilpotent ideals. -/
-theorem Ideal.IsNilpotent.induction_on (hI : IsNilpotent I)
-    {P : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, Prop}
-    (h₁ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, I ^ 2 = ⊥ → P I)
-    (h₂ :
-      ∀ ⦃S : Type _⦄ [CommRing S],
-        ∀ I J : Ideal S, I ≤ J → P I → P (J.map (Ideal.Quotient.mk I)) → P J) :
-    P I := by
-  obtain ⟨n, hI : I ^ n = ⊥⟩ := hI
-  revert S
-  apply Nat.strong_induction_on n
-  clear n
-  intro n H S _ I hI
-  by_cases hI' : I = ⊥
-  · subst hI'
-    apply h₁
-    rw [← Ideal.zero_eq_bot, zero_pow]
-    exact zero_lt_two
-  cases n
-  · rw [pow_zero, Ideal.one_eq_top] at hI
-    haveI := subsingleton_of_bot_eq_top hI.symm
-    exact (hI' (Subsingleton.elim _ _)).elim
-  cases n
-  · rw [pow_one] at hI
-    exact (hI' hI).elim
-  apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)
-  · apply H n.succ _ (I ^ 2)
-    · rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one, Nat.succ_eq_add_one]
-      exact Ideal.pow_le_pow (by linarith)
-    · exact le_refl n.succ.succ
-  · apply h₁
-    rw [← Ideal.map_pow, Ideal.map_quotient_self]
-#align ideal.is_nilpotent.induction_on Ideal.IsNilpotent.induction_on
-
-theorem IsNilpotent.isUnit_quotient_mk_iff {R : Type _} [CommRing R] {I : Ideal R}
-    (hI : IsNilpotent I) {x : R} : IsUnit (Ideal.Quotient.mk I x) ↔ IsUnit x :=
-  by
-  refine' ⟨_, fun h => h.map I.Quotient.mk⟩
-  revert x
-  apply Ideal.IsNilpotent.induction_on I hI <;> clear hI I
-  swap
-  · introv e h₁ h₂ h₃
-    apply h₁
-    apply h₂
-    exact
-      h₃.map
-        ((DoubleQuot.quotQuotEquivQuotSup I J).trans
-              (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom
-  · introv e H
-    skip
-    obtain ⟨y, hy⟩ := Ideal.Quotient.mk_surjective (↑H.unit⁻¹ : S ⧸ I)
-    have : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by
-      rw [map_one, _root_.map_mul, hy, IsUnit.mul_val_inv]
-    rw [Ideal.Quotient.eq] at this
-    have : (x * y - 1) ^ 2 = 0 := by
-      rw [← Ideal.mem_bot, ← e]
-      exact Ideal.pow_mem_pow this _
-    have : x * (y * (2 - x * y)) = 1 :=
-      by
-      rw [eq_comm, ← sub_eq_zero, ← this]
-      ring
-    exact isUnit_of_mul_eq_one _ _ this
-#align is_nilpotent.is_unit_quotient_mk_iff IsNilpotent.isUnit_quotient_mk_iff
-
-end Ideal
-

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

@@ -52,7 +52,7 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) :=
   by
   obtain ⟨n, hn⟩ := h
   use n
-  rw [neg_pow, hn, mul_zero]
+  rw [neg_pow, hn, MulZeroClass.mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
 @[simp]
@@ -162,17 +162,17 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   rw [h_comm.add_pow']
   apply Finset.sum_eq_zero
   rintro ⟨i, j⟩ hij
-  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
+  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, MulZeroClass.mul_zero]
   cases' Nat.le_or_le_of_add_eq_add_pred (finset.nat.mem_antidiagonal.mp hij) with hi hj
-  · rw [pow_eq_zero_of_le hi hn, zero_mul]
-  · rw [pow_eq_zero_of_le hj hm, mul_zero]
+  · rw [pow_eq_zero_of_le hi hn, MulZeroClass.zero_mul]
+  · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) :=
   by
   obtain ⟨n, hn⟩ := h
   use n
-  rw [h_comm.mul_pow, hn, zero_mul]
+  rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
 
 theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/62e8311c791f02c47451bf14aa2501048e7c2f33

@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module ring_theory.nilpotent
-! leanprover-community/mathlib commit 79de90f7beca025f469dcda978ae655c4d985946
+! leanprover-community/mathlib commit e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Choose.Sum
 import Mathbin.Algebra.Algebra.Bilinear
-import Mathbin.RingTheory.Ideal.Operations
+import Mathbin.RingTheory.Ideal.QuotientOperations
 
 /-!
 # Nilpotent elements

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

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -122,7 +122,7 @@ theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero {m n k : ℕ}
   rintro ⟨i, j⟩ hij
   suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
   by_cases hi : m ≤ i
-  rw [pow_eq_zero_of_le hi hx, zero_mul]
+  · rw [pow_eq_zero_of_le hi hx, zero_mul]
   rw [pow_eq_zero_of_le ?_ hy, mul_zero]
   linarith [Finset.mem_antidiagonal.mp hij]
 

Similar to #12486, which did this for Algebra.Algebra.Basic.

Splits Algebra.Module.Defs off Algebra.Module.Basic. Most imports only need the Defs file, which has significantly smaller imports. The remaining Algebra.Module.Basic is now a grab-bag of unrelated results, and should probably be split further or rehomed.

This is mostly motivated by the wasted effort during minimization upon encountering Algebra.Module.Basic.

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

@@ -6,7 +6,7 @@ Authors: Oliver Nash
 import Mathlib.Algebra.Associated
 import Mathlib.Algebra.GeomSum
 import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
-import Mathlib.Algebra.Module.Basic
+import Mathlib.Algebra.Module.Defs
 import Mathlib.Algebra.SMulWithZero
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Data.Nat.Lattice

Mathlib.RingTheory.Nilpotent has a few very simple definitions (Mathlib.Data.Nat.Lattice is sufficient to state them), but needs some pretty heavy imports (ideals, linear algebra) towards the end. This change moves the heavier parts into a new file.

Mathlib.RingTheory.Nilpotent has a few very simple definitions (Mathlib.Data.Nat.Lattice is sufficient to state them), but needs some pretty heavy imports (ideals, linear algebra) towards the end. This change moves the heavier parts into a new file.

Mathlib.RingTheory.Nilpotent has a few very simple definitions (Mathlib.Data.Nat.Lattice is sufficient to state them), but needs some pretty heavy imports (ideals, linear algebra) towards the end. This change moves the heavier parts into a new file.

This is based on seeing the import RingTheory.Ideal.Operations → LinearAlgebra.Basis on the longest pole. It feels like Ideal.Operations is a bit of a chokepoint for compiling Mathlib since it imports many files and is imported by many files. So splitting out a few obvious parts should help with compile times. Moreover, there are a bunch of imports that I could remove and have the file still compile: presumably these are (were) transitive dependencies that shake does not remove.

The following results and their corollaries were split off:

• Ideal.basisSpanSingleton
• Basis.mem_ideal_iff
• Ideal.colon

In particular, now Ideal.Operations should no longer need to know about Basis or submodule quotients.

@@ -3,13 +3,14 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
-import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
 import Mathlib.Algebra.GeomSum
-import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
 import Mathlib.Algebra.GroupPower.Ring
-import Mathlib.RingTheory.Ideal.Operations
+import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
+import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.LinearAlgebra.Matrix.ToLin
+import Mathlib.LinearAlgebra.Quotient
+import Mathlib.RingTheory.Ideal.Operations
 
 #align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
 

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

@@ -71,7 +71,7 @@ lemma IsNilpotent.pow_succ (n : ℕ) {S : Type*} [MonoidWithZero S] {x : S}
 theorem  IsNilpotent.of_pow [MonoidWithZero R] {x : R} {m : ℕ}
     (h : IsNilpotent (x ^ m)) : IsNilpotent x := by
   obtain ⟨n, h⟩ := h
-  use (m*n)
+  use m*n
   rw [← h, pow_mul x m n]
 
 lemma IsNilpotent.pow_of_pos {n} {S : Type*} [MonoidWithZero S] {x : S}

• add_pow_mem_of_pow_mem_of_le says that (a + b) ^ k belongs to an ideal I if a ^ m and b ^ n belong to that ideal, and m + n ≤ k + 1.
• Commute.add_pow_of_add_le_succ_eq_zero_of_pow_eq_zero says that (a + b) ^ k = 0 if a ^ m = 0, b^ n = 0, and m + n ≤ k + 1.
• this is used to simplify the definition of docs#Ideal.radical and the definition of docs#Commute.isNilpotent_add

@@ -311,17 +311,27 @@ section Semiring
 
 variable [Semiring R] (h_comm : Commute x y)
 
-theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y) := by
-  obtain ⟨n, hn⟩ := hx
-  obtain ⟨m, hm⟩ := hy
-  use n + m - 1
+theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero {m n k : ℕ}
+    (hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
+    (x + y) ^ k = 0 := by
   rw [h_comm.add_pow']
   apply Finset.sum_eq_zero
   rintro ⟨i, j⟩ hij
   suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
-  cases' Nat.le_or_le_of_add_eq_add_pred (Finset.mem_antidiagonal.mp hij) with hi hj
-  · rw [pow_eq_zero_of_le hi hn, zero_mul]
-  · rw [pow_eq_zero_of_le hj hm, mul_zero]
+  by_cases hi : m ≤ i
+  rw [pow_eq_zero_of_le hi hx, zero_mul]
+  rw [pow_eq_zero_of_le ?_ hy, mul_zero]
+  linarith [Finset.mem_antidiagonal.mp hij]
+
+theorem add_pow_add_eq_zero_of_pow_eq_zero {m n : ℕ}
+    (hx : x ^ m = 0) (hy : y ^ n = 0) :
+    (x + y) ^ (m + n - 1) = 0 :=
+  h_comm.add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero hx hy <| by rw [← Nat.sub_le_iff_le_add]
+
+theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y) := by
+  obtain ⟨n, hn⟩ := hx
+  obtain ⟨m, hm⟩ := hy
+  exact ⟨_, add_pow_add_eq_zero_of_pow_eq_zero h_comm hn hm⟩
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
 protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}

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

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

where the latter is more natural

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

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

but it seems to no longer apply.

Remarks on the PR :

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

@@ -471,7 +471,7 @@ lemma NoZeroSMulDivisors.isReduced (R M : Type*)
     exact eq_zero_of_zero_eq_one hk.symm x
   · obtain ⟨m : M, hm : m ≠ 0⟩ := exists_ne (0 : M)
     have : x ^ (k + 1) • m = 0 := by simp only [hk, zero_smul]
-    rw [pow_succ, mul_smul] at this
+    rw [pow_succ', mul_smul] at this
     rcases eq_zero_or_eq_zero_of_smul_eq_zero this with rfl | hx
     · rfl
     · exact ih <| (eq_zero_or_eq_zero_of_smul_eq_zero hx).resolve_right hm

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -452,7 +452,6 @@ end LinearMap
 namespace Module.End
 
 variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
-
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
 
 theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by

• Prove isSemisimple_of_mem_adjoin: if two commuting endomorphisms of a finite-dimensional vector space over a perfect field are both semisimple, then every endomorphism in the algebra generated by them (in particular their product and sum) is semisimple.

• In the same file LinearAlgebra/Semisimple.lean, eq_zero_of_isNilpotent_isSemisimple and isSemisimple_of_squarefree_aeval_eq_zero are golfed, and IsSemisimple.minpoly_squarefree is proved

RingTheory/SimpleModule.lean:

• Define IsSemisimpleRing R to mean that R is a semisimple R-module. add properties of simple modules and a characterization (they are exactly the quotients of the ring by maximal left ideals).

• The annihilator of a semisimple module is a radical ideal.

• Any module over a semisimple ring is semisimple.

• A finite product of semisimple rings is semisimple.

• Any quotient of a semisimple ring is semisimple.

• Add Artin--Wedderburn as a TODO (proof_wanted).

• Order/Atoms.lean: add the instance from IsSimpleOrder to ComplementedLattice, so that IsSimpleModule → IsSemisimpleModule is automatically inferred.

Prerequisites for showing a product of semisimple rings is semisimple:

• Algebra/Module/Submodule/Map.lean: generalize orderIsoMapComap so that it only requires RingHomSurjective rather than RingHomInvPair

• Algebra/Ring/CompTypeclasses.lean, Mathlib/Algebra/Ring/Pi.lean, Algebra/Ring/Prod.lean: add RingHomSurjective instances

RingTheory/Artinian.lean:

• quotNilradicalEquivPi: the quotient of a commutative Artinian ring R by its nilradical is isomorphic to the (finite) product of its quotients by maximal ideals (therefore a product of fields). equivPi: if the ring is moreover reduced, then the ring itself is a product of fields. Deduce that R is a semisimple ring and both R and R[X] are decomposition monoids. Requires RingEquiv.quotientBot in RingTheory/Ideal/QuotientOperations.lean.

• Data/Polynomial/Eval.lean: the polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings. (Used to show R[X] is a decomposition monoid.)

Other necessary results:

• FieldTheory/Minpoly/Field.lean: the minimal polynomial of an element in a reduced algebra over a field is radical.

• RingTheory/PowerBasis.lean: generalize PowerBasis.finiteDimensional and rename it to .finite.

Annihilator stuff, some of which do not end up being used:

• RingTheory/Ideal/Operations.lean: define Module.annihilator and redefine Submodule.annihilator in terms of it; add lemmas, including one that says an arbitrary intersection of radical ideals is radical. The new lemma Ideal.isRadical_iff_pow_one_lt depends on pow_imp_self_of_one_lt in Mathlib/Data/Nat/Interval.lean, which is also used to golf the proof of isRadical_iff_pow_one_lt.

• Algebra/Module/Torsion.lean: add a lemma and an instance (unused)

• Data/Polynomial/Module/Basic.lean: add a def (unused) and a lemma

• LinearAlgebra/AnnihilatingPolynomial.lean: add lemma span_minpoly_eq_annihilator

Some results about idempotent linear maps (projections) and idempotent elements, used to show that any (left) ideal in a semisimple ring is spanned by an idempotent element (unused):

• LinearAlgebra/Projection.lean: add def isIdempotentElemEquiv

• LinearAlgebra/Span.lean: add two lemmas

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -291,9 +291,7 @@ theorem isRadical_iff_span_singleton [CommSemiring R] :
 
 theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
     IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x :=
-  ⟨fun h x => h k x, fun h =>
-    k.cauchy_induction_mul (fun n h x hd => h x <| (pow_succ' x n).symm ▸ hd.mul_right x) 0 hk
-      (fun x hd => pow_one x ▸ hd) fun n _ hn x hd => h x <| hn _ <| (pow_mul x k n).subst hd⟩
+  ⟨(· k), k.pow_imp_self_of_one_lt hk _ fun _ _ h ↦ .inl (dvd_mul_of_dvd_left h _)⟩
 #align is_radical_iff_pow_one_lt isRadical_iff_pow_one_lt
 
 theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -163,7 +163,7 @@ noncomputable def nilpotencyClass : ℕ := sInf {k | x ^ k = 0}
 @[simp] lemma nilpotencyClass_eq_zero_of_subsingleton [Subsingleton R] :
     nilpotencyClass x = 0 := by
   let s : Set ℕ := {k | x ^ k = 0}
-  suffices s = univ by change sInf _ = 0; simp [this]
+  suffices s = univ by change sInf _ = 0; simp [s] at this; simp [this]
   exact eq_univ_iff_forall.mpr fun k ↦ Subsingleton.elim _ _
 
 lemma isNilpotent_of_pos_nilpotencyClass (hx : 0 < nilpotencyClass x) :
@@ -187,7 +187,7 @@ lemma nilpotencyClass_eq_succ_iff {k : ℕ} :
     nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 := by
   let s : Set ℕ := {k | x ^ k = 0}
   have : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := fun k₁ k₂ h_le hk₁ ↦ pow_eq_zero_of_le h_le hk₁
-  simp [nilpotencyClass, Nat.sInf_upward_closed_eq_succ_iff this]
+  simp [s, nilpotencyClass, Nat.sInf_upward_closed_eq_succ_iff this]
 
 @[simp] lemma nilpotencyClass_zero [Nontrivial R] :
     nilpotencyClass (0 : R) = 1 :=

• Introduce typeclass DecompositionMonoid, which says every element in the monoid is primal, i.e., whenever an element divides a product b * c, it can be factored into a product such that the factors divides b and c respectively. A domain is called pre-Schreier if its multiplicative monoid is a decomposition monoid, and these are more general than GCD domains.

• Show that any GCDMonoid is a DecompositionMonoid. In order for lemmas about DecompositionMonoids to automatically apply to UniqueFactorizationMonoids, we add instances from UniqueFactorizationMonoid α to Nonempty (NormalizedGCDMonoid α) to Nonempty (GCDMonoid α) to DecompositionMonoid α. (Zulip) See the bottom of message for an updated diagram of classes and instances.

• Introduce binary predicate IsRelPrime which says that the only common divisors of the two elements are units. Replace previous occurrences in mathlib by this predicate.

• Duplicate all lemmas about IsCoprime in Coprime/Basic (except three lemmas about smul) to IsRelPrime. Due to import constraints, they are spread into three files Algebra/Divisibility/Units (including key lemmas assuming DecompositionMonoid), GroupWithZero/Divisibility, and Coprime/Basic.

• Show IsCoprime always imply IsRelPrime and is equivalent to it in Bezout rings. To reduce duplication, the definition of Bezout rings and the GCDMonoid instance are moved from RingTheory/Bezout to RingTheory/PrincipalIdealDomain, and some results in PrincipalIdealDomain are generalized to Bezout rings.

• Remove the recently added file Squarefree/UniqueFactorizationMonoid and place the results appropriately within Squarefree/Basic. All results are generalized to DecompositionMonoid or weaker except the last one.

Zulip

With this PR, all the following instances (indicated by arrows) now work; this PR fills the central part.

                                                                          EuclideanDomain (bundled)
                                                                              ↙          ↖
                                                                 IsPrincipalIdealRing ← Field (bundled)
                                                                            ↓             ↓
         NormalizationMonoid ←          NormalizedGCDMonoid → GCDMonoid  IsBezout ← ValuationRing ← DiscreteValuationRing
                   ↓                             ↓                 ↘       ↙
Nonempty NormalizationMonoid ← Nonempty NormalizedGCDMonoid →  Nonempty GCDMonoid → IsIntegrallyClosed
                                                 ↑                    ↓
                    WfDvdMonoid ← UniqueFactorizationMonoid → DecompositionMonoid
                                                 ↑
                                       IsPrincipalIdealRing

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Oliver Nash <github@olivernash.org>

@@ -261,12 +261,23 @@ theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [
   rfl
 #align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjective
 
+instance [Zero R] [Pow R ℕ] [Zero S] [Pow S ℕ] [IsReduced R] [IsReduced S] : IsReduced (R × S) where
+  eq_zero _ := fun ⟨n, hn⟩ ↦ have hn := Prod.ext_iff.1 hn
+    Prod.ext (IsReduced.eq_zero _ ⟨n, hn.1⟩) (IsReduced.eq_zero _ ⟨n, hn.2⟩)
+
+instance (ι) (R : ι → Type*) [∀ i, Zero (R i)] [∀ i, Pow (R i) ℕ]
+    [∀ i, IsReduced (R i)] : IsReduced (∀ i, R i) where
+  eq_zero _ := fun ⟨n, hn⟩ ↦ funext fun i ↦ IsReduced.eq_zero _ ⟨n, congr_fun hn i⟩
+
 /-- An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
   divides a power of `x`. -/
 def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=
   ∀ (n : ℕ) (x), y ∣ x ^ n → y ∣ x
 #align is_radical IsRadical
 
+theorem Prime.isRadical [CommMonoidWithZero R] {y : R} (hy : Prime y) : IsRadical y :=
+  fun _ _ ↦ hy.dvd_of_dvd_pow
+
 theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by
   simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
   exact forall_swap
@@ -290,6 +301,12 @@ theorem isReduced_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
   simp_rw [← zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]
 #align is_reduced_iff_pow_one_lt isReduced_iff_pow_one_lt
 
+theorem IsRadical.of_dvd [CancelCommMonoidWithZero R] {x y : R} (hy : IsRadical y) (h0 : y ≠ 0)
+    (hxy : x ∣ y) : IsRadical x := (isRadical_iff_pow_one_lt 2 one_lt_two).2 <| by
+  obtain ⟨z, rfl⟩ := hxy
+  refine fun w dvd ↦ ((mul_dvd_mul_iff_right <| right_ne_zero_of_mul h0).mp <| hy 2 _ ?_)
+  rw [mul_pow, sq z]; exact mul_dvd_mul dvd (dvd_mul_left z z)
+
 namespace Commute
 
 section Semiring

@@ -458,5 +458,6 @@ lemma NoZeroSMulDivisors.isReduced (R M : Type*)
   · obtain ⟨m : M, hm : m ≠ 0⟩ := exists_ne (0 : M)
     have : x ^ (k + 1) • m = 0 := by simp only [hk, zero_smul]
     rw [pow_succ, mul_smul] at this
-    rcases eq_zero_or_eq_zero_of_smul_eq_zero this with rfl | hx; rfl
-    exact ih <| (eq_zero_or_eq_zero_of_smul_eq_zero hx).resolve_right hm
+    rcases eq_zero_or_eq_zero_of_smul_eq_zero this with rfl | hx
+    · rfl
+    · exact ih <| (eq_zero_or_eq_zero_of_smul_eq_zero hx).resolve_right hm

This is just a modified version of the code provided by Antoine Chambert-Loir here: https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/jordan-chevalley.20decomposition/near/411402670

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

@@ -109,22 +109,45 @@ lemma IsNilpotent.map_iff [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Typ
     IsNilpotent (f r) ↔ IsNilpotent r :=
   ⟨fun ⟨k, hk⟩ ↦ ⟨k, (map_eq_zero_iff f hf).mp <| by rwa [map_pow]⟩, fun h ↦ h.map f⟩
 
-theorem IsNilpotent.sub_one_isUnit [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
-  obtain ⟨n, hn⟩ := hnil
-  refine' ⟨⟨r - 1, -∑ i in Finset.range n, r ^ i, _, _⟩, rfl⟩
-  · rw [mul_neg, mul_geom_sum, hn]
-    simp
-  · rw [neg_mul, geom_sum_mul, hn]
-    simp
-
-theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilpotent r)
-    (hru : Commute r (↑u⁻¹ : R)) : IsUnit (u + r) := by
-  rw [← Units.isUnit_mul_units _ u⁻¹, add_mul, Units.mul_inv, ← IsUnit.neg_iff, add_comm, neg_add,
-    ← sub_eq_add_neg]
+theorem IsUnit.isNilpotent_mul_unit_of_commute_iff [MonoidWithZero R] {r u : R}
+    (hu : IsUnit u) (h_comm : Commute r u) :
+    IsNilpotent (r * u) ↔ IsNilpotent r :=
+  exists_congr fun n ↦ by rw [h_comm.mul_pow, (hu.pow n).mul_left_eq_zero]
+
+theorem IsUnit.isNilpotent_unit_mul_of_commute_iff [MonoidWithZero R] {r u : R}
+    (hu : IsUnit u) (h_comm : Commute r u) :
+    IsNilpotent (u * r) ↔ IsNilpotent r :=
+  h_comm ▸ hu.isNilpotent_mul_unit_of_commute_iff h_comm
+
+theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
   obtain ⟨n, hn⟩ := hnil
-  refine' IsNilpotent.sub_one_isUnit ⟨n, _⟩
-  rw [neg_pow, hru.mul_pow, hn]
-  simp
+  refine ⟨⟨r - 1, -∑ i in Finset.range n, r ^ i, ?_, ?_⟩, rfl⟩
+  · simp [mul_geom_sum, hn]
+  · simp [geom_sum_mul, hn]
+
+theorem IsNilpotent.isUnit_one_sub [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 - r) := by
+  rw [← IsUnit.neg_iff, neg_sub]
+  exact isUnit_sub_one hnil
+
+theorem IsNilpotent.isUnit_add_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r + 1) := by
+  rw [← IsUnit.neg_iff, neg_add']
+  exact isUnit_sub_one hnil.neg
+
+theorem IsNilpotent.isUnit_one_add [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (1 + r) :=
+  add_comm r 1 ▸ isUnit_add_one hnil
+
+theorem IsNilpotent.isUnit_add_left_of_commute [Ring R] {r u : R}
+    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
+    IsUnit (u + r) := by
+  rw [← Units.isUnit_mul_units _ hu.unit⁻¹, add_mul, IsUnit.mul_val_inv]
+  replace h_comm : Commute r (↑hu.unit⁻¹) := Commute.units_inv_right h_comm
+  refine IsNilpotent.isUnit_one_add ?_
+  exact (hu.unit⁻¹.isUnit.isNilpotent_mul_unit_of_commute_iff h_comm).mpr hnil
+
+theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
+    (hnil : IsNilpotent r) (hu : IsUnit u) (h_comm : Commute r u) :
+    IsUnit (r + u) :=
+  add_comm r u ▸ hnil.isUnit_add_left_of_commute hu h_comm
 
 section NilpotencyClass
 

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>

@@ -98,13 +98,14 @@ lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero
   rw [smul_pow, ha, smul_zero]
 
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
-    [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
+    [FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
+    IsNilpotent (f r) := by
   use hr.choose
   rw [← map_pow, hr.choose_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
 
 lemma IsNilpotent.map_iff [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
-    [MonoidWithZeroHomClass F R S] {f : F} (hf : Function.Injective f) :
+    [FunLike F R S] [MonoidWithZeroHomClass F R S] {f : F} (hf : Function.Injective f) :
     IsNilpotent (f r) ↔ IsNilpotent r :=
   ⟨fun ⟨k, hk⟩ ↦ ⟨k, (map_eq_zero_iff f hf).mp <| by rwa [map_pow]⟩, fun h ↦ h.map f⟩
 
@@ -220,7 +221,8 @@ theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x
 #align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zero
 
 theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
-    [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
+    [FunLike F R S] [MonoidWithZeroHomClass F R S]
+    (f : F) (hf : Function.Injective f) [IsReduced S] :
     IsReduced R := by
   constructor
   intro x hx
@@ -230,7 +232,7 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
 #align is_reduced_of_injective isReduced_of_injective
 
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
-    [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
+    [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
     (RingHom.ker f).IsRadical ↔ IsReduced S := by
   simp_rw [isReduced_iff, hf.forall, IsNilpotent, ← map_pow, ← RingHom.mem_ker]
   rfl

• depends on: #9830
• depends on: #9772
• depends on: #9996

This checks files for unused imports. The output here is piped through gh-problem-matcher-wrap so that it will show up as annotations.

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

@@ -5,9 +5,10 @@ Authors: Oliver Nash
 -/
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
+import Mathlib.Algebra.GeomSum
 import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
+import Mathlib.Algebra.GroupPower.Ring
 import Mathlib.RingTheory.Ideal.Operations
-import Mathlib.Algebra.GeomSum
 import Mathlib.LinearAlgebra.Matrix.ToLin
 
 #align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"

@@ -28,7 +28,7 @@ import Mathlib.LinearAlgebra.Matrix.ToLin
 
 universe u v
 
-open BigOperators
+open BigOperators Function Set
 
 variable {R S : Type*} {x y : R}
 
@@ -45,6 +45,9 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
   ⟨n, e⟩
 #align is_nilpotent.mk IsNilpotent.mk
 
+@[simp] lemma isNilpotent_of_subsingleton [Zero R] [Pow R ℕ] [Subsingleton R] : IsNilpotent x :=
+  ⟨0, Subsingleton.elim _ _⟩
+
 @[simp] theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
@@ -121,6 +124,74 @@ theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilp
   rw [neg_pow, hru.mul_pow, hn]
   simp
 
+section NilpotencyClass
+
+section ZeroPow
+
+variable [Zero R] [Pow R ℕ]
+
+variable (x) in
+/-- If `x` is nilpotent, the nilpotency class is the smallest natural number `k` such that
+`x ^ k = 0`. If `x` is not nilpotent, the nilpotency class takes the junk value `0`. -/
+noncomputable def nilpotencyClass : ℕ := sInf {k | x ^ k = 0}
+
+@[simp] lemma nilpotencyClass_eq_zero_of_subsingleton [Subsingleton R] :
+    nilpotencyClass x = 0 := by
+  let s : Set ℕ := {k | x ^ k = 0}
+  suffices s = univ by change sInf _ = 0; simp [this]
+  exact eq_univ_iff_forall.mpr fun k ↦ Subsingleton.elim _ _
+
+lemma isNilpotent_of_pos_nilpotencyClass (hx : 0 < nilpotencyClass x) :
+    IsNilpotent x := by
+  let s : Set ℕ := {k | x ^ k = 0}
+  change s.Nonempty
+  change 0 < sInf s at hx
+  by_contra contra
+  simp [not_nonempty_iff_eq_empty.mp contra] at hx
+
+lemma pow_nilpotencyClass (hx : IsNilpotent x) : x ^ (nilpotencyClass x) = 0 :=
+  Nat.sInf_mem hx
+
+end ZeroPow
+
+section MonoidWithZero
+
+variable [MonoidWithZero R]
+
+lemma nilpotencyClass_eq_succ_iff {k : ℕ} :
+    nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 := by
+  let s : Set ℕ := {k | x ^ k = 0}
+  have : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := fun k₁ k₂ h_le hk₁ ↦ pow_eq_zero_of_le h_le hk₁
+  simp [nilpotencyClass, Nat.sInf_upward_closed_eq_succ_iff this]
+
+@[simp] lemma nilpotencyClass_zero [Nontrivial R] :
+    nilpotencyClass (0 : R) = 1 :=
+  nilpotencyClass_eq_succ_iff.mpr <| by constructor <;> simp
+
+@[simp] lemma pos_nilpotencyClass_iff [Nontrivial R] :
+    0 < nilpotencyClass x ↔ IsNilpotent x := by
+  refine ⟨isNilpotent_of_pos_nilpotencyClass, fun hx ↦ Nat.pos_of_ne_zero fun hx' ↦ ?_⟩
+  replace hx := pow_nilpotencyClass hx
+  rw [hx', pow_zero] at hx
+  exact one_ne_zero hx
+
+lemma pow_pred_nilpotencyClass [Nontrivial R] (hx : IsNilpotent x) :
+    x ^ (nilpotencyClass x - 1) ≠ 0 :=
+  (nilpotencyClass_eq_succ_iff.mp <| Nat.eq_add_of_sub_eq (pos_nilpotencyClass_iff.mpr hx) rfl).2
+
+lemma eq_zero_of_nilpotencyClass_eq_one (hx : nilpotencyClass x = 1) :
+    x = 0 := by
+  have : IsNilpotent x := isNilpotent_of_pos_nilpotencyClass (hx ▸ one_pos)
+  rw [← pow_nilpotencyClass this, hx, pow_one]
+
+@[simp] lemma nilpotencyClass_eq_one [Nontrivial R] :
+    nilpotencyClass x = 1 ↔ x = 0 :=
+  ⟨eq_zero_of_nilpotencyClass_eq_one, fun hx ↦ hx ▸ nilpotencyClass_zero⟩
+
+end MonoidWithZero
+
+end NilpotencyClass
+
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
 @[mk_iff]
 class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where

• @[mk_iff] class MyPred now generates myPred_iff, not MyPred_iff
• add Lean.Name.decapitalize
• fix indentation and a few typos in the docs/comments.

Partially addresses issue #9129

@@ -122,7 +122,7 @@ theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilp
   simp
 
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
-@[mk_iff isReduced_iff]
+@[mk_iff]
 class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where
   /-- A reduced structure has no nonzero nilpotent elements. -/
   eq_zero : ∀ x : R, IsNilpotent x → x = 0

Changed IsNilpotent.pow to IsNilpotent.pow_succ. Added IsNilpotent.of_pow and IsNilpotent.pow_iff_pos.

Co-authored-by: Xavier Xarles <56635243+XavierXarles@users.noreply.github.com> Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>

@@ -58,17 +58,28 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
   rw [neg_pow, hn, mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
-lemma IsNilpotent.pow {n : ℕ} {S : Type*} [MonoidWithZero S] {x : S}
+lemma IsNilpotent.pow_succ (n : ℕ) {S : Type*} [MonoidWithZero S] {x : S}
     (hx : IsNilpotent x) : IsNilpotent (x ^ n.succ) := by
   obtain ⟨N,hN⟩ := hx
   use N
   rw [← pow_mul, Nat.succ_mul, pow_add, hN, mul_zero]
 
+theorem  IsNilpotent.of_pow [MonoidWithZero R] {x : R} {m : ℕ}
+    (h : IsNilpotent (x ^ m)) : IsNilpotent x := by
+  obtain ⟨n, h⟩ := h
+  use (m*n)
+  rw [← h, pow_mul x m n]
+
 lemma IsNilpotent.pow_of_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
     (hx : IsNilpotent x) (hn : n ≠ 0) : IsNilpotent (x ^ n) := by
   cases n with
   | zero => contradiction
-  | succ => exact IsNilpotent.pow hx
+  | succ => exact  IsNilpotent.pow_succ _ hx
+
+@[simp]
+lemma IsNilpotent.pow_iff_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
+    (hn : n ≠ 0) : IsNilpotent (x ^ n) ↔ IsNilpotent x :=
+ ⟨fun h => of_pow h, fun h => pow_of_pos h hn⟩
 
 @[simp]
 theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=

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

@@ -5,7 +5,7 @@ Authors: Oliver Nash
 -/
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
-import Mathlib.GroupTheory.Submonoid.ZeroDivisors
+import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
 import Mathlib.RingTheory.Ideal.Operations
 import Mathlib.Algebra.GeomSum
 import Mathlib.LinearAlgebra.Matrix.ToLin

@@ -205,7 +205,9 @@ protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
     (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
     IsNilpotent (∑ i in s, f i) := by
   classical
-  induction' s using Finset.induction with j s hj ih; simp
+  induction s using Finset.induction with
+  | empty => simp
+  | @insert j s hj ih => ?_
   rw [Finset.sum_insert hj]
   apply Commute.isNilpotent_add
   · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))

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

@@ -62,7 +62,7 @@ lemma IsNilpotent.pow {n : ℕ} {S : Type*} [MonoidWithZero S] {x : S}
     (hx : IsNilpotent x) : IsNilpotent (x ^ n.succ) := by
   obtain ⟨N,hN⟩ := hx
   use N
-  rw [←pow_mul, Nat.succ_mul, pow_add, hN, mul_zero]
+  rw [← pow_mul, Nat.succ_mul, pow_add, hN, mul_zero]
 
 lemma IsNilpotent.pow_of_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
     (hx : IsNilpotent x) (hn : n ≠ 0) : IsNilpotent (x ^ n) := by

Since we have already proved Cartan subalgebras of Lie algebras with non-singular Killing forms are Abelian, the changes here mean that the following now works without any assumptions on characteristic:

example {K L : Type*} [Field K] [LieRing L] [LieAlgebra K L]
    [FiniteDimensional K L] [LieAlgebra.IsKilling K L]
    (H : LieSubalgebra K L) [H.IsCartanSubalgebra] :
    LieModule.LinearWeights K H L := inferInstance

@@ -75,6 +75,13 @@ theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩
 #align is_nilpotent_neg_iff isNilpotent_neg_iff
 
+lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]
+    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :
+    IsNilpotent (t • a) := by
+  obtain ⟨k, ha⟩ := ha
+  use k
+  rw [smul_pow, ha, smul_zero]
+
 theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
   use hr.choose

@@ -49,6 +49,9 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
 
+theorem not_isNilpotent_one [MonoidWithZero R] [Nontrivial R] :
+    ¬ IsNilpotent (1 : R) := fun ⟨_, H⟩ ↦ zero_ne_one (H.symm.trans (one_pow _))
+
 theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
   obtain ⟨n, hn⟩ := h
   use n

@@ -78,6 +78,11 @@ theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*
   rw [← map_pow, hr.choose_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
 
+lemma IsNilpotent.map_iff [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
+    [MonoidWithZeroHomClass F R S] {f : F} (hf : Function.Injective f) :
+    IsNilpotent (f r) ↔ IsNilpotent r :=
+  ⟨fun ⟨k, hk⟩ ↦ ⟨k, (map_eq_zero_iff f hf).mp <| by rwa [map_pow]⟩, fun h ↦ h.map f⟩
+
 theorem IsNilpotent.sub_one_isUnit [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
   obtain ⟨n, hn⟩ := hnil
   refine' ⟨⟨r - 1, -∑ i in Finset.range n, r ^ i, _, _⟩, rfl⟩
@@ -322,3 +327,16 @@ theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) :=
 #align module.End.is_nilpotent.mapq Module.End.IsNilpotent.mapQ
 
 end Module.End
+
+lemma NoZeroSMulDivisors.isReduced (R M : Type*)
+    [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M] [NoZeroSMulDivisors R M] :
+    IsReduced R := by
+  refine ⟨fun x ⟨k, hk⟩ ↦ ?_⟩
+  induction' k with k ih
+  · rw [Nat.zero_eq, pow_zero] at hk
+    exact eq_zero_of_zero_eq_one hk.symm x
+  · obtain ⟨m : M, hm : m ≠ 0⟩ := exists_ne (0 : M)
+    have : x ^ (k + 1) • m = 0 := by simp only [hk, zero_smul]
+    rw [pow_succ, mul_smul] at this
+    rcases eq_zero_or_eq_zero_of_smul_eq_zero this with rfl | hx; rfl
+    exact ih <| (eq_zero_or_eq_zero_of_smul_eq_zero hx).resolve_right hm

We define a type class Finset.HasAntidiagonal A which contains a function antidiagonal : A → Finset (A × A) such that antidiagonal n is the Finset of all pairs adding to n, as witnessed by mem_antidiagonal.

When A is a canonically ordered add monoid with locally finite order this typeclass can be instantiated with Finset.antidiagonalOfLocallyFinite. This applies in particular when A is ℕ, more generally or σ →₀ ℕ, or even ι →₀ A under the additional assumption OrderedSub A that make it a canonically ordered add monoid. (In fact, we would just need an AddMonoid with a compatible order, finite Iic, such that if a + b = n, then a, b ≤ n, and any finiteness condition would be OK.)

For computational reasons it is better to manually provide instances for ℕ and σ →₀ ℕ, to avoid quadratic runtime performance. These instances are provided as Finset.Nat.instHasAntidiagonal and Finsupp.instHasAntidiagonal. This is why Finset.antidiagonalOfLocallyFinite is an abbrev and not an instance.

This definition does not exactly match with that of Multiset.antidiagonal defined in Mathlib.Data.Multiset.Antidiagonal, because of the multiplicities. Indeed, by counting multiplicities, Multiset α is equivalent to α →₀ ℕ, but Finset.antidiagonal and Multiset.antidiagonal will return different objects. For example, for s : Multiset ℕ := {0,0,0}, Multiset.antidiagonal s has 8 elements but Finset.antidiagonal s has only 4.

def s : Multiset ℕ := {0, 0, 0}
#eval (Finset.antidiagonal s).card -- 4
#eval Multiset.card (Multiset.antidiagonal s) -- 8

TODO

• Define HasMulAntidiagonal (for monoids). For PNat, we will recover the set of divisors of a strictly positive integer.

This closes #7917

Co-authored by: María Inés de Frutos-Fernández <mariaines.dff@gmail.com> and Eric Wieser <efw27@cam.ac.uk>

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -181,7 +181,7 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   apply Finset.sum_eq_zero
   rintro ⟨i, j⟩ hij
   suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
-  cases' Nat.le_or_le_of_add_eq_add_pred (Finset.Nat.mem_antidiagonal.mp hij) with hi hj
+  cases' Nat.le_or_le_of_add_eq_add_pred (Finset.mem_antidiagonal.mp hij) with hi hj
   · rw [pow_eq_zero_of_le hi hn, zero_mul]
   · rw [pow_eq_zero_of_le hj hm, mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add

We have turned to Type* instead of Type _, but many of them remained in mathlib because the straight replacement did not work. In general, having Type _ before the colon is a code smell, though, as it hides which types should be in the same universe and which shouldn't, and is not very robust.

This PR replaces most of the remaining Type _ before the colon (except those in category theory) by Type* or Type u. This has uncovered a few bugs (where declarations were not as polymorphic as they should be).

I had to increase heartbeats at two places when replacing Type _ by Type*, but I think it's worth it as it's really more robust.

@@ -186,7 +186,7 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   · rw [pow_eq_zero_of_le hj hm, mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
-protected lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
+protected lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
     (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
     IsNilpotent (∑ i in s, f i) := by
   classical
@@ -244,7 +244,7 @@ section CommSemiring
 
 variable [CommSemiring R] {x y : R}
 
-lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
+lemma isNilpotent_sum {ι : Type*} {s : Finset ι} {f : ι → R}
     (hnp : ∀ i ∈ s, IsNilpotent (f i)) :
     IsNilpotent (∑ i in s, f i) :=
   Commute.isNilpotent_sum hnp fun _ _ _ _ ↦ Commute.all _ _

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

@@ -87,7 +87,7 @@ theorem IsNilpotent.sub_one_isUnit [Ring R] {r : R} (hnil : IsNilpotent r) : IsU
     simp
 
 theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilpotent r)
-  (hru : Commute r (↑u⁻¹ : R)) : IsUnit (u + r) := by
+    (hru : Commute r (↑u⁻¹ : R)) : IsUnit (u + r) := by
   rw [← Units.isUnit_mul_units _ u⁻¹, add_mul, Units.mul_inv, ← IsUnit.neg_iff, add_comm, neg_add,
     ← sub_eq_add_neg]
   obtain ⟨n, hn⟩ := hnil

add lemma saying that powers of nilpotent elements are nilpotent.

@@ -55,6 +55,18 @@ theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
   rw [neg_pow, hn, mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
+lemma IsNilpotent.pow {n : ℕ} {S : Type*} [MonoidWithZero S] {x : S}
+    (hx : IsNilpotent x) : IsNilpotent (x ^ n.succ) := by
+  obtain ⟨N,hN⟩ := hx
+  use N
+  rw [←pow_mul, Nat.succ_mul, pow_add, hN, mul_zero]
+
+lemma IsNilpotent.pow_of_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
+    (hx : IsNilpotent x) (hn : n ≠ 0) : IsNilpotent (x ^ n) := by
+  cases n with
+  | zero => contradiction
+  | succ => exact IsNilpotent.pow hx
+
 @[simp]
 theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩

A simple replacement: R S : Type u changes to R S : Type*. Before this change, the example below failed:

variable {A B : Type*} [MonoidWithZero A] [MonoidWithZero B]

example {a : A} (ha : IsNilpotent a) (f : A →*₀ B) : IsNilpotent (f a) := by
  exact ha.map f

@@ -30,7 +30,7 @@ universe u v
 
 open BigOperators
 
-variable {R S : Type u} {x y : R}
+variable {R S : Type*} {x y : R}
 
 /-- An element is said to be nilpotent if some natural-number-power of it equals zero.
 

@@ -8,6 +8,7 @@ import Mathlib.Algebra.Algebra.Bilinear
 import Mathlib.GroupTheory.Submonoid.ZeroDivisors
 import Mathlib.RingTheory.Ideal.Operations
 import Mathlib.Algebra.GeomSum
+import Mathlib.LinearAlgebra.Matrix.ToLin
 
 #align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
 
@@ -284,6 +285,16 @@ theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilp
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
 
+variable {R}
+variable {ι M : Type*} [Fintype ι] [DecidableEq ι] [AddCommMonoid M] [Module R M]
+
+@[simp]
+lemma isNilpotent_toMatrix_iff (b : Basis ι R M) (f : M →ₗ[R] M) :
+    IsNilpotent (toMatrix b b f) ↔ IsNilpotent f := by
+  refine' exists_congr fun k ↦ _
+  rw [toMatrix_pow]
+  exact (toMatrix b b).map_eq_zero_iff
+
 end LinearMap
 
 namespace Module.End

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).

@@ -51,7 +51,7 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
 theorem IsNilpotent.neg [Ring R] (h : IsNilpotent x) : IsNilpotent (-x) := by
   obtain ⟨n, hn⟩ := h
   use n
-  rw [neg_pow, hn, MulZeroClass.mul_zero]
+  rw [neg_pow, hn, mul_zero]
 #align is_nilpotent.neg IsNilpotent.neg
 
 @[simp]
@@ -167,10 +167,10 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   rw [h_comm.add_pow']
   apply Finset.sum_eq_zero
   rintro ⟨i, j⟩ hij
-  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, MulZeroClass.mul_zero]
+  suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
   cases' Nat.le_or_le_of_add_eq_add_pred (Finset.Nat.mem_antidiagonal.mp hij) with hi hj
-  · rw [pow_eq_zero_of_le hi hn, MulZeroClass.zero_mul]
-  · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
+  · rw [pow_eq_zero_of_le hi hn, zero_mul]
+  · rw [pow_eq_zero_of_le hj hm, mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
 protected lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
@@ -188,7 +188,7 @@ protected lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) := by
   obtain ⟨n, hn⟩ := h
   use n
-  rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
+  rw [h_comm.mul_pow, hn, zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
 
 protected lemma isNilpotent_mul_left_iff (hy : y ∈ nonZeroDivisorsLeft R) :

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

This has nice performance benefits.

@@ -59,7 +59,7 @@ theorem isNilpotent_neg_iff [Ring R] : IsNilpotent (-x) ↔ IsNilpotent x :=
   ⟨fun h => neg_neg x ▸ h.neg, fun h => h.neg⟩
 #align is_nilpotent_neg_iff isNilpotent_neg_iff
 
-theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type _}
+theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
     [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) : IsNilpotent (f r) := by
   use hr.choose
   rw [← map_pow, hr.choose_spec, map_zero]
@@ -84,7 +84,7 @@ theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilp
 
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
 @[mk_iff isReduced_iff]
-class IsReduced (R : Type _) [Zero R] [Pow R ℕ] : Prop where
+class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where
   /-- A reduced structure has no nonzero nilpotent elements. -/
   eq_zero : ∀ x : R, IsNilpotent x → x = 0
 #align is_reduced IsReduced
@@ -108,7 +108,7 @@ theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x
   ⟨fun h => h.eq_zero, fun h => h.symm ▸ IsNilpotent.zero⟩
 #align is_nilpotent_iff_eq_zero isNilpotent_iff_eq_zero
 
-theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _}
+theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
     [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective f) [IsReduced S] :
     IsReduced R := by
   constructor
@@ -237,7 +237,7 @@ lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
   Commute.isNilpotent_sum hnp fun _ _ _ _ ↦ Commute.all _ _
 
 /-- The nilradical of a commutative semiring is the ideal of nilpotent elements. -/
-def nilradical (R : Type _) [CommSemiring R] : Ideal R :=
+def nilradical (R : Type*) [CommSemiring R] : Ideal R :=
   (0 : Ideal R).radical
 #align nilradical nilradical
 
@@ -245,7 +245,7 @@ theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
 
-theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
+theorem nilradical_eq_sInf (R : Type*) [CommSemiring R] :
     nilradical R = sInf { J : Ideal R | J.IsPrime } :=
   (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
 #align nilradical_eq_Inf nilradical_eq_sInf
@@ -260,7 +260,7 @@ theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :
 #align nilradical_le_prime nilradical_le_prime
 
 @[simp]
-theorem nilradical_eq_zero (R : Type _) [CommSemiring R] [IsReduced R] : nilradical R = 0 :=
+theorem nilradical_eq_zero (R : Type*) [CommSemiring R] [IsReduced R] : nilradical R = 0 :=
   Ideal.ext fun _ => isNilpotent_iff_eq_zero
 #align nilradical_eq_zero nilradical_eq_zero
 

@@ -5,6 +5,7 @@ Authors: Oliver Nash
 -/
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
+import Mathlib.GroupTheory.Submonoid.ZeroDivisors
 import Mathlib.RingTheory.Ideal.Operations
 import Mathlib.Algebra.GeomSum
 
@@ -43,7 +44,7 @@ theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) :
   ⟨n, e⟩
 #align is_nilpotent.mk IsNilpotent.mk
 
-theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
+@[simp] theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) :=
   ⟨1, pow_one 0⟩
 #align is_nilpotent.zero IsNilpotent.zero
 
@@ -172,17 +173,43 @@ theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent
   · rw [pow_eq_zero_of_le hj hm, MulZeroClass.mul_zero]
 #align commute.is_nilpotent_add Commute.isNilpotent_add
 
+protected lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
+    (hnp : ∀ i ∈ s, IsNilpotent (f i)) (h_comm : ∀ i j, i ∈ s → j ∈ s → Commute (f i) (f j)) :
+    IsNilpotent (∑ i in s, f i) := by
+  classical
+  induction' s using Finset.induction with j s hj ih; simp
+  rw [Finset.sum_insert hj]
+  apply Commute.isNilpotent_add
+  · exact Commute.sum_right _ _ _ (fun i hi ↦ h_comm _ _ (by simp) (by simp [hi]))
+  · apply hnp; simp
+  · exact ih (fun i hi ↦ hnp i (by simp [hi]))
+      (fun i j hi hj ↦ h_comm i j (by simp [hi]) (by simp [hj]))
+
 theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) := by
   obtain ⟨n, hn⟩ := h
   use n
   rw [h_comm.mul_pow, hn, MulZeroClass.zero_mul]
 #align commute.is_nilpotent_mul_left Commute.isNilpotent_mul_left
 
+protected lemma isNilpotent_mul_left_iff (hy : y ∈ nonZeroDivisorsLeft R) :
+    IsNilpotent (x * y) ↔ IsNilpotent x := by
+  refine' ⟨_, h_comm.isNilpotent_mul_left⟩
+  rintro ⟨k, hk⟩
+  rw [mul_pow h_comm] at hk
+  exact ⟨k, (nonZeroDivisorsLeft R).pow_mem hy k _ hk⟩
+
 theorem isNilpotent_mul_right (h : IsNilpotent y) : IsNilpotent (x * y) := by
   rw [h_comm.eq]
   exact h_comm.symm.isNilpotent_mul_left h
 #align commute.is_nilpotent_mul_right Commute.isNilpotent_mul_right
 
+protected lemma isNilpotent_mul_right_iff (hx : x ∈ nonZeroDivisorsRight R) :
+    IsNilpotent (x * y) ↔ IsNilpotent y := by
+  refine' ⟨_, h_comm.isNilpotent_mul_right⟩
+  rintro ⟨k, hk⟩
+  rw [mul_pow h_comm] at hk
+  exact ⟨k, (nonZeroDivisorsRight R).pow_mem hx k _ hk⟩
+
 end Semiring
 
 section Ring
@@ -202,7 +229,12 @@ end Commute
 
 section CommSemiring
 
-variable [CommSemiring R]
+variable [CommSemiring R] {x y : R}
+
+lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}
+    (hnp : ∀ i ∈ s, IsNilpotent (f i)) :
+    IsNilpotent (∑ i in s, f i) :=
+  Commute.isNilpotent_sum hnp fun _ _ _ _ ↦ Commute.all _ _
 
 /-- The nilradical of a commutative semiring is the ideal of nilpotent elements. -/
 def nilradical (R : Type _) [CommSemiring R] : Ideal R :=

• depends on: #5966

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

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module ring_theory.nilpotent
-! leanprover-community/mathlib commit da420a8c6dd5bdfb85c4ced85c34388f633bc6ff
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
 import Mathlib.RingTheory.Ideal.Operations
 import Mathlib.Algebra.GeomSum
 
+#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
+
 /-!
 # Nilpotent elements
 

We proved that a polynomial is a unit if and only if all of its coefficients are nilpotent, except the constant term which is a unit.

Co-authored-by: Cyprien Chauveau cyprien.chauveau@etu.u-paris.fr Co-authored-by: Lucas Pouillart lucas.pouillart@etu.u-paris.fr

Co-authored-by: EmilieUthaiwat <102412311+EmilieUthaiwat@users.noreply.github.com>

@@ -11,6 +11,7 @@ Authors: Oliver Nash
 import Mathlib.Data.Nat.Choose.Sum
 import Mathlib.Algebra.Algebra.Bilinear
 import Mathlib.RingTheory.Ideal.Operations
+import Mathlib.Algebra.GeomSum
 
 /-!
 # Nilpotent elements
@@ -28,6 +29,8 @@ import Mathlib.RingTheory.Ideal.Operations
 
 universe u v
 
+open BigOperators
+
 variable {R S : Type u} {x y : R}
 
 /-- An element is said to be nilpotent if some natural-number-power of it equals zero.
@@ -64,6 +67,23 @@ theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type
   rw [← map_pow, hr.choose_spec, map_zero]
 #align is_nilpotent.map IsNilpotent.map
 
+theorem IsNilpotent.sub_one_isUnit [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by
+  obtain ⟨n, hn⟩ := hnil
+  refine' ⟨⟨r - 1, -∑ i in Finset.range n, r ^ i, _, _⟩, rfl⟩
+  · rw [mul_neg, mul_geom_sum, hn]
+    simp
+  · rw [neg_mul, geom_sum_mul, hn]
+    simp
+
+theorem Commute.IsNilpotent.add_isUnit [Ring R] {r : R} {u : Rˣ} (hnil : IsNilpotent r)
+  (hru : Commute r (↑u⁻¹ : R)) : IsUnit (u + r) := by
+  rw [← Units.isUnit_mul_units _ u⁻¹, add_mul, Units.mul_inv, ← IsUnit.neg_iff, add_comm, neg_add,
+    ← sub_eq_add_neg]
+  obtain ⟨n, hn⟩ := hnil
+  refine' IsNilpotent.sub_one_isUnit ⟨n, _⟩
+  rw [neg_pow, hru.mul_pow, hn]
+  simp
+
 /-- A structure that has zero and pow is reduced if it has no nonzero nilpotent elements. -/
 @[mk_iff isReduced_iff]
 class IsReduced (R : Type _) [Zero R] [Pow R ℕ] : Prop where

Changes are of the form

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

@@ -224,14 +224,14 @@ variable (R) {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A]
 @[simp]
 theorem isNilpotent_mulLeft_iff (a : A) : IsNilpotent (mulLeft R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
-      simp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn⊢ <;>
+      simp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_left_iff LinearMap.isNilpotent_mulLeft_iff
 
 @[simp]
 theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by
   constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
-      simp only [mulRight_eq_zero_iff, pow_mulRight] at hn⊢ <;>
+      simp only [mulRight_eq_zero_iff, pow_mulRight] at hn ⊢ <;>
     exact hn
 #align linear_map.is_nilpotent_mul_right_iff LinearMap.isNilpotent_mulRight_iff
 

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

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

@@ -196,18 +196,18 @@ theorem mem_nilradical : x ∈ nilradical R ↔ IsNilpotent x :=
   Iff.rfl
 #align mem_nilradical mem_nilradical
 
-theorem nilradical_eq_infₛ (R : Type _) [CommSemiring R] :
-    nilradical R = infₛ { J : Ideal R | J.IsPrime } :=
-  (Ideal.radical_eq_infₛ ⊥).trans <| by simp_rw [and_iff_right bot_le]
-#align nilradical_eq_Inf nilradical_eq_infₛ
+theorem nilradical_eq_sInf (R : Type _) [CommSemiring R] :
+    nilradical R = sInf { J : Ideal R | J.IsPrime } :=
+  (Ideal.radical_eq_sInf ⊥).trans <| by simp_rw [and_iff_right bot_le]
+#align nilradical_eq_Inf nilradical_eq_sInf
 
 theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J := by
-  rw [← mem_nilradical, nilradical_eq_infₛ, Submodule.mem_infₛ]
+  rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]
   rfl
 #align nilpotent_iff_mem_prime nilpotent_iff_mem_prime
 
 theorem nilradical_le_prime (J : Ideal R) [H : J.IsPrime] : nilradical R ≤ J :=
-  (nilradical_eq_infₛ R).symm ▸ infₛ_le H
+  (nilradical_eq_sInf R).symm ▸ sInf_le H
 #align nilradical_le_prime nilradical_le_prime
 
 @[simp]

We make sure that the canonical path from NonAssocSemiring to Ring passes through Semiring, as this is a path which is followed all the time in linear algebra where the defining semilinear map σ : R →+* S depends on the NonAssocSemiring structure of R and S while the module definition depends on the Semiring structure.

Tt is not currently possible to adjust priorities by hand (see lean4#2115). Instead, the last declared instance is used, so we make sure that Semiring is declared after NonAssocRing, so that Semiring -> NonAssocSemiring is tried before NonAssocRing -> NonAssocSemiring.

@@ -100,9 +100,6 @@ theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type _
   exact (hx.map f).eq_zero
 #align is_reduced_of_injective isReduced_of_injective
 
--- Porting note: Added etaExperiment line to synthesize RingHomClass
-set_option synthInstance.etaExperiment true
-
 theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
     [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
     (RingHom.ker f).IsRadical ↔ IsReduced S := by
@@ -110,8 +107,6 @@ theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [
   rfl
 #align ring_hom.ker_is_radical_iff_reduced_of_surjective RingHom.ker_isRadical_iff_reduced_of_surjective
 
-set_option synthInstance.etaExperiment false
-
 /-- An element `y` in a monoid is radical if for any element `x`, `y` divides `x` whenever it
   divides a power of `x`. -/
 def IsRadical [Dvd R] [Pow R ℕ] (y : R) : Prop :=