ring_theory.perfection ⟷ Mathlib.RingTheory.Perfection

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

@@ -7,7 +7,7 @@ import Algebra.CharP.Pi
 import Algebra.CharP.Quotient
 import Algebra.CharP.Subring
 import Algebra.Ring.Pi
-import Analysis.SpecialFunctions.Pow.Nnreal
+import Analysis.SpecialFunctions.Pow.NNReal
 import FieldTheory.PerfectClosure
 import RingTheory.Localization.FractionRing
 import RingTheory.Subring.Basic
@@ -672,7 +672,7 @@ theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0)
     exact coeff_nat_find_add_ne_zero k
   erw [ih (coeff_nat_find_add_ne_zero k), ← hx, ← coeff_pow_p, RingHom.map_pow, ← hx, ←
     RingHom.map_pow, ModP.preVal_mk h1, ModP.preVal_mk h2, RingHom.map_pow, v.map_pow, ← pow_mul,
-    pow_succ]
+    pow_succ']
   rfl
 #align pre_tilt.val_aux_eq PreTilt.valAux_eq
 -/

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

@@ -561,10 +561,10 @@ theorem preVal_add (x y : ModP K v O hv p) :
 theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x ≠ 0 :=
   by
   refine'
-    ⟨fun h hx => by rw [hx, pre_val_zero] at h ; exact not_lt_zero' h, fun h =>
+    ⟨fun h hx => by rw [hx, pre_val_zero] at h; exact not_lt_zero' h, fun h =>
       lt_of_not_le fun hp => h _⟩
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
-  rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp 
+  rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp
   rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]; exact hp
 #align mod_p.v_p_lt_pre_val ModP.v_p_lt_preVal
 -/
@@ -572,7 +572,7 @@ theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x 
 #print ModP.preVal_eq_zero /-
 theorem preVal_eq_zero {x : ModP K v O hv p} : preVal K v O hv p x = 0 ↔ x = 0 :=
   ⟨fun hvx =>
-    by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0 ;
+    by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0;
       exact not_lt_zero' hx0,
     fun hx => hx.symm ▸ preVal_zero⟩
 #align mod_p.pre_val_eq_zero ModP.preVal_eq_zero
@@ -598,10 +598,10 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (Nat.cast_pos.2 hp.1.Pos)
-  rw [← RingHom.map_mul]; rw [← RingHom.map_pow] at hx hy 
+  rw [← RingHom.map_mul]; rw [← RingHom.map_pow] at hx hy
   rw [← v_p_lt_val hv] at hx hy ⊢
   rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
-    mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.NeZero : (p : ℝ) ≠ 0), rpow_one] at hx hy 
+    mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.NeZero : (p : ℝ) ≠ 0), rpow_one] at hx hy
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)
   by_cases hvp : v p = 0; · rw [hvp]; exact zero_le _; replace hvp := zero_lt_iff.2 hvp
   conv_lhs => rw [← rpow_one (v p)]; rw [← rpow_add (ne_of_gt hvp)]
@@ -764,7 +764,7 @@ theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   by_cases hf0 : f = 0; · rw [hf0]; exact iff_of_true (Valuation.map_zero _) rfl
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := Classical.not_forall.1 fun h => hf0 <| Perfection.ext h
   show val_aux K v O hv p f = 0 ↔ f = 0; refine' iff_of_false (fun hvf => hn _) hf0
-  rw [val_aux_eq hn] at hvf ; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf 
+  rw [val_aux_eq hn] at hvf; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf
 #align pre_tilt.map_eq_zero PreTilt.map_eq_zero
 -/
 

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

@@ -735,8 +735,8 @@ theorem valAux_add (f g : PreTilt K v O hv p) :
     le_max_iff.1
       (ModP.preVal_add (coeff _ _ (max (max m n) k) f) (coeff _ _ (max (max m n) k) g)) with
     h h
-  · exact le_max_of_le_left (pow_le_pow_of_le_left' h _)
-  · exact le_max_of_le_right (pow_le_pow_of_le_left' h _)
+  · exact le_max_of_le_left (pow_le_pow_left' h _)
+  · exact le_max_of_le_right (pow_le_pow_left' h _)
 #align pre_tilt.val_aux_add PreTilt.valAux_add
 -/
 

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

@@ -700,8 +700,8 @@ theorem valAux_mul (f g : PreTilt K v O hv p) :
   by
   by_cases hf : f = 0; · rw [hf, MulZeroClass.zero_mul, val_aux_zero, MulZeroClass.zero_mul]
   by_cases hg : g = 0; · rw [hg, MulZeroClass.mul_zero, val_aux_zero, MulZeroClass.mul_zero]
-  obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
-  obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
+  obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := Classical.not_forall.1 fun h => hf <| Perfection.ext h
+  obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := Classical.not_forall.1 fun h => hg <| Perfection.ext h
   replace hm := coeff_ne_zero_of_le hm (le_max_left m n)
   replace hn := coeff_ne_zero_of_le hn (le_max_right m n)
   have hfg : coeff _ _ (max m n + 1) (f * g) ≠ 0 :=
@@ -722,9 +722,10 @@ theorem valAux_add (f g : PreTilt K v O hv p) :
   by_cases hf : f = 0; · rw [hf, zero_add, val_aux_zero, max_eq_right]; exact zero_le _
   by_cases hg : g = 0; · rw [hg, add_zero, val_aux_zero, max_eq_left]; exact zero_le _
   by_cases hfg : f + g = 0; · rw [hfg, val_aux_zero]; exact zero_le _
-  replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
-  replace hg : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
-  replace hfg : ∃ n, coeff _ _ n (f + g) ≠ 0 := not_forall.1 fun h => hfg <| Perfection.ext h
+  replace hf : ∃ n, coeff _ _ n f ≠ 0 := Classical.not_forall.1 fun h => hf <| Perfection.ext h
+  replace hg : ∃ n, coeff _ _ n g ≠ 0 := Classical.not_forall.1 fun h => hg <| Perfection.ext h
+  replace hfg : ∃ n, coeff _ _ n (f + g) ≠ 0 :=
+    Classical.not_forall.1 fun h => hfg <| Perfection.ext h
   obtain ⟨m, hm⟩ := hf; obtain ⟨n, hn⟩ := hg; obtain ⟨k, hk⟩ := hfg
   replace hm := coeff_ne_zero_of_le hm (le_trans (le_max_left m n) (le_max_left _ k))
   replace hn := coeff_ne_zero_of_le hn (le_trans (le_max_right m n) (le_max_left _ k))
@@ -761,7 +762,7 @@ variable {K v O hv p}
 theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   by
   by_cases hf0 : f = 0; · rw [hf0]; exact iff_of_true (Valuation.map_zero _) rfl
-  obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf0 <| Perfection.ext h
+  obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := Classical.not_forall.1 fun h => hf0 <| Perfection.ext h
   show val_aux K v O hv p f = 0 ↔ f = 0; refine' iff_of_false (fun hvf => hn _) hf0
   rw [val_aux_eq hn] at hvf ; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf 
 #align pre_tilt.map_eq_zero PreTilt.map_eq_zero

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

@@ -3,15 +3,15 @@ Copyright (c) 2020 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
-import Mathbin.Algebra.CharP.Pi
-import Mathbin.Algebra.CharP.Quotient
-import Mathbin.Algebra.CharP.Subring
-import Mathbin.Algebra.Ring.Pi
-import Mathbin.Analysis.SpecialFunctions.Pow.Nnreal
-import Mathbin.FieldTheory.PerfectClosure
-import Mathbin.RingTheory.Localization.FractionRing
-import Mathbin.RingTheory.Subring.Basic
-import Mathbin.RingTheory.Valuation.Integers
+import Algebra.CharP.Pi
+import Algebra.CharP.Quotient
+import Algebra.CharP.Subring
+import Algebra.Ring.Pi
+import Analysis.SpecialFunctions.Pow.Nnreal
+import FieldTheory.PerfectClosure
+import RingTheory.Localization.FractionRing
+import RingTheory.Subring.Basic
+import RingTheory.Valuation.Integers
 
 #align_import ring_theory.perfection from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
 

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

@@ -43,7 +43,7 @@ def Monoid.perfection (M : Type u₁) [CommMonoid M] (p : ℕ) : Submonoid (ℕ
     where
   carrier := {f | ∀ n, f (n + 1) ^ p = f n}
   one_mem' n := one_pow _
-  mul_mem' f g hf hg n := (mul_pow _ _ _).trans <| congr_arg₂ _ (hf n) (hg n)
+  hMul_mem' f g hf hg n := (mul_pow _ _ _).trans <| congr_arg₂ _ (hf n) (hg n)
 #align monoid.perfection Monoid.perfection
 -/
 
@@ -243,7 +243,7 @@ def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Typ
           rw [← f.map_pow, Function.iterate_succ_apply', pthRoot_pow_p]⟩
       map_one' := ext fun n => (congr_arg f <| RingHom.iterate_map_one _ _).trans f.map_one
       map_mul' := fun x y =>
-        ext fun n => (congr_arg f <| RingHom.iterate_map_mul _ _ _ _).trans <| f.map_mul _ _
+        ext fun n => (congr_arg f <| RingHom.iterate_map_hMul _ _ _ _).trans <| f.map_hMul _ _
       map_zero' := ext fun n => (congr_arg f <| RingHom.iterate_map_zero _ _).trans f.map_zero
       map_add' := fun x y =>
         ext fun n => (congr_arg f <| RingHom.iterate_map_add _ _ _ _).trans <| f.map_add _ _ }
@@ -275,7 +275,7 @@ def map (φ : R →+* S) : Ring.Perfection R p →+* Ring.Perfection S p
     where
   toFun f := ⟨fun n => φ (coeff R p n f), fun n => by rw [← φ.map_pow, coeff_pow_p']⟩
   map_one' := Subtype.eq <| funext fun n => φ.map_one
-  map_mul' f g := Subtype.eq <| funext fun n => φ.map_mul _ _
+  map_mul' f g := Subtype.eq <| funext fun n => φ.map_hMul _ _
   map_zero' := Subtype.eq <| funext fun n => φ.map_zero
   map_add' f g := Subtype.eq <| funext fun n => φ.map_add _ _
 #align perfection.map Perfection.map

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

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module ring_theory.perfection
-! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.CharP.Pi
 import Mathbin.Algebra.CharP.Quotient
@@ -18,6 +13,8 @@ import Mathbin.RingTheory.Localization.FractionRing
 import Mathbin.RingTheory.Subring.Basic
 import Mathbin.RingTheory.Valuation.Integers
 
+#align_import ring_theory.perfection from "leanprover-community/mathlib"@"087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197"
+
 /-!
 # Ring Perfection and Tilt
 

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

@@ -62,6 +62,7 @@ def Ring.perfectionSubsemiring (R : Type u₁) [CommSemiring R] (p : ℕ) [hp :
 #align ring.perfection_subsemiring Ring.perfectionSubsemiring
 -/
 
+#print Ring.perfectionSubring /-
 /-- The perfection of a ring `R` with characteristic `p`, as a subring,
 defined to be the projective limit of `R` using the Frobenius maps `R → R`
 indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. -/
@@ -70,6 +71,7 @@ def Ring.perfectionSubring (R : Type u₁) [CommRing R] (p : ℕ) [hp : Fact p.P
   (Ring.perfectionSubsemiring R p).toSubring fun n => by
     simp_rw [← frobenius_def, Pi.neg_apply, Pi.one_apply, RingHom.map_neg, RingHom.map_one]
 #align ring.perfection_subring Ring.perfectionSubring
+-/
 
 #print Ring.Perfection /-
 /-- The perfection of a ring `R` with characteristic `p`,
@@ -84,21 +86,23 @@ namespace Perfection
 
 variable (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact p.Prime] [CharP R p]
 
-include hp
-
 instance : CommSemiring (Ring.Perfection R p) :=
   (Ring.perfectionSubsemiring R p).toCommSemiring
 
 instance : CharP (Ring.Perfection R p) p :=
   CharP.subsemiring (ℕ → R) p (Ring.perfectionSubsemiring R p)
 
+#print Perfection.ring /-
 instance ring (R : Type u₁) [CommRing R] [CharP R p] : Ring (Ring.Perfection R p) :=
   (Ring.perfectionSubring R p).toRing
 #align perfection.ring Perfection.ring
+-/
 
+#print Perfection.commRing /-
 instance commRing (R : Type u₁) [CommRing R] [CharP R p] : CommRing (Ring.Perfection R p) :=
   (Ring.perfectionSubring R p).toCommRing
 #align perfection.comm_ring Perfection.commRing
+-/
 
 instance : Inhabited (Ring.Perfection R p) :=
   ⟨0⟩
@@ -117,10 +121,12 @@ def coeff (n : ℕ) : Ring.Perfection R p →+* R
 
 variable {R p}
 
+#print Perfection.ext /-
 @[ext]
 theorem ext {f g : Ring.Perfection R p} (h : ∀ n, coeff R p n f = coeff R p n g) : f = g :=
   Subtype.eq <| funext h
 #align perfection.ext Perfection.ext
+-/
 
 variable (R p)
 
@@ -138,38 +144,52 @@ def pthRoot : Ring.Perfection R p →+* Ring.Perfection R p
 
 variable {R p}
 
+#print Perfection.coeff_mk /-
 @[simp]
 theorem coeff_mk (f : ℕ → R) (hf) (n : ℕ) : coeff R p n ⟨f, hf⟩ = f n :=
   rfl
 #align perfection.coeff_mk Perfection.coeff_mk
+-/
 
+#print Perfection.coeff_pthRoot /-
 theorem coeff_pthRoot (f : Ring.Perfection R p) (n : ℕ) :
     coeff R p n (pthRoot R p f) = coeff R p (n + 1) f :=
   rfl
 #align perfection.coeff_pth_root Perfection.coeff_pthRoot
+-/
 
+#print Perfection.coeff_pow_p /-
 theorem coeff_pow_p (f : Ring.Perfection R p) (n : ℕ) : coeff R p (n + 1) (f ^ p) = coeff R p n f :=
   by rw [RingHom.map_pow]; exact f.2 n
 #align perfection.coeff_pow_p Perfection.coeff_pow_p
+-/
 
+#print Perfection.coeff_pow_p' /-
 theorem coeff_pow_p' (f : Ring.Perfection R p) (n : ℕ) : coeff R p (n + 1) f ^ p = coeff R p n f :=
   f.2 n
 #align perfection.coeff_pow_p' Perfection.coeff_pow_p'
+-/
 
+#print Perfection.coeff_frobenius /-
 theorem coeff_frobenius (f : Ring.Perfection R p) (n : ℕ) :
     coeff R p (n + 1) (frobenius _ p f) = coeff R p n f := by apply coeff_pow_p f n
 #align perfection.coeff_frobenius Perfection.coeff_frobenius
+-/
 
+#print Perfection.coeff_iterate_frobenius /-
 -- `coeff_pow_p f n` also works but is slow!
 theorem coeff_iterate_frobenius (f : Ring.Perfection R p) (n m : ℕ) :
     coeff R p (n + m) ((frobenius _ p^[m]) f) = coeff R p n f :=
   Nat.recOn m rfl fun m ih => by erw [Function.iterate_succ_apply', coeff_frobenius, ih]
 #align perfection.coeff_iterate_frobenius Perfection.coeff_iterate_frobenius
+-/
 
+#print Perfection.coeff_iterate_frobenius' /-
 theorem coeff_iterate_frobenius' (f : Ring.Perfection R p) (n m : ℕ) (hmn : m ≤ n) :
     coeff R p n ((frobenius _ p^[m]) f) = coeff R p (n - m) f :=
   Eq.symm <| (coeff_iterate_frobenius _ _ m).symm.trans <| (tsub_add_cancel_of_le hmn).symm ▸ rfl
 #align perfection.coeff_iterate_frobenius' Perfection.coeff_iterate_frobenius'
+-/
 
 #print Perfection.pthRoot_frobenius /-
 theorem pthRoot_frobenius : (pthRoot R p).comp (frobenius _ p) = RingHom.id _ :=
@@ -187,16 +207,20 @@ theorem frobenius_pthRoot : (frobenius _ p).comp (pthRoot R p) = RingHom.id _ :=
 #align perfection.frobenius_pth_root Perfection.frobenius_pthRoot
 -/
 
+#print Perfection.coeff_add_ne_zero /-
 theorem coeff_add_ne_zero {f : Ring.Perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
     coeff R p (n + k) f ≠ 0 :=
   Nat.recOn k hfn fun k ih h => ih <| by erw [← coeff_pow_p, RingHom.map_pow, h, zero_pow hp.1.Pos]
 #align perfection.coeff_add_ne_zero Perfection.coeff_add_ne_zero
+-/
 
+#print Perfection.coeff_ne_zero_of_le /-
 theorem coeff_ne_zero_of_le {f : Ring.Perfection R p} {m n : ℕ} (hfm : coeff R p m f ≠ 0)
     (hmn : m ≤ n) : coeff R p n f ≠ 0 :=
   let ⟨k, hk⟩ := Nat.exists_eq_add_of_le hmn
   hk.symm ▸ coeff_add_ne_zero hfm k
 #align perfection.coeff_ne_zero_of_le Perfection.coeff_ne_zero_of_le
+-/
 
 variable (R p)
 
@@ -237,11 +261,13 @@ def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Typ
 #align perfection.lift Perfection.lift
 -/
 
+#print Perfection.hom_ext /-
 theorem hom_ext {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂}
     [CommSemiring S] [CharP S p] {f g : R →+* Ring.Perfection S p}
     (hfg : ∀ x, coeff S p 0 (f x) = coeff S p 0 (g x)) : f = g :=
   (lift p R S).symm.Injective <| RingHom.ext hfg
 #align perfection.hom_ext Perfection.hom_ext
+-/
 
 variable {R} {S : Type u₂} [CommSemiring S] [CharP S p]
 
@@ -258,10 +284,12 @@ def map (φ : R →+* S) : Ring.Perfection R p →+* Ring.Perfection S p
 #align perfection.map Perfection.map
 -/
 
+#print Perfection.coeff_map /-
 theorem coeff_map (φ : R →+* S) (f : Ring.Perfection R p) (n : ℕ) :
     coeff S p n (map p φ f) = φ (coeff R p n f) :=
   rfl
 #align perfection.coeff_map Perfection.coeff_map
+-/
 
 end Perfection
 
@@ -284,6 +312,7 @@ variable {R : Type u₁} [CommSemiring R] [CharP R p]
 
 variable {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p]
 
+#print PerfectionMap.mk' /-
 /-- Create a `perfection_map` from an isomorphism to the perfection. -/
 @[simps]
 theorem mk' {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : Perfection.lift p P R f = g) :
@@ -298,6 +327,7 @@ theorem mk' {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : Perfection.
         show Perfection.coeff R p n (Perfection.lift p P R f x) = Perfection.coeff R p n ⟨y, hy⟩ by
           rw [hfg, ← coeFn_coeBase, hx]⟩ }
 #align perfection_map.mk' PerfectionMap.mk'
+-/
 
 variable (p R P)
 
@@ -322,6 +352,7 @@ theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
 
 variable {p R P}
 
+#print PerfectionMap.equiv /-
 /-- A perfection map induces an isomorphism to the prefection. -/
 noncomputable def equiv {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring.Perfection R p :=
   RingEquiv.ofBijective (Perfection.lift p P R π)
@@ -329,6 +360,7 @@ noncomputable def equiv {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring
       let ⟨x, hx⟩ := m.Surjective f.1 f.2
       ⟨x, Perfection.ext <| hx⟩⟩
 #align perfection_map.equiv PerfectionMap.equiv
+-/
 
 #print PerfectionMap.equiv_apply /-
 theorem equiv_apply {π : P →+* R} (m : PerfectionMap p π) (x : P) :
@@ -337,25 +369,33 @@ theorem equiv_apply {π : P →+* R} (m : PerfectionMap p π) (x : P) :
 #align perfection_map.equiv_apply PerfectionMap.equiv_apply
 -/
 
+#print PerfectionMap.comp_equiv /-
 theorem comp_equiv {π : P →+* R} (m : PerfectionMap p π) (x : P) :
     Perfection.coeff R p 0 (m.Equiv x) = π x :=
   rfl
 #align perfection_map.comp_equiv PerfectionMap.comp_equiv
+-/
 
+#print PerfectionMap.comp_equiv' /-
 theorem comp_equiv' {π : P →+* R} (m : PerfectionMap p π) :
     (Perfection.coeff R p 0).comp ↑m.Equiv = π :=
   RingHom.ext fun x => rfl
 #align perfection_map.comp_equiv' PerfectionMap.comp_equiv'
+-/
 
+#print PerfectionMap.comp_symm_equiv /-
 theorem comp_symm_equiv {π : P →+* R} (m : PerfectionMap p π) (f : Ring.Perfection R p) :
     π (m.Equiv.symm f) = Perfection.coeff R p 0 f :=
   (m.comp_equiv _).symm.trans <| congr_arg _ <| m.Equiv.apply_symm_apply f
 #align perfection_map.comp_symm_equiv PerfectionMap.comp_symm_equiv
+-/
 
+#print PerfectionMap.comp_symm_equiv' /-
 theorem comp_symm_equiv' {π : P →+* R} (m : PerfectionMap p π) :
     π.comp ↑m.Equiv.symm = Perfection.coeff R p 0 :=
   RingHom.ext m.comp_symm_equiv
 #align perfection_map.comp_symm_equiv' PerfectionMap.comp_symm_equiv'
+-/
 
 variable (p R P)
 
@@ -384,11 +424,13 @@ noncomputable def lift [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP
 
 variable {R p}
 
+#print PerfectionMap.hom_ext /-
 theorem hom_ext [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {P : Type u₃}
     [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π)
     {f g : R →+* P} (hfg : ∀ x, π (f x) = π (g x)) : f = g :=
   (lift p R S P π m).symm.Injective <| RingHom.ext hfg
 #align perfection_map.hom_ext PerfectionMap.hom_ext
+-/
 
 variable {R P} (p) {S : Type u₂} [CommSemiring S] [CharP S p]
 
@@ -410,10 +452,12 @@ theorem comp_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n :
 #align perfection_map.comp_map PerfectionMap.comp_map
 -/
 
+#print PerfectionMap.map_map /-
 theorem map_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
     (φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) :=
   RingHom.ext_iff.1 (comp_map p m n φ) x
 #align perfection_map.map_map PerfectionMap.map_map
+-/
 
 #print PerfectionMap.map_eq_map /-
 -- Why is this slow?
@@ -433,13 +477,13 @@ variable (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O)
 
 variable (p : ℕ)
 
-include hv
-
+#print ModP /-
 /-- `O/(p)` for `O`, ring of integers of `K`. -/
 @[nolint unused_arguments has_nonempty_instance]
 def ModP :=
   O ⧸ (Ideal.span {p} : Ideal O)
 #align mod_p ModP
+-/
 
 variable [hp : Fact p.Prime] [hvp : Fact (v p ≠ 1)]
 
@@ -448,8 +492,6 @@ namespace ModP
 instance : CommRing (ModP K v O hv p) :=
   Ideal.Quotient.commRing _
 
-include hp hvp
-
 instance : CharP (ModP K v O hv p) p :=
   CharP.quotient O p <| mt hv.one_of_isUnit <| (map_natCast (algebraMap O K) p).symm ▸ hvp.1
 
@@ -460,16 +502,17 @@ section Classical
 
 attribute [local instance] Classical.dec
 
-omit hp hvp
-
+#print ModP.preVal /-
 /-- For a field `K` with valuation `v : K → ℝ≥0` and ring of integers `O`,
 a function `O/(p) → ℝ≥0` that sends `0` to `0` and `x + (p)` to `v(x)` as long as `x ∉ (p)`. -/
 noncomputable def preVal (x : ModP K v O hv p) : ℝ≥0 :=
   if x = 0 then 0 else v (algebraMap O K x.out')
 #align mod_p.pre_val ModP.preVal
+-/
 
 variable {K v O hv p}
 
+#print ModP.preVal_mk /-
 theorem preVal_mk {x : O} (hx : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0) :
     preVal K v O hv p (Ideal.Quotient.mk _ x) = v (algebraMap O K x) :=
   by
@@ -482,11 +525,15 @@ theorem preVal_mk {x : O} (hx : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0)
   exact fun hprx =>
     hx (Ideal.Quotient.eq_zero_iff_mem.2 <| Ideal.mem_span_singleton.2 <| dvd_of_mul_left_dvd hprx)
 #align mod_p.pre_val_mk ModP.preVal_mk
+-/
 
+#print ModP.preVal_zero /-
 theorem preVal_zero : preVal K v O hv p 0 = 0 :=
   if_pos rfl
 #align mod_p.pre_val_zero ModP.preVal_zero
+-/
 
+#print ModP.preVal_mul /-
 theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
     preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y :=
   by
@@ -497,7 +544,9 @@ theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
   rw [← RingHom.map_mul] at hxy0 ⊢
   rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, RingHom.map_mul, v.map_mul]
 #align mod_p.pre_val_mul ModP.preVal_mul
+-/
 
+#print ModP.preVal_add /-
 theorem preVal_add (x y : ModP K v O hv p) :
     preVal K v O hv p (x + y) ≤ max (preVal K v O hv p x) (preVal K v O hv p y) :=
   by
@@ -509,7 +558,9 @@ theorem preVal_add (x y : ModP K v O hv p) :
   rw [← RingHom.map_add] at hxy0 ⊢
   rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, RingHom.map_add]; exact v.map_add _ _
 #align mod_p.pre_val_add ModP.preVal_add
+-/
 
+#print ModP.v_p_lt_preVal /-
 theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x ≠ 0 :=
   by
   refine'
@@ -519,28 +570,32 @@ theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x 
   rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp 
   rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]; exact hp
 #align mod_p.v_p_lt_pre_val ModP.v_p_lt_preVal
+-/
 
+#print ModP.preVal_eq_zero /-
 theorem preVal_eq_zero {x : ModP K v O hv p} : preVal K v O hv p x = 0 ↔ x = 0 :=
   ⟨fun hvx =>
     by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0 ;
       exact not_lt_zero' hx0,
     fun hx => hx.symm ▸ preVal_zero⟩
 #align mod_p.pre_val_eq_zero ModP.preVal_eq_zero
+-/
 
 variable (hv hvp)
 
+#print ModP.v_p_lt_val /-
 theorem v_p_lt_val {x : O} :
     v p < v (algebraMap O K x) ↔ (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0 := by
   rw [lt_iff_not_le, not_iff_not, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd,
     Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
 #align mod_p.v_p_lt_val ModP.v_p_lt_val
+-/
 
 open NNReal
 
 variable {hv} (hvp)
 
-include hp
-
+#print ModP.mul_ne_zero_of_pow_p_ne_zero /-
 theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
     x * y ≠ 0 := by
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -556,18 +611,19 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   refine' rpow_le_rpow_of_exponent_ge hvp (map_natCast (algebraMap O K) p ▸ hv.2 _) _
   rw [← add_div, div_le_one (Nat.cast_pos.2 hp.1.Pos : 0 < (p : ℝ))]; exact_mod_cast hp.1.two_le
 #align mod_p.mul_ne_zero_of_pow_p_ne_zero ModP.mul_ne_zero_of_pow_p_ne_zero
+-/
 
 end Classical
 
 end ModP
 
+#print PreTilt /-
 /-- Perfection of `O/(p)` where `O` is the ring of integers of `K`. -/
 @[nolint has_nonempty_instance]
 def PreTilt :=
   Ring.Perfection (ModP K v O hv p) p
 #align pre_tilt PreTilt
-
-include hp hvp
+-/
 
 namespace PreTilt
 
@@ -583,6 +639,7 @@ open scoped Classical
 
 open Perfection
 
+#print PreTilt.valAux /-
 /-- The valuation `Perfection(O/(p)) → ℝ≥0` as a function.
 Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
 otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
@@ -591,14 +648,18 @@ noncomputable def valAux (f : PreTilt K v O hv p) : ℝ≥0 :=
     ModP.preVal K v O hv p (coeff _ _ (Nat.find h) f) ^ p ^ Nat.find h
   else 0
 #align pre_tilt.val_aux PreTilt.valAux
+-/
 
 variable {K v O hv p}
 
+#print PreTilt.coeff_nat_find_add_ne_zero /-
 theorem coeff_nat_find_add_ne_zero {f : PreTilt K v O hv p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
     coeff _ _ (Nat.find h + k) f ≠ 0 :=
   coeff_add_ne_zero (Nat.find_spec h) k
 #align pre_tilt.coeff_nat_find_add_ne_zero PreTilt.coeff_nat_find_add_ne_zero
+-/
 
+#print PreTilt.valAux_eq /-
 theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
     valAux K v O hv p f = ModP.preVal K v O hv p (coeff _ _ n f) ^ p ^ n :=
   by
@@ -617,11 +678,15 @@ theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0)
     pow_succ]
   rfl
 #align pre_tilt.val_aux_eq PreTilt.valAux_eq
+-/
 
+#print PreTilt.valAux_zero /-
 theorem valAux_zero : valAux K v O hv p 0 = 0 :=
   dif_neg fun ⟨n, hn⟩ => hn rfl
 #align pre_tilt.val_aux_zero PreTilt.valAux_zero
+-/
 
+#print PreTilt.valAux_one /-
 theorem valAux_one : valAux K v O hv p 1 = 1 :=
   (valAux_eq <| show coeff (ModP K v O hv p) p 0 1 ≠ 0 from one_ne_zero).trans <|
     by
@@ -630,7 +695,9 @@ theorem valAux_one : valAux K v O hv p 1 = 1 :=
     change (1 : ModP K v O hv p) ≠ 0
     exact one_ne_zero
 #align pre_tilt.val_aux_one PreTilt.valAux_one
+-/
 
+#print PreTilt.valAux_mul /-
 theorem valAux_mul (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f * g) = valAux K v O hv p f * valAux K v O hv p g :=
   by
@@ -649,7 +716,9 @@ theorem valAux_mul (f g : PreTilt K v O hv p) :
   rw [val_aux_eq (coeff_add_ne_zero hm 1), val_aux_eq (coeff_add_ne_zero hn 1), val_aux_eq hfg]
   rw [RingHom.map_mul] at hfg ⊢; rw [ModP.preVal_mul hfg, mul_pow]
 #align pre_tilt.val_aux_mul PreTilt.valAux_mul
+-/
 
+#print PreTilt.valAux_add /-
 theorem valAux_add (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f + g) ≤ max (valAux K v O hv p f) (valAux K v O hv p g) :=
   by
@@ -671,9 +740,11 @@ theorem valAux_add (f g : PreTilt K v O hv p) :
   · exact le_max_of_le_left (pow_le_pow_of_le_left' h _)
   · exact le_max_of_le_right (pow_le_pow_of_le_left' h _)
 #align pre_tilt.val_aux_add PreTilt.valAux_add
+-/
 
 variable (K v O hv p)
 
+#print PreTilt.val /-
 /-- The valuation `Perfection(O/(p)) → ℝ≥0`.
 Given `f ∈ Perfection(O/(p))`, if `f = 0` then output `0`;
 otherwise output `pre_val(f(n))^(p^n)` for any `n` such that `f(n) ≠ 0`. -/
@@ -685,9 +756,11 @@ noncomputable def val : Valuation (PreTilt K v O hv p) ℝ≥0
   map_zero' := valAux_zero
   map_add_le_max' := valAux_add
 #align pre_tilt.val PreTilt.val
+-/
 
 variable {K v O hv p}
 
+#print PreTilt.map_eq_zero /-
 theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   by
   by_cases hf0 : f = 0; · rw [hf0]; exact iff_of_true (Valuation.map_zero _) rfl
@@ -695,6 +768,7 @@ theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   show val_aux K v O hv p f = 0 ↔ f = 0; refine' iff_of_false (fun hvf => hn _) hf0
   rw [val_aux_eq hn] at hvf ; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf 
 #align pre_tilt.map_eq_zero PreTilt.map_eq_zero
+-/
 
 end Classical
 
@@ -709,6 +783,7 @@ instance : IsDomain (PreTilt K v O hv p) :=
 
 end PreTilt
 
+#print Tilt /-
 /-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
 [scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`,
 this is implemented as the fraction field of the perfection of `O/(p)`. -/
@@ -716,6 +791,7 @@ this is implemented as the fraction field of the perfection of `O/(p)`. -/
 def Tilt :=
   FractionRing (PreTilt K v O hv p)
 #align tilt Tilt
+-/
 
 namespace Tilt
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 
 ! This file was ported from Lean 3 source module ring_theory.perfection
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
+! leanprover-community/mathlib commit 087c325ae0ab42dbdd5dee55bc37d3d5a0bf2197
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -21,6 +21,9 @@ import Mathbin.RingTheory.Valuation.Integers
 /-!
 # Ring Perfection and Tilt
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define the perfection of a ring of characteristic p, and the tilt of a field
 given a valuation to `ℝ≥0`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677

@@ -35,6 +35,7 @@ universe u₁ u₂ u₃ u₄
 
 open scoped NNReal
 
+#print Monoid.perfection /-
 /-- The perfection of a monoid `M`, defined to be the projective limit of `M`
 using the `p`-th power maps `M → M` indexed by the natural numbers, implemented as
 `{ f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }`. -/
@@ -44,7 +45,9 @@ def Monoid.perfection (M : Type u₁) [CommMonoid M] (p : ℕ) : Submonoid (ℕ
   one_mem' n := one_pow _
   mul_mem' f g hf hg n := (mul_pow _ _ _).trans <| congr_arg₂ _ (hf n) (hg n)
 #align monoid.perfection Monoid.perfection
+-/
 
+#print Ring.perfectionSubsemiring /-
 /-- The perfection of a ring `R` with characteristic `p`, as a subsemiring,
 defined to be the projective limit of `R` using the Frobenius maps `R → R`
 indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }`. -/
@@ -54,6 +57,7 @@ def Ring.perfectionSubsemiring (R : Type u₁) [CommSemiring R] (p : ℕ) [hp :
     zero_mem' := fun n => zero_pow <| hp.1.Pos
     add_mem' := fun f g hf hg n => (frobenius_add R p _ _).trans <| congr_arg₂ _ (hf n) (hg n) }
 #align ring.perfection_subsemiring Ring.perfectionSubsemiring
+-/
 
 /-- The perfection of a ring `R` with characteristic `p`, as a subring,
 defined to be the projective limit of `R` using the Frobenius maps `R → R`
@@ -64,12 +68,14 @@ def Ring.perfectionSubring (R : Type u₁) [CommRing R] (p : ℕ) [hp : Fact p.P
     simp_rw [← frobenius_def, Pi.neg_apply, Pi.one_apply, RingHom.map_neg, RingHom.map_one]
 #align ring.perfection_subring Ring.perfectionSubring
 
+#print Ring.Perfection /-
 /-- The perfection of a ring `R` with characteristic `p`,
 defined to be the projective limit of `R` using the Frobenius maps `R → R`
 indexed by the natural numbers, implemented as `{f : ℕ → R // ∀ n, f (n + 1) ^ p = f n}`. -/
 def Ring.Perfection (R : Type u₁) [CommSemiring R] (p : ℕ) : Type u₁ :=
   { f // ∀ n : ℕ, (f : ℕ → R) (n + 1) ^ p = f n }
 #align ring.perfection Ring.Perfection
+-/
 
 namespace Perfection
 
@@ -94,6 +100,7 @@ instance commRing (R : Type u₁) [CommRing R] [CharP R p] : CommRing (Ring.Perf
 instance : Inhabited (Ring.Perfection R p) :=
   ⟨0⟩
 
+#print Perfection.coeff /-
 /-- The `n`-th coefficient of an element of the perfection. -/
 def coeff (n : ℕ) : Ring.Perfection R p →+* R
     where
@@ -103,6 +110,7 @@ def coeff (n : ℕ) : Ring.Perfection R p →+* R
   map_zero' := rfl
   map_add' f g := rfl
 #align perfection.coeff Perfection.coeff
+-/
 
 variable {R p}
 
@@ -113,6 +121,7 @@ theorem ext {f g : Ring.Perfection R p} (h : ∀ n, coeff R p n f = coeff R p n
 
 variable (R p)
 
+#print Perfection.pthRoot /-
 /-- The `p`-th root of an element of the perfection. -/
 def pthRoot : Ring.Perfection R p →+* Ring.Perfection R p
     where
@@ -122,6 +131,7 @@ def pthRoot : Ring.Perfection R p →+* Ring.Perfection R p
   map_zero' := rfl
   map_add' f g := rfl
 #align perfection.pth_root Perfection.pthRoot
+-/
 
 variable {R p}
 
@@ -158,17 +168,21 @@ theorem coeff_iterate_frobenius' (f : Ring.Perfection R p) (n m : ℕ) (hmn : m
   Eq.symm <| (coeff_iterate_frobenius _ _ m).symm.trans <| (tsub_add_cancel_of_le hmn).symm ▸ rfl
 #align perfection.coeff_iterate_frobenius' Perfection.coeff_iterate_frobenius'
 
+#print Perfection.pthRoot_frobenius /-
 theorem pthRoot_frobenius : (pthRoot R p).comp (frobenius _ p) = RingHom.id _ :=
   RingHom.ext fun x =>
     ext fun n => by rw [RingHom.comp_apply, RingHom.id_apply, coeff_pth_root, coeff_frobenius]
 #align perfection.pth_root_frobenius Perfection.pthRoot_frobenius
+-/
 
+#print Perfection.frobenius_pthRoot /-
 theorem frobenius_pthRoot : (frobenius _ p).comp (pthRoot R p) = RingHom.id _ :=
   RingHom.ext fun x =>
     ext fun n => by
       rw [RingHom.comp_apply, RingHom.id_apply, RingHom.map_frobenius, coeff_pth_root, ←
         RingHom.map_frobenius, coeff_frobenius]
 #align perfection.frobenius_pth_root Perfection.frobenius_pthRoot
+-/
 
 theorem coeff_add_ne_zero {f : Ring.Perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
     coeff R p (n + k) f ≠ 0 :=
@@ -183,13 +197,16 @@ theorem coeff_ne_zero_of_le {f : Ring.Perfection R p} {m n : ℕ} (hfm : coeff R
 
 variable (R p)
 
+#print Perfection.perfectRing /-
 instance perfectRing : PerfectRing (Ring.Perfection R p) p
     where
   pthRoot' := pthRoot R p
   frobenius_pthRoot' := congr_fun <| congr_arg RingHom.toFun <| @frobenius_pthRoot R _ p _ _
   pth_root_frobenius' := congr_fun <| congr_arg RingHom.toFun <| @pthRoot_frobenius R _ p _ _
 #align perfection.perfect_ring Perfection.perfectRing
+-/
 
+#print Perfection.lift /-
 /-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
 any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* perfection S p`. -/
 @[simps]
@@ -215,6 +232,7 @@ def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Typ
           rw [← coeff_iterate_frobenius _ 0 n, zero_add, ← RingHom.map_iterate_frobenius,
             right_inverse_pth_root_frobenius.iterate]
 #align perfection.lift Perfection.lift
+-/
 
 theorem hom_ext {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂}
     [CommSemiring S] [CharP S p] {f g : R →+* Ring.Perfection S p}
@@ -224,6 +242,7 @@ theorem hom_ext {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {
 
 variable {R} {S : Type u₂} [CommSemiring S] [CharP S p]
 
+#print Perfection.map /-
 /-- A ring homomorphism `R →+* S` induces `perfection R p →+* perfection S p` -/
 @[simps]
 def map (φ : R →+* S) : Ring.Perfection R p →+* Ring.Perfection S p
@@ -234,6 +253,7 @@ def map (φ : R →+* S) : Ring.Perfection R p →+* Ring.Perfection S p
   map_zero' := Subtype.eq <| funext fun n => φ.map_zero
   map_add' f g := Subtype.eq <| funext fun n => φ.map_add _ _
 #align perfection.map Perfection.map
+-/
 
 theorem coeff_map (φ : R →+* S) (f : Ring.Perfection R p) (n : ℕ) :
     coeff S p n (map p φ f) = φ (coeff R p n f) :=
@@ -242,6 +262,7 @@ theorem coeff_map (φ : R →+* S) (f : Ring.Perfection R p) (n : ℕ) :
 
 end Perfection
 
+#print PerfectionMap /-
 /-- A perfection map to a ring of characteristic `p` is a map that is isomorphic
 to its perfection. -/
 @[nolint has_nonempty_instance]
@@ -250,6 +271,7 @@ structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R
   Injective : ∀ ⦃x y : P⦄, (∀ n, π ((pthRoot P p^[n]) x) = π ((pthRoot P p^[n]) y)) → x = y
   Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n, π ((pthRoot P p^[n]) x) = f n
 #align perfection_map PerfectionMap
+-/
 
 namespace PerfectionMap
 
@@ -276,11 +298,14 @@ theorem mk' {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : Perfection.
 
 variable (p R P)
 
+#print PerfectionMap.of /-
 /-- The canonical perfection map from the perfection of a ring. -/
 theorem of : PerfectionMap p (Perfection.coeff R p 0) :=
   mk' (RingEquiv.refl _) <| (Equiv.apply_eq_iff_eq_symm_apply _).2 rfl
 #align perfection_map.of PerfectionMap.of
+-/
 
+#print PerfectionMap.id /-
 /-- For a perfect ring, it itself is the perfection. -/
 theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
   { Injective := fun x y hxy => hxy 0
@@ -290,6 +315,7 @@ theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
           Nat.recOn n rfl fun n ih =>
             injective_pow_p p <| by rw [Function.iterate_succ_apply', pthRoot_pow_p _, ih, hf]⟩ }
 #align perfection_map.id PerfectionMap.id
+-/
 
 variable {p R P}
 
@@ -301,10 +327,12 @@ noncomputable def equiv {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring
       ⟨x, Perfection.ext <| hx⟩⟩
 #align perfection_map.equiv PerfectionMap.equiv
 
+#print PerfectionMap.equiv_apply /-
 theorem equiv_apply {π : P →+* R} (m : PerfectionMap p π) (x : P) :
     m.Equiv x = Perfection.lift p P R π x :=
   rfl
 #align perfection_map.equiv_apply PerfectionMap.equiv_apply
+-/
 
 theorem comp_equiv {π : P →+* R} (m : PerfectionMap p π) (x : P) :
     Perfection.coeff R p 0 (m.Equiv x) = π x :=
@@ -328,6 +356,7 @@ theorem comp_symm_equiv' {π : P →+* R} (m : PerfectionMap p π) :
 
 variable (p R P)
 
+#print PerfectionMap.lift /-
 /-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
 any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* P`,
 where `P` is any perfection of `S`. -/
@@ -348,6 +377,7 @@ noncomputable def lift [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP
           show Perfection.lift p R S (π.comp f) x = RingHom.comp (↑m.Equiv) f x from
             RingHom.ext_iff.1 ((Perfection.lift p R S).apply_eq_iff_eq_symm_apply.2 rfl) _
 #align perfection_map.lift PerfectionMap.lift
+-/
 
 variable {R p}
 
@@ -361,28 +391,34 @@ variable {R P} (p) {S : Type u₂} [CommSemiring S] [CharP S p]
 
 variable {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p]
 
+#print PerfectionMap.map /-
 /-- A ring homomorphism `R →+* S` induces `P →+* Q`, a map of the respective perfections. -/
 @[nolint unused_arguments]
 noncomputable def map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
     (φ : R →+* S) : P →+* Q :=
   lift p P S Q σ n <| φ.comp π
 #align perfection_map.map PerfectionMap.map
+-/
 
+#print PerfectionMap.comp_map /-
 theorem comp_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
     (φ : R →+* S) : σ.comp (map p m n φ) = φ.comp π :=
   (lift p P S Q σ n).symm_apply_apply _
 #align perfection_map.comp_map PerfectionMap.comp_map
+-/
 
 theorem map_map {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ)
     (φ : R →+* S) (x : P) : σ (map p m n φ x) = φ (π x) :=
   RingHom.ext_iff.1 (comp_map p m n φ) x
 #align perfection_map.map_map PerfectionMap.map_map
 
+#print PerfectionMap.map_eq_map /-
 -- Why is this slow?
 theorem map_eq_map (φ : R →+* S) :
     @map p _ R _ _ _ _ _ _ S _ _ _ _ _ _ _ (of p R) _ (of p S) φ = Perfection.map p φ :=
   hom_ext _ (of p S) fun f => by rw [map_map, Perfection.coeff_map]
 #align perfection_map.map_eq_map PerfectionMap.map_eq_map
+-/
 
 end PerfectionMap
 

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

@@ -40,7 +40,7 @@ using the `p`-th power maps `M → M` indexed by the natural numbers, implemente
 `{ f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }`. -/
 def Monoid.perfection (M : Type u₁) [CommMonoid M] (p : ℕ) : Submonoid (ℕ → M)
     where
-  carrier := { f | ∀ n, f (n + 1) ^ p = f n }
+  carrier := {f | ∀ n, f (n + 1) ^ p = f n}
   one_mem' n := one_pow _
   mul_mem' f g hf hg n := (mul_pow _ _ _).trans <| congr_arg₂ _ (hf n) (hg n)
 #align monoid.perfection Monoid.perfection

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

@@ -246,7 +246,7 @@ end Perfection
 to its perfection. -/
 @[nolint has_nonempty_instance]
 structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R] [CharP R p]
-  {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
+    {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
   Injective : ∀ ⦃x y : P⦄, (∀ n, π ((pthRoot P p^[n]) x) = π ((pthRoot P p^[n]) y)) → x = y
   Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n, π ((pthRoot P p^[n]) x) = f n
 #align perfection_map PerfectionMap
@@ -455,7 +455,7 @@ theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
   have hy0 : y ≠ 0 := mt (by rintro rfl; rw [MulZeroClass.mul_zero]) hxy0
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
-  rw [← RingHom.map_mul] at hxy0⊢
+  rw [← RingHom.map_mul] at hxy0 ⊢
   rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, RingHom.map_mul, v.map_mul]
 #align mod_p.pre_val_mul ModP.preVal_mul
 
@@ -467,23 +467,23 @@ theorem preVal_add (x y : ModP K v O hv p) :
   by_cases hxy0 : x + y = 0; · rw [hxy0, pre_val_zero]; exact zero_le _
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
-  rw [← RingHom.map_add] at hxy0⊢
+  rw [← RingHom.map_add] at hxy0 ⊢
   rw [pre_val_mk hx0, pre_val_mk hy0, pre_val_mk hxy0, RingHom.map_add]; exact v.map_add _ _
 #align mod_p.pre_val_add ModP.preVal_add
 
 theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x ≠ 0 :=
   by
   refine'
-    ⟨fun h hx => by rw [hx, pre_val_zero] at h; exact not_lt_zero' h, fun h =>
+    ⟨fun h hx => by rw [hx, pre_val_zero] at h ; exact not_lt_zero' h, fun h =>
       lt_of_not_le fun hp => h _⟩
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
-  rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp
+  rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp 
   rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]; exact hp
 #align mod_p.v_p_lt_pre_val ModP.v_p_lt_preVal
 
 theorem preVal_eq_zero {x : ModP K v O hv p} : preVal K v O hv p x = 0 ↔ x = 0 :=
   ⟨fun hvx =>
-    by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0;
+    by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0 ;
       exact not_lt_zero' hx0,
     fun hx => hx.symm ▸ preVal_zero⟩
 #align mod_p.pre_val_eq_zero ModP.preVal_eq_zero
@@ -507,10 +507,10 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (Nat.cast_pos.2 hp.1.Pos)
-  rw [← RingHom.map_mul]; rw [← RingHom.map_pow] at hx hy
-  rw [← v_p_lt_val hv] at hx hy⊢
+  rw [← RingHom.map_mul]; rw [← RingHom.map_pow] at hx hy 
+  rw [← v_p_lt_val hv] at hx hy ⊢
   rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
-    mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.NeZero : (p : ℝ) ≠ 0), rpow_one] at hx hy
+    mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.NeZero : (p : ℝ) ≠ 0), rpow_one] at hx hy 
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)
   by_cases hvp : v p = 0; · rw [hvp]; exact zero_le _; replace hvp := zero_lt_iff.2 hvp
   conv_lhs => rw [← rpow_one (v p)]; rw [← rpow_add (ne_of_gt hvp)]
@@ -608,7 +608,7 @@ theorem valAux_mul (f g : PreTilt K v O hv p) :
     · rw [← RingHom.map_pow, coeff_pow_p f]; assumption
     · rw [← RingHom.map_pow, coeff_pow_p g]; assumption
   rw [val_aux_eq (coeff_add_ne_zero hm 1), val_aux_eq (coeff_add_ne_zero hn 1), val_aux_eq hfg]
-  rw [RingHom.map_mul] at hfg⊢; rw [ModP.preVal_mul hfg, mul_pow]
+  rw [RingHom.map_mul] at hfg ⊢; rw [ModP.preVal_mul hfg, mul_pow]
 #align pre_tilt.val_aux_mul PreTilt.valAux_mul
 
 theorem valAux_add (f g : PreTilt K v O hv p) :
@@ -654,7 +654,7 @@ theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   by_cases hf0 : f = 0; · rw [hf0]; exact iff_of_true (Valuation.map_zero _) rfl
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf0 <| Perfection.ext h
   show val_aux K v O hv p f = 0 ↔ f = 0; refine' iff_of_false (fun hvf => hn _) hf0
-  rw [val_aux_eq hn] at hvf; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf
+  rw [val_aux_eq hn] at hvf ; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf 
 #align pre_tilt.map_eq_zero PreTilt.map_eq_zero
 
 end Classical
@@ -664,7 +664,7 @@ instance : IsDomain (PreTilt K v O hv p) :=
   haveI : Nontrivial (PreTilt K v O hv p) := ⟨(CharP.nontrivial_of_char_ne_one hp.1.ne_one).1⟩
   haveI : NoZeroDivisors (PreTilt K v O hv p) :=
     ⟨fun f g hfg => by
-      simp_rw [← map_eq_zero] at hfg⊢; contrapose! hfg; rw [Valuation.map_mul]
+      simp_rw [← map_eq_zero] at hfg ⊢; contrapose! hfg; rw [Valuation.map_mul]
       exact mul_ne_zero hfg.1 hfg.2⟩
   exact NoZeroDivisors.to_isDomain _
 

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

@@ -33,7 +33,7 @@ Define the valuation on the tilt, and define a characteristic predicate for the
 
 universe u₁ u₂ u₃ u₄
 
-open NNReal
+open scoped NNReal
 
 /-- The perfection of a monoid `M`, defined to be the projective limit of `M`
 using the `p`-th power maps `M → M` indexed by the natural numbers, implemented as
@@ -540,7 +540,7 @@ instance : CharP (PreTilt K v O hv p) p :=
 
 section Classical
 
-open Classical
+open scoped Classical
 
 open Perfection
 

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

@@ -136,9 +136,7 @@ theorem coeff_pthRoot (f : Ring.Perfection R p) (n : ℕ) :
 #align perfection.coeff_pth_root Perfection.coeff_pthRoot
 
 theorem coeff_pow_p (f : Ring.Perfection R p) (n : ℕ) : coeff R p (n + 1) (f ^ p) = coeff R p n f :=
-  by
-  rw [RingHom.map_pow]
-  exact f.2 n
+  by rw [RingHom.map_pow]; exact f.2 n
 #align perfection.coeff_pow_p Perfection.coeff_pow_p
 
 theorem coeff_pow_p' (f : Ring.Perfection R p) (n : ℕ) : coeff R p (n + 1) f ^ p = coeff R p n f :=
@@ -453,18 +451,8 @@ theorem preVal_zero : preVal K v O hv p 0 = 0 :=
 theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
     preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y :=
   by
-  have hx0 : x ≠ 0 :=
-    mt
-      (by
-        rintro rfl
-        rw [MulZeroClass.zero_mul])
-      hxy0
-  have hy0 : y ≠ 0 :=
-    mt
-      (by
-        rintro rfl
-        rw [MulZeroClass.mul_zero])
-      hxy0
+  have hx0 : x ≠ 0 := mt (by rintro rfl; rw [MulZeroClass.zero_mul]) hxy0
+  have hy0 : y ≠ 0 := mt (by rintro rfl; rw [MulZeroClass.mul_zero]) hxy0
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   rw [← RingHom.map_mul] at hxy0⊢
@@ -474,15 +462,9 @@ theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
 theorem preVal_add (x y : ModP K v O hv p) :
     preVal K v O hv p (x + y) ≤ max (preVal K v O hv p x) (preVal K v O hv p y) :=
   by
-  by_cases hx0 : x = 0;
-  · rw [hx0, zero_add]
-    exact le_max_right _ _
-  by_cases hy0 : y = 0;
-  · rw [hy0, add_zero]
-    exact le_max_left _ _
-  by_cases hxy0 : x + y = 0;
-  · rw [hxy0, pre_val_zero]
-    exact zero_le _
+  by_cases hx0 : x = 0; · rw [hx0, zero_add]; exact le_max_right _ _
+  by_cases hy0 : y = 0; · rw [hy0, add_zero]; exact le_max_left _ _
+  by_cases hxy0 : x + y = 0; · rw [hxy0, pre_val_zero]; exact zero_le _
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   rw [← RingHom.map_add] at hxy0⊢
@@ -492,9 +474,8 @@ theorem preVal_add (x y : ModP K v O hv p) :
 theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x ≠ 0 :=
   by
   refine'
-    ⟨fun h hx => by
-      rw [hx, pre_val_zero] at h
-      exact not_lt_zero' h, fun h => lt_of_not_le fun hp => h _⟩
+    ⟨fun h hx => by rw [hx, pre_val_zero] at h; exact not_lt_zero' h, fun h =>
+      lt_of_not_le fun hp => h _⟩
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   rw [pre_val_mk h, ← map_natCast (algebraMap O K) p, hv.le_iff_dvd] at hp
   rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]; exact hp
@@ -502,9 +483,7 @@ theorem v_p_lt_preVal {x : ModP K v O hv p} : v p < preVal K v O hv p x ↔ x 
 
 theorem preVal_eq_zero {x : ModP K v O hv p} : preVal K v O hv p x = 0 ↔ x = 0 :=
   ⟨fun hvx =>
-    by_contradiction fun hx0 : x ≠ 0 =>
-      by
-      rw [← v_p_lt_pre_val, hvx] at hx0
+    by_contradiction fun hx0 : x ≠ 0 => by rw [← v_p_lt_pre_val, hvx] at hx0;
       exact not_lt_zero' hx0,
     fun hx => hx.symm ▸ preVal_zero⟩
 #align mod_p.pre_val_eq_zero ModP.preVal_eq_zero
@@ -533,9 +512,7 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
     mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.NeZero : (p : ℝ) ≠ 0), rpow_one] at hx hy
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)
-  by_cases hvp : v p = 0;
-  · rw [hvp]
-    exact zero_le _; replace hvp := zero_lt_iff.2 hvp
+  by_cases hvp : v p = 0; · rw [hvp]; exact zero_le _; replace hvp := zero_lt_iff.2 hvp
   conv_lhs => rw [← rpow_one (v p)]; rw [← rpow_add (ne_of_gt hvp)]
   refine' rpow_le_rpow_of_exponent_ge hvp (map_natCast (algebraMap O K) p ▸ hv.2 _) _
   rw [← add_div, div_le_one (Nat.cast_pos.2 hp.1.Pos : 0 < (p : ℝ))]; exact_mod_cast hp.1.two_le
@@ -589,8 +566,7 @@ theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0)
   have h : ∃ n, coeff _ _ n f ≠ 0 := ⟨n, hfn⟩
   rw [val_aux, dif_pos h]
   obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le (Nat.find_min' h hfn)
-  induction' k with k ih
-  · rfl
+  induction' k with k ih; · rfl
   obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff _ _ (Nat.find h + k + 1) f)
   have h1 : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0 := hx.symm ▸ hfn
   have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP K v O hv p) ≠ 0 :=
@@ -619,10 +595,8 @@ theorem valAux_one : valAux K v O hv p 1 = 1 :=
 theorem valAux_mul (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f * g) = valAux K v O hv p f * valAux K v O hv p g :=
   by
-  by_cases hf : f = 0
-  · rw [hf, MulZeroClass.zero_mul, val_aux_zero, MulZeroClass.zero_mul]
-  by_cases hg : g = 0
-  · rw [hg, MulZeroClass.mul_zero, val_aux_zero, MulZeroClass.mul_zero]
+  by_cases hf : f = 0; · rw [hf, MulZeroClass.zero_mul, val_aux_zero, MulZeroClass.zero_mul]
+  by_cases hg : g = 0; · rw [hg, MulZeroClass.mul_zero, val_aux_zero, MulZeroClass.mul_zero]
   obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
   replace hm := coeff_ne_zero_of_le hm (le_max_left m n)
@@ -631,27 +605,18 @@ theorem valAux_mul (f g : PreTilt K v O hv p) :
     by
     rw [RingHom.map_mul]
     refine' ModP.mul_ne_zero_of_pow_p_ne_zero _ _
-    · rw [← RingHom.map_pow, coeff_pow_p f]
-      assumption
-    · rw [← RingHom.map_pow, coeff_pow_p g]
-      assumption
+    · rw [← RingHom.map_pow, coeff_pow_p f]; assumption
+    · rw [← RingHom.map_pow, coeff_pow_p g]; assumption
   rw [val_aux_eq (coeff_add_ne_zero hm 1), val_aux_eq (coeff_add_ne_zero hn 1), val_aux_eq hfg]
-  rw [RingHom.map_mul] at hfg⊢
-  rw [ModP.preVal_mul hfg, mul_pow]
+  rw [RingHom.map_mul] at hfg⊢; rw [ModP.preVal_mul hfg, mul_pow]
 #align pre_tilt.val_aux_mul PreTilt.valAux_mul
 
 theorem valAux_add (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f + g) ≤ max (valAux K v O hv p f) (valAux K v O hv p g) :=
   by
-  by_cases hf : f = 0;
-  · rw [hf, zero_add, val_aux_zero, max_eq_right]
-    exact zero_le _
-  by_cases hg : g = 0;
-  · rw [hg, add_zero, val_aux_zero, max_eq_left]
-    exact zero_le _
-  by_cases hfg : f + g = 0;
-  · rw [hfg, val_aux_zero]
-    exact zero_le _
+  by_cases hf : f = 0; · rw [hf, zero_add, val_aux_zero, max_eq_right]; exact zero_le _
+  by_cases hg : g = 0; · rw [hg, add_zero, val_aux_zero, max_eq_left]; exact zero_le _
+  by_cases hfg : f + g = 0; · rw [hfg, val_aux_zero]; exact zero_le _
   replace hf : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
   replace hg : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
   replace hfg : ∃ n, coeff _ _ n (f + g) ≠ 0 := not_forall.1 fun h => hfg <| Perfection.ext h
@@ -686,9 +651,7 @@ variable {K v O hv p}
 
 theorem map_eq_zero {f : PreTilt K v O hv p} : val K v O hv p f = 0 ↔ f = 0 :=
   by
-  by_cases hf0 : f = 0;
-  · rw [hf0]
-    exact iff_of_true (Valuation.map_zero _) rfl
+  by_cases hf0 : f = 0; · rw [hf0]; exact iff_of_true (Valuation.map_zero _) rfl
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf0 <| Perfection.ext h
   show val_aux K v O hv p f = 0 ↔ f = 0; refine' iff_of_false (fun hvf => hn _) hf0
   rw [val_aux_eq hn] at hvf; replace hvf := pow_eq_zero hvf; rwa [ModP.preVal_eq_zero] at hvf
@@ -701,9 +664,7 @@ instance : IsDomain (PreTilt K v O hv p) :=
   haveI : Nontrivial (PreTilt K v O hv p) := ⟨(CharP.nontrivial_of_char_ne_one hp.1.ne_one).1⟩
   haveI : NoZeroDivisors (PreTilt K v O hv p) :=
     ⟨fun f g hfg => by
-      simp_rw [← map_eq_zero] at hfg⊢
-      contrapose! hfg
-      rw [Valuation.map_mul]
+      simp_rw [← map_eq_zero] at hfg⊢; contrapose! hfg; rw [Valuation.map_mul]
       exact mul_ne_zero hfg.1 hfg.2⟩
   exact NoZeroDivisors.to_isDomain _
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 
 ! This file was ported from Lean 3 source module ring_theory.perfection
-! leanprover-community/mathlib commit b1d911acd60ab198808e853292106ee352b648ea
+! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,7 +12,7 @@ import Mathbin.Algebra.CharP.Pi
 import Mathbin.Algebra.CharP.Quotient
 import Mathbin.Algebra.CharP.Subring
 import Mathbin.Algebra.Ring.Pi
-import Mathbin.Analysis.SpecialFunctions.Pow
+import Mathbin.Analysis.SpecialFunctions.Pow.Nnreal
 import Mathbin.FieldTheory.PerfectClosure
 import Mathbin.RingTheory.Localization.FractionRing
 import Mathbin.RingTheory.Subring.Basic

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

@@ -457,13 +457,13 @@ theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
     mt
       (by
         rintro rfl
-        rw [zero_mul])
+        rw [MulZeroClass.zero_mul])
       hxy0
   have hy0 : y ≠ 0 :=
     mt
       (by
         rintro rfl
-        rw [mul_zero])
+        rw [MulZeroClass.mul_zero])
       hxy0
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
@@ -620,9 +620,9 @@ theorem valAux_mul (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f * g) = valAux K v O hv p f * valAux K v O hv p g :=
   by
   by_cases hf : f = 0
-  · rw [hf, zero_mul, val_aux_zero, zero_mul]
+  · rw [hf, MulZeroClass.zero_mul, val_aux_zero, MulZeroClass.zero_mul]
   by_cases hg : g = 0
-  · rw [hg, mul_zero, val_aux_zero, mul_zero]
+  · rw [hg, MulZeroClass.mul_zero, val_aux_zero, MulZeroClass.mul_zero]
   obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
   replace hm := coeff_ne_zero_of_le hm (le_max_left m n)

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

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

@@ -477,7 +477,7 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   rw [← (Ideal.Quotient.mk (Ideal.span {(p : O)})).map_mul]
   rw [← (Ideal.Quotient.mk (Ideal.span {(p : O)})).map_pow] at hx hy
   rw [← v_p_lt_val hv] at hx hy ⊢
-  rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
+  rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_natCast, ← rpow_mul,
     mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.ne_zero : (p : ℝ) ≠ 0), rpow_one] at hx hy
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)
   by_cases hvp : v p = 0

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -229,7 +229,7 @@ end Perfection
 
 /-- A perfection map to a ring of characteristic `p` is a map that is isomorphic
 to its perfection. -/
--- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
+-- @[nolint has_nonempty_instance] -- Porting note(#5171): This linter does not exist yet.
 structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R] [CharP R p]
     {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
   injective : ∀ ⦃x y : P⦄,
@@ -372,7 +372,7 @@ variable (p : ℕ)
 
 -- Porting note: Specified all arguments explicitly
 /-- `O/(p)` for `O`, ring of integers of `K`. -/
-@[nolint unusedArguments] -- Porting note: Removed `nolint has_nonempty_instance`
+@[nolint unusedArguments] -- Porting note(#5171): removed `nolint has_nonempty_instance`
 def ModP (K : Type u₁) [Field K] (v : Valuation K ℝ≥0) (O : Type u₂) [CommRing O] [Algebra O K]
     (_ : v.Integers O) (p : ℕ) :=
   O ⧸ (Ideal.span {(p : O)} : Ideal O)
@@ -494,7 +494,7 @@ end Classical
 end ModP
 
 /-- Perfection of `O/(p)` where `O` is the ring of integers of `K`. -/
--- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
+-- @[nolint has_nonempty_instance] -- Porting note(#5171): This linter does not exist yet.
 def PreTilt :=
   Ring.Perfection (ModP K v O hv p) p
 #align pre_tilt PreTilt
@@ -638,7 +638,7 @@ end PreTilt
 /-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in
 [scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`,
 this is implemented as the fraction field of the perfection of `O/(p)`. -/
--- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
+-- @[nolint has_nonempty_instance] -- Porting note(#5171): This linter does not exist yet.
 def Tilt :=
   FractionRing (PreTilt K v O hv p)
 #align tilt Tilt

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.

@@ -543,7 +543,7 @@ theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0)
     exact coeff_nat_find_add_ne_zero k
   erw [ih (coeff_nat_find_add_ne_zero k), ← hx, ← coeff_pow_p, RingHom.map_pow, ← hx,
     ← RingHom.map_pow, ModP.preVal_mk h1, ModP.preVal_mk h2, RingHom.map_pow, v.map_pow, ← pow_mul,
-    pow_succ]
+    pow_succ']
   rfl
 #align pre_tilt.val_aux_eq PreTilt.valAux_eq
 

This PR uses the HomClass architecture to generalize the map_iterate statements in Algebra/GroupPower/IterateHom.lean.

@@ -190,10 +190,10 @@ noncomputable def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing
   toFun f :=
     { toFun := fun r => ⟨fun n => f (((frobeniusEquiv R p).symm : R →+* R)^[n] r),
         fun n => by erw [← f.map_pow, Function.iterate_succ_apply', frobeniusEquiv_symm_pow_p]⟩
-      map_one' := ext fun n => (congr_arg f <| RingHom.iterate_map_one _ _).trans f.map_one
+      map_one' := ext fun n => (congr_arg f <| iterate_map_one _ _).trans f.map_one
       map_mul' := fun x y =>
         ext fun n => (congr_arg f <| iterate_map_mul _ _ _ _).trans <| f.map_mul _ _
-      map_zero' := ext fun n => (congr_arg f <| RingHom.iterate_map_zero _ _).trans f.map_zero
+      map_zero' := ext fun n => (congr_arg f <| iterate_map_zero _ _).trans f.map_zero
       map_add' := fun x y =>
         ext fun n => (congr_arg f <| iterate_map_add _ _ _ _).trans <| f.map_add _ _ }
   invFun := RingHom.comp <| coeff S p 0

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)

@@ -241,9 +241,7 @@ structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R
 namespace PerfectionMap
 
 variable {p : ℕ} [Fact p.Prime]
-
 variable {R : Type u₁} [CommSemiring R] [CharP R p]
-
 variable {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p]
 
 /-- Create a `PerfectionMap` from an isomorphism to the perfection. -/
@@ -340,9 +338,7 @@ theorem hom_ext [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {
 #align perfection_map.hom_ext PerfectionMap.hom_ext
 
 variable {P} (p)
-
 variable {S : Type u₂} [CommSemiring S] [CharP S p]
-
 variable {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p]
 
 /-- A ring homomorphism `R →+* S` induces `P →+* Q`, a map of the respective perfections. -/
@@ -371,9 +367,7 @@ end PerfectionMap
 section Perfectoid
 
 variable (K : Type u₁) [Field K] (v : Valuation K ℝ≥0)
-
 variable (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O)
-
 variable (p : ℕ)
 
 -- Porting note: Specified all arguments explicitly

@@ -480,8 +480,9 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (Nat.cast_pos.2 hp.1.pos)
-  erw [← RingHom.map_mul]; erw [← RingHom.map_pow] at hx hy
-  erw [← v_p_lt_val hv] at hx hy ⊢
+  rw [← (Ideal.Quotient.mk (Ideal.span {(p : O)})).map_mul]
+  rw [← (Ideal.Quotient.mk (Ideal.span {(p : O)})).map_pow] at hx hy
+  rw [← v_p_lt_val hv] at hx hy ⊢
   rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
     mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.ne_zero : (p : ℝ) ≠ 0), rpow_one] at hx hy
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)

This involves moving lemmas from Algebra.GroupPower.Ring to Algebra.GroupWithZero.Basic and changing some 0 < n assumptions to n ≠ 0.

From LeanAPAP

@@ -47,7 +47,7 @@ indexed by the natural numbers, implemented as `{ f : ℕ → R | ∀ n, f (n +
 def Ring.perfectionSubsemiring (R : Type u₁) [CommSemiring R] (p : ℕ) [hp : Fact p.Prime]
     [CharP R p] : Subsemiring (ℕ → R) :=
   { Monoid.perfection R p with
-    zero_mem' := fun _ => zero_pow <| hp.1.pos
+    zero_mem' := fun _ ↦ zero_pow hp.1.ne_zero
     add_mem' := fun hf hg n => (frobenius_add R p _ _).trans <| congr_arg₂ _ (hf n) (hg n) }
 #align ring.perfection_subsemiring Ring.perfectionSubsemiring
 
@@ -163,7 +163,8 @@ theorem frobenius_pthRoot : (frobenius _ p).comp (pthRoot R p) = RingHom.id _ :=
 
 theorem coeff_add_ne_zero {f : Ring.Perfection R p} {n : ℕ} (hfn : coeff R p n f ≠ 0) (k : ℕ) :
     coeff R p (n + k) f ≠ 0 :=
-  Nat.recOn k hfn fun k ih h => ih <| by erw [← coeff_pow_p, RingHom.map_pow, h, zero_pow hp.1.pos]
+  Nat.recOn k hfn fun k ih h => ih <| by
+    erw [← coeff_pow_p, RingHom.map_pow, h, zero_pow hp.1.ne_zero]
 #align perfection.coeff_add_ne_zero Perfection.coeff_add_ne_zero
 
 theorem coeff_ne_zero_of_le {f : Ring.Perfection R p} {m n : ℕ} (hfm : coeff R p m f ≠ 0)

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -177,8 +177,8 @@ variable (R p)
 instance perfectRing : PerfectRing (Ring.Perfection R p) p where
   bijective_frobenius := Function.bijective_iff_has_inverse.mpr
     ⟨pthRoot R p,
-     FunLike.congr_fun <| @frobenius_pthRoot R _ p _ _,
-     FunLike.congr_fun <| @pthRoot_frobenius R _ p _ _⟩
+     DFunLike.congr_fun <| @frobenius_pthRoot R _ p _ _,
+     DFunLike.congr_fun <| @pthRoot_frobenius R _ p _ _⟩
 #align perfection.perfect_ring Perfection.perfectRing
 
 /-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,

The names for lemmas about monotonicity of (a ^ ·) and (· ^ n) were a mess. This PR tidies up everything related by following the naming convention for (a * ·) and (· * b). Namely, (a ^ ·) is pow_right and (· ^ n) is pow_left in lemma names. All lemma renames follow the corresponding multiplication lemma names closely.

Renames

Algebra.GroupPower.Order

• pow_mono → pow_right_mono
• pow_le_pow → pow_le_pow_right
• pow_le_pow_of_le_left → pow_le_pow_left
• pow_lt_pow_of_lt_left → pow_lt_pow_left
• strictMonoOn_pow → pow_left_strictMonoOn
• pow_strictMono_right → pow_right_strictMono
• pow_lt_pow → pow_lt_pow_right
• pow_lt_pow_iff → pow_lt_pow_iff_right
• pow_le_pow_iff → pow_le_pow_iff_right
• self_lt_pow → lt_self_pow
• strictAnti_pow → pow_right_strictAnti
• pow_lt_pow_iff_of_lt_one → pow_lt_pow_iff_right_of_lt_one
• pow_lt_pow_of_lt_one → pow_lt_pow_right_of_lt_one
• lt_of_pow_lt_pow → lt_of_pow_lt_pow_left
• le_of_pow_le_pow → le_of_pow_le_pow_left
• pow_lt_pow₀ → pow_lt_pow_right₀

Algebra.GroupPower.CovariantClass

• pow_le_pow_of_le_left' → pow_le_pow_left'
• nsmul_le_nsmul_of_le_right → nsmul_le_nsmul_right
• pow_lt_pow' → pow_lt_pow_right'
• nsmul_lt_nsmul → nsmul_lt_nsmul_left
• pow_strictMono_left → pow_right_strictMono'
• nsmul_strictMono_right → nsmul_left_strictMono
• StrictMono.pow_right' → StrictMono.pow_const
• StrictMono.nsmul_left → StrictMono.const_nsmul
• pow_strictMono_right' → pow_left_strictMono
• nsmul_strictMono_left → nsmul_right_strictMono
• Monotone.pow_right → Monotone.pow_const
• Monotone.nsmul_left → Monotone.const_nsmul
• lt_of_pow_lt_pow' → lt_of_pow_lt_pow_left'
• lt_of_nsmul_lt_nsmul → lt_of_nsmul_lt_nsmul_right
• pow_le_pow' → pow_le_pow_right'
• nsmul_le_nsmul → nsmul_le_nsmul_left
• pow_le_pow_of_le_one' → pow_le_pow_right_of_le_one'
• nsmul_le_nsmul_of_nonpos → nsmul_le_nsmul_left_of_nonpos
• le_of_pow_le_pow' → le_of_pow_le_pow_left'
• le_of_nsmul_le_nsmul' → le_of_nsmul_le_nsmul_right'
• pow_le_pow_iff' → pow_le_pow_iff_right'
• nsmul_le_nsmul_iff → nsmul_le_nsmul_iff_left
• pow_lt_pow_iff' → pow_lt_pow_iff_right'
• nsmul_lt_nsmul_iff → nsmul_lt_nsmul_iff_left

Data.Nat.Pow

• Nat.pow_lt_pow_of_lt_left → Nat.pow_lt_pow_left
• Nat.pow_le_iff_le_left → Nat.pow_le_pow_iff_left
• Nat.pow_lt_iff_lt_left → Nat.pow_lt_pow_iff_left

Lemmas added

• pow_le_pow_iff_left
• pow_lt_pow_iff_left
• pow_right_injective
• pow_right_inj
• Nat.pow_le_pow_left to have the correct name since Nat.pow_le_pow_of_le_left is in Std.
• Nat.pow_le_pow_right to have the correct name since Nat.pow_le_pow_of_le_right is in Std.

Lemmas removed

• self_le_pow was a duplicate of le_self_pow.
• Nat.pow_lt_pow_of_lt_right is defeq to pow_lt_pow_right.
• Nat.pow_right_strictMono is defeq to pow_right_strictMono.
• Nat.pow_le_iff_le_right is defeq to pow_le_pow_iff_right.
• Nat.pow_lt_iff_lt_right is defeq to pow_lt_pow_iff_right.

Other changes

• A bunch of proofs have been golfed.
• Some lemma assumptions have been turned from 0 < n or 1 ≤ n to n ≠ 0.
• A few Nat lemmas have been protected.
• One docstring has been fixed.

@@ -600,8 +600,8 @@ theorem valAux_add (f g : PreTilt K v O hv p) :
   rw [valAux_eq hm, valAux_eq hn, valAux_eq hk, RingHom.map_add]
   cases' le_max_iff.1
       (ModP.preVal_add (coeff _ _ (max (max m n) k) f) (coeff _ _ (max (max m n) k) g)) with h h
-  · exact le_max_of_le_left (pow_le_pow_of_le_left' h _)
-  · exact le_max_of_le_right (pow_le_pow_of_le_left' h _)
+  · exact le_max_of_le_left (pow_le_pow_left' h _)
+  · exact le_max_of_le_right (pow_le_pow_left' h _)
 #align pre_tilt.val_aux_add PreTilt.valAux_add
 
 variable (K v O hv p)

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

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

@@ -490,7 +490,7 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   conv_lhs => rw [← rpow_one (v p)]
   rw [← rpow_add (ne_of_gt hvp)]
   refine' rpow_le_rpow_of_exponent_ge hvp (map_natCast (algebraMap O K) p ▸ hv.2 _) _
-  rw [← add_div, div_le_one (Nat.cast_pos.2 hp.1.pos : 0 < (p : ℝ))]; exact_mod_cast hp.1.two_le
+  rw [← add_div, div_le_one (Nat.cast_pos.2 hp.1.pos : 0 < (p : ℝ))]; exact mod_cast hp.1.two_le
 #align mod_p.mul_ne_zero_of_pow_p_ne_zero ModP.mul_ne_zero_of_pow_p_ne_zero
 
 end Classical

This eliminates (fun a ↦ β) α in the type when applying a FunLike.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -428,7 +428,7 @@ theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
   have hy0 : y ≠ 0 := mt (by rintro rfl; rw [mul_zero]) hxy0
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
-  rw [← RingHom.map_mul] at hxy0 ⊢
+  rw [← map_mul (Ideal.Quotient.mk (Ideal.span {↑p})) r s] at hxy0 ⊢
   rw [preVal_mk hx0, preVal_mk hy0, preVal_mk hxy0, RingHom.map_mul, v.map_mul]
 #align mod_p.pre_val_mul ModP.preVal_mul
 
@@ -442,7 +442,7 @@ theorem preVal_add (x y : ModP K v O hv p) :
   · rw [hxy0, preVal_zero]; exact zero_le _
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
-  rw [← RingHom.map_add] at hxy0 ⊢
+  rw [← map_add (Ideal.Quotient.mk (Ideal.span {↑p})) r s] at hxy0 ⊢
   rw [preVal_mk hx0, preVal_mk hy0, preVal_mk hxy0, RingHom.map_add]; exact v.map_add _ _
 #align mod_p.pre_val_add ModP.preVal_add
 
@@ -479,8 +479,8 @@ theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP K v O hv p} (hx : x ^ p ≠ 0)
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   have h1p : (0 : ℝ) < 1 / p := one_div_pos.2 (Nat.cast_pos.2 hp.1.pos)
-  rw [← RingHom.map_mul]; rw [← RingHom.map_pow] at hx hy
-  rw [← v_p_lt_val hv] at hx hy ⊢
+  erw [← RingHom.map_mul]; erw [← RingHom.map_pow] at hx hy
+  erw [← v_p_lt_val hv] at hx hy ⊢
   rw [RingHom.map_pow, v.map_pow, ← rpow_lt_rpow_iff h1p, ← rpow_nat_cast, ← rpow_mul,
     mul_one_div_cancel (Nat.cast_ne_zero.2 hp.1.ne_zero : (p : ℝ) ≠ 0), rpow_one] at hx hy
   rw [RingHom.map_mul, v.map_mul]; refine' lt_of_le_of_lt _ (mul_lt_mul₀ hx hy)
@@ -540,7 +540,7 @@ theorem valAux_eq {f : PreTilt K v O hv p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0)
   obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le (Nat.find_min' h hfn)
   induction' k with k ih
   · rfl
-  obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff _ _ (Nat.find h + k + 1) f)
+  obtain ⟨x, hx⟩ := Ideal.Quotient.mk_surjective (coeff (ModP K v O hv p) p (Nat.find h + k + 1) f)
   have h1 : (Ideal.Quotient.mk _ x : ModP K v O hv p) ≠ 0 := hx.symm ▸ hfn
   have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP K v O hv p) ≠ 0 := by
     erw [RingHom.map_pow, hx, ← RingHom.map_pow, coeff_pow_p]

Purely cosmetic PR.

@@ -327,7 +327,7 @@ noncomputable def lift [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP
   right_inv f := by
     exact RingHom.ext fun x => m.equiv.injective <| (m.equiv.apply_symm_apply _).trans
       <| show Perfection.lift p R S (π.comp f) x = RingHom.comp (↑m.equiv) f x from
-        RingHom.ext_iff.1 (by rw [Equiv.apply_eq_iff_eq_symm_apply]; rfl ) _
+        RingHom.ext_iff.1 (by rw [Equiv.apply_eq_iff_eq_symm_apply]; rfl) _
 #align perfection_map.lift PerfectionMap.lift
 
 variable {R p}

@@ -231,9 +231,9 @@ to its perfection. -/
 -- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
 structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R] [CharP R p]
     {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
-  Injective : ∀ ⦃x y : P⦄,
+  injective : ∀ ⦃x y : P⦄,
     (∀ n, π (((frobeniusEquiv P p).symm)^[n] x) = π (((frobeniusEquiv P p).symm)^[n] y)) → x = y
-  Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n,
+  surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n,
     π (((frobeniusEquiv P p).symm)^[n] x) = f n
 #align perfection_map PerfectionMap
 
@@ -249,11 +249,11 @@ variable {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p]
 @[simps]
 theorem mk' {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : Perfection.lift p P R f = g) :
     PerfectionMap p f :=
-  { Injective := fun x y hxy =>
+  { injective := fun x y hxy =>
       g.injective <|
         (RingHom.ext_iff.1 hfg x).symm.trans <|
           Eq.symm <| (RingHom.ext_iff.1 hfg y).symm.trans <| Perfection.ext fun n => (hxy n).symm
-    Surjective := fun y hy =>
+    surjective := fun y hy =>
       let ⟨x, hx⟩ := g.surjective ⟨y, hy⟩
       ⟨x, fun n =>
         show Perfection.coeff R p n (Perfection.lift p P R f x) = Perfection.coeff R p n ⟨y, hy⟩ by
@@ -269,8 +269,8 @@ theorem of : PerfectionMap p (Perfection.coeff R p 0) :=
 
 /-- For a perfect ring, it itself is the perfection. -/
 theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
-  { Injective := fun x y hxy => hxy 0
-    Surjective := fun f hf =>
+  { injective := fun x y hxy => hxy 0
+    surjective := fun f hf =>
       ⟨f 0, fun n =>
         show ((frobeniusEquiv R p).symm)^[n] (f 0) = f n from
           Nat.recOn n rfl fun n ih => injective_pow_p R p <| by
@@ -282,8 +282,8 @@ variable {p R P}
 /-- A perfection map induces an isomorphism to the perfection. -/
 noncomputable def equiv {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring.Perfection R p :=
   RingEquiv.ofBijective (Perfection.lift p P R π)
-    ⟨fun _ _ hxy => m.Injective fun n => (congr_arg (Perfection.coeff R p n) hxy : _), fun f =>
-      let ⟨x, hx⟩ := m.Surjective f.1 f.2
+    ⟨fun _ _ hxy => m.injective fun n => (congr_arg (Perfection.coeff R p n) hxy : _), fun f =>
+      let ⟨x, hx⟩ := m.surjective f.1 f.2
       ⟨x, Perfection.ext <| hx⟩⟩
 #align perfection_map.equiv PerfectionMap.equiv
 

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).

@@ -424,8 +424,8 @@ theorem preVal_zero : preVal K v O hv p 0 = 0 :=
 
 theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
     preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y := by
-  have hx0 : x ≠ 0 := mt (by rintro rfl; rw [MulZeroClass.zero_mul]) hxy0
-  have hy0 : y ≠ 0 := mt (by rintro rfl; rw [MulZeroClass.mul_zero]) hxy0
+  have hx0 : x ≠ 0 := mt (by rintro rfl; rw [zero_mul]) hxy0
+  have hy0 : y ≠ 0 := mt (by rintro rfl; rw [mul_zero]) hxy0
   obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
   obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
   rw [← RingHom.map_mul] at hxy0 ⊢
@@ -566,9 +566,9 @@ theorem valAux_one : valAux K v O hv p 1 = 1 :=
 theorem valAux_mul (f g : PreTilt K v O hv p) :
     valAux K v O hv p (f * g) = valAux K v O hv p f * valAux K v O hv p g := by
   by_cases hf : f = 0
-  · rw [hf, MulZeroClass.zero_mul, valAux_zero, MulZeroClass.zero_mul]
+  · rw [hf, zero_mul, valAux_zero, zero_mul]
   by_cases hg : g = 0
-  · rw [hg, MulZeroClass.mul_zero, valAux_zero, MulZeroClass.mul_zero]
+  · rw [hg, mul_zero, valAux_zero, mul_zero]
   obtain ⟨m, hm⟩ : ∃ n, coeff _ _ n f ≠ 0 := not_forall.1 fun h => hf <| Perfection.ext h
   obtain ⟨n, hn⟩ : ∃ n, coeff _ _ n g ≠ 0 := not_forall.1 fun h => hg <| Perfection.ext h
   replace hm := coeff_ne_zero_of_le hm (le_max_left m n)

The main changes are:

• we replace the data-bearing PerfectRing typeclass with a Prop-valued (non-constructive) version,
• we introduce a new typeclass PerfectField,
• we add a proof that a perfect field of positive characteristic has surjective Frobenius map,
• we add some basic facts such as perfection of finite rings / fields and products of perfect rings.

@@ -8,7 +8,7 @@ import Mathlib.Algebra.CharP.Quotient
 import Mathlib.Algebra.CharP.Subring
 import Mathlib.Algebra.Ring.Pi
 import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
-import Mathlib.FieldTheory.PerfectClosure
+import Mathlib.FieldTheory.Perfect
 import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.RingTheory.Subring.Basic
 import Mathlib.RingTheory.Valuation.Integers
@@ -175,31 +175,32 @@ theorem coeff_ne_zero_of_le {f : Ring.Perfection R p} {m n : ℕ} (hfm : coeff R
 variable (R p)
 
 instance perfectRing : PerfectRing (Ring.Perfection R p) p where
-  pthRoot' := pthRoot R p
-  frobenius_pthRoot' := FunLike.congr_fun <| @frobenius_pthRoot R _ p _ _
-  pthRoot_frobenius' := FunLike.congr_fun <| @pthRoot_frobenius R _ p _ _
+  bijective_frobenius := Function.bijective_iff_has_inverse.mpr
+    ⟨pthRoot R p,
+     FunLike.congr_fun <| @frobenius_pthRoot R _ p _ _,
+     FunLike.congr_fun <| @pthRoot_frobenius R _ p _ _⟩
 #align perfection.perfect_ring Perfection.perfectRing
 
 /-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
 any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* Perfection S p`. -/
 @[simps]
-def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S]
-    [CharP S p] : (R →+* S) ≃ (R →+* Ring.Perfection S p) where
+noncomputable def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p]
+    (S : Type u₂) [CommSemiring S] [CharP S p] : (R →+* S) ≃ (R →+* Ring.Perfection S p) where
   toFun f :=
-    { toFun := fun r => ⟨fun n => f ((_root_.pthRoot R p)^[n] r),
-        fun n => by rw [← f.map_pow, Function.iterate_succ_apply', pthRoot_pow_p]⟩
+    { toFun := fun r => ⟨fun n => f (((frobeniusEquiv R p).symm : R →+* R)^[n] r),
+        fun n => by erw [← f.map_pow, Function.iterate_succ_apply', frobeniusEquiv_symm_pow_p]⟩
       map_one' := ext fun n => (congr_arg f <| RingHom.iterate_map_one _ _).trans f.map_one
       map_mul' := fun x y =>
-        ext fun n => (congr_arg f <| RingHom.iterate_map_mul _ _ _ _).trans <| f.map_mul _ _
+        ext fun n => (congr_arg f <| iterate_map_mul _ _ _ _).trans <| f.map_mul _ _
       map_zero' := ext fun n => (congr_arg f <| RingHom.iterate_map_zero _ _).trans f.map_zero
       map_add' := fun x y =>
-        ext fun n => (congr_arg f <| RingHom.iterate_map_add _ _ _ _).trans <| f.map_add _ _ }
+        ext fun n => (congr_arg f <| iterate_map_add _ _ _ _).trans <| f.map_add _ _ }
   invFun := RingHom.comp <| coeff S p 0
   left_inv f := RingHom.ext fun r => rfl
   right_inv f := RingHom.ext fun r => ext fun n =>
-    show coeff S p 0 (f ((_root_.pthRoot R p)^[n] r)) = coeff S p n (f r) by
+    show coeff S p 0 (f (((frobeniusEquiv R p).symm)^[n] r)) = coeff S p n (f r) by
       rw [← coeff_iterate_frobenius _ 0 n, zero_add, ← RingHom.map_iterate_frobenius,
-        rightInverse_pthRoot_frobenius.iterate]
+        Function.RightInverse.iterate (frobenius_apply_frobeniusEquiv_symm R p) n]
 
 theorem hom_ext {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂}
     [CommSemiring S] [CharP S p] {f g : R →+* Ring.Perfection S p}
@@ -230,8 +231,10 @@ to its perfection. -/
 -- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
 structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R] [CharP R p]
     {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
-  Injective : ∀ ⦃x y : P⦄, (∀ n, π ((pthRoot P p)^[n] x) = π ((pthRoot P p)^[n] y)) → x = y
-  Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n, π ((pthRoot P p)^[n] x) = f n
+  Injective : ∀ ⦃x y : P⦄,
+    (∀ n, π (((frobeniusEquiv P p).symm)^[n] x) = π (((frobeniusEquiv P p).symm)^[n] y)) → x = y
+  Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n,
+    π (((frobeniusEquiv P p).symm)^[n] x) = f n
 #align perfection_map PerfectionMap
 
 namespace PerfectionMap
@@ -269,9 +272,9 @@ theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
   { Injective := fun x y hxy => hxy 0
     Surjective := fun f hf =>
       ⟨f 0, fun n =>
-        show (pthRoot R p)^[n] (f 0) = f n from
-          Nat.recOn n rfl fun n ih =>
-            injective_pow_p p <| by rw [Function.iterate_succ_apply', pthRoot_pow_p _, ih, hf]⟩ }
+        show ((frobeniusEquiv R p).symm)^[n] (f 0) = f n from
+          Nat.recOn n rfl fun n ih => injective_pow_p R p <| by
+            rw [Function.iterate_succ_apply', frobeniusEquiv_symm_pow_p, ih, hf]⟩ }
 #align perfection_map.id PerfectionMap.id
 
 variable {p R P}

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module ring_theory.perfection
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.CharP.Pi
 import Mathlib.Algebra.CharP.Quotient
@@ -18,6 +13,8 @@ import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.RingTheory.Subring.Basic
 import Mathlib.RingTheory.Valuation.Integers
 
+#align_import ring_theory.perfection from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
+
 /-!
 # Ring Perfection and Tilt
 

@@ -143,12 +143,12 @@ theorem coeff_frobenius (f : Ring.Perfection R p) (n : ℕ) :
 
 -- `coeff_pow_p f n` also works but is slow!
 theorem coeff_iterate_frobenius (f : Ring.Perfection R p) (n m : ℕ) :
-    coeff R p (n + m) ((frobenius _ p^[m]) f) = coeff R p n f :=
+    coeff R p (n + m) ((frobenius _ p)^[m] f) = coeff R p n f :=
   Nat.recOn m rfl fun m ih => by erw [Function.iterate_succ_apply', coeff_frobenius, ih]
 #align perfection.coeff_iterate_frobenius Perfection.coeff_iterate_frobenius
 
 theorem coeff_iterate_frobenius' (f : Ring.Perfection R p) (n m : ℕ) (hmn : m ≤ n) :
-    coeff R p n ((frobenius _ p^[m]) f) = coeff R p (n - m) f :=
+    coeff R p n ((frobenius _ p)^[m] f) = coeff R p (n - m) f :=
   Eq.symm <| (coeff_iterate_frobenius _ _ m).symm.trans <| (tsub_add_cancel_of_le hmn).symm ▸ rfl
 #align perfection.coeff_iterate_frobenius' Perfection.coeff_iterate_frobenius'
 
@@ -189,7 +189,7 @@ any homomorphism `R →+* S` can be lifted to a homomorphism `R →+* Perfection
 def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S]
     [CharP S p] : (R →+* S) ≃ (R →+* Ring.Perfection S p) where
   toFun f :=
-    { toFun := fun r => ⟨fun n => f ((_root_.pthRoot R p^[n]) r),
+    { toFun := fun r => ⟨fun n => f ((_root_.pthRoot R p)^[n] r),
         fun n => by rw [← f.map_pow, Function.iterate_succ_apply', pthRoot_pow_p]⟩
       map_one' := ext fun n => (congr_arg f <| RingHom.iterate_map_one _ _).trans f.map_one
       map_mul' := fun x y =>
@@ -200,7 +200,7 @@ def lift (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Typ
   invFun := RingHom.comp <| coeff S p 0
   left_inv f := RingHom.ext fun r => rfl
   right_inv f := RingHom.ext fun r => ext fun n =>
-    show coeff S p 0 (f ((_root_.pthRoot R p^[n]) r)) = coeff S p n (f r) by
+    show coeff S p 0 (f ((_root_.pthRoot R p)^[n] r)) = coeff S p n (f r) by
       rw [← coeff_iterate_frobenius _ 0 n, zero_add, ← RingHom.map_iterate_frobenius,
         rightInverse_pthRoot_frobenius.iterate]
 
@@ -233,8 +233,8 @@ to its perfection. -/
 -- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.
 structure PerfectionMap (p : ℕ) [Fact p.Prime] {R : Type u₁} [CommSemiring R] [CharP R p]
     {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) : Prop where
-  Injective : ∀ ⦃x y : P⦄, (∀ n, π ((pthRoot P p^[n]) x) = π ((pthRoot P p^[n]) y)) → x = y
-  Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n, π ((pthRoot P p^[n]) x) = f n
+  Injective : ∀ ⦃x y : P⦄, (∀ n, π ((pthRoot P p)^[n] x) = π ((pthRoot P p)^[n] y)) → x = y
+  Surjective : ∀ f : ℕ → R, (∀ n, f (n + 1) ^ p = f n) → ∃ x : P, ∀ n, π ((pthRoot P p)^[n] x) = f n
 #align perfection_map PerfectionMap
 
 namespace PerfectionMap
@@ -272,7 +272,7 @@ theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
   { Injective := fun x y hxy => hxy 0
     Surjective := fun f hf =>
       ⟨f 0, fun n =>
-        show (pthRoot R p^[n]) (f 0) = f n from
+        show (pthRoot R p)^[n] (f 0) = f n from
           Nat.recOn n rfl fun n ih =>
             injective_pow_p p <| by rw [Function.iterate_succ_apply', pthRoot_pow_p _, ih, hf]⟩ }
 #align perfection_map.id PerfectionMap.id

These are all doc fixes

@@ -279,7 +279,7 @@ theorem id [PerfectRing R p] : PerfectionMap p (RingHom.id R) :=
 
 variable {p R P}
 
-/-- A perfection map induces an isomorphism to the prefection. -/
+/-- A perfection map induces an isomorphism to the perfection. -/
 noncomputable def equiv {π : P →+* R} (m : PerfectionMap p π) : P ≃+* Ring.Perfection R p :=
   RingEquiv.ofBijective (Perfection.lift p P R π)
     ⟨fun _ _ hxy => m.Injective fun n => (congr_arg (Perfection.coeff R p n) hxy : _), fun f =>

Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com>