ring_theory.witt_vector.witt_polynomial

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

c3019c79

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

38df578a

bump to nightly-2024-04-16-08

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

7b30d7e5

Diff

@@ -122,7 +122,7 @@ variable {R} {S : Type _} [CommRing S]
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
   by
-  rw [wittPolynomial, RingHom.map_sum, wittPolynomial, sum_congr rfl]
+  rw [wittPolynomial, map_sum, wittPolynomial, sum_congr rfl]
   intro i hi
   rw [map_monomial, RingHom.map_pow, map_natCast]
 #align map_witt_polynomial map_wittPolynomial
@@ -135,7 +135,7 @@ variable (R)
 theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
     constantCoeff (wittPolynomial p R n) = 0 :=
   by
-  simp only [wittPolynomial, RingHom.map_sum, constant_coeff_monomial]
+  simp only [wittPolynomial, map_sum, constant_coeff_monomial]
   rw [sum_eq_zero]
   rintro i hi
   rw [if_neg]
@@ -257,7 +257,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
   by
   apply Nat.strong_induction_on n <;> clear n
   intro n IH
-  rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, RingHom.map_sum, constant_coeff_C,
+  rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constant_coeff_C,
     sum_eq_zero]
   · simp only [constant_coeff_X, sub_zero, MulZeroClass.mul_zero]
   · intro m H

38df578a

bump to nightly-2024-04-07-08

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

b03b8656

Diff

@@ -5,9 +5,9 @@ Authors: Johan Commelin, Robert Y. Lewis
 -/
 import Algebra.CharP.Invertible
 import Data.Fintype.BigOperators
-import Data.MvPolynomial.Degrees
-import Data.MvPolynomial.CommRing
-import Data.MvPolynomial.Expand
+import Algebra.MvPolynomial.Degrees
+import Algebra.MvPolynomial.CommRing
+import Algebra.MvPolynomial.Expand
 import Data.ZMod.Basic
 
 #align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"

38df578a

bump to nightly-2024-04-06-08

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

e34029c9

Diff

@@ -5,10 +5,10 @@ Authors: Johan Commelin, Robert Y. Lewis
 -/
 import Algebra.CharP.Invertible
 import Data.Fintype.BigOperators
-import Data.MvPolynomial.Variables
+import Data.MvPolynomial.Degrees
 import Data.MvPolynomial.CommRing
 import Data.MvPolynomial.Expand
-import Data.Zmod.Basic
+import Data.ZMod.Basic
 
 #align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
 
@@ -178,7 +178,7 @@ theorem wittPolynomial_zmod_self (n : ℕ) :
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
     MulZeroClass.zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
-  rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ]
+  rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ']
   congr
   rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
@@ -350,7 +350,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
   rw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind₁_X_right, alg_hom_C, AlgHom.map_sum]
   have : W_ R n - ∑ i in range n, C (p ^ i : R) * X i ^ p ^ (n - i) = C (p ^ n : R) * X n := by
     simp only [wittPolynomial_eq_sum_C_mul_X_pow, tsub_self, sum_range_succ_comm, pow_one,
-      add_sub_cancel, pow_zero]
+      add_sub_cancel_right, pow_zero]
   rw [sum_congr rfl, this]
   ·-- this is really slow for some reason
     rw [mul_right_comm, ← C_mul, ← mul_pow, mul_invOf_self, one_pow, C_1, one_mul]

38df578a

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -180,7 +180,7 @@ theorem wittPolynomial_zmod_self (n : ℕ) :
   intro k hk
   rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ]
   congr
-  rw [mem_range] at hk 
+  rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
 #align witt_polynomial_zmod_self wittPolynomial_zmod_self
 -/
@@ -261,7 +261,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
     sum_eq_zero]
   · simp only [constant_coeff_X, sub_zero, MulZeroClass.mul_zero]
   · intro m H
-    rw [mem_range] at H 
+    rw [mem_range] at H
     simp only [RingHom.map_mul, RingHom.map_pow, constant_coeff_C, IH m H]
     rw [zero_pow, MulZeroClass.mul_zero]
     apply pow_pos hp.1.Pos
@@ -297,14 +297,14 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
   all_goals
     intro H
     replace H := vars_sum_subset _ _ H
-    rw [mem_bUnion] at H 
+    rw [mem_bUnion] at H
     rcases H with ⟨j, hj, H⟩
-    rw [vars_C_mul] at H 
+    rw [vars_C_mul] at H
     swap
     · apply pow_ne_zero; exact_mod_cast hp.1.NeZero
-    rw [mem_range] at hj 
+    rw [mem_range] at hj
     replace H := (ih j hj).2 (vars_pow _ _ H)
-    rw [mem_range] at H 
+    rw [mem_range] at H
   · rw [mem_range]
     exact lt_of_lt_of_le H hj
   · exact lt_irrefl n (lt_of_lt_of_le H hj)
@@ -355,7 +355,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
   ·-- this is really slow for some reason
     rw [mul_right_comm, ← C_mul, ← mul_pow, mul_invOf_self, one_pow, C_1, one_mul]
   · intro i h
-    rw [mem_range] at h 
+    rw [mem_range] at h
     simp only [AlgHom.map_mul, AlgHom.map_pow, alg_hom_C, H i h]
 #align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfW
 -/

38df578a

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,12 +3,12 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Robert Y. Lewis
 -/
-import Mathbin.Algebra.CharP.Invertible
-import Mathbin.Data.Fintype.BigOperators
-import Mathbin.Data.MvPolynomial.Variables
-import Mathbin.Data.MvPolynomial.CommRing
-import Mathbin.Data.MvPolynomial.Expand
-import Mathbin.Data.Zmod.Basic
+import Algebra.CharP.Invertible
+import Data.Fintype.BigOperators
+import Data.MvPolynomial.Variables
+import Data.MvPolynomial.CommRing
+import Data.MvPolynomial.Expand
+import Data.Zmod.Basic
 
 #align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"

38df578a

bump to nightly-2023-08-07-11

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

d9d9267a

Diff

@@ -166,12 +166,12 @@ theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ 
 #align aeval_witt_polynomial aeval_wittPolynomial
 -/
 
-#print wittPolynomial_zMod_self /-
+#print wittPolynomial_zmod_self /-
 /-- Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
 by expanding the `n`th Witt polynomial by `p`.
 -/
 @[simp]
-theorem wittPolynomial_zMod_self (n : ℕ) :
+theorem wittPolynomial_zmod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) :=
   by
   simp only [wittPolynomial_eq_sum_C_mul_X_pow]
@@ -182,7 +182,7 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   congr
   rw [mem_range] at hk 
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
-#align witt_polynomial_zmod_self wittPolynomial_zMod_self
+#align witt_polynomial_zmod_self wittPolynomial_zmod_self
 -/
 
 section PPrime

38df578a

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Robert Y. Lewis
-
-! This file was ported from Lean 3 source module ring_theory.witt_vector.witt_polynomial
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.CharP.Invertible
 import Mathbin.Data.Fintype.BigOperators
@@ -15,6 +10,8 @@ import Mathbin.Data.MvPolynomial.CommRing
 import Mathbin.Data.MvPolynomial.Expand
 import Mathbin.Data.Zmod.Basic
 
+#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
 /-!
 # Witt polynomials

38df578a

bump to nightly-2023-06-28-19

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

6577c9bd

Diff

@@ -89,15 +89,15 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
 #align witt_polynomial wittPolynomial
 -/
 
-#print wittPolynomial_eq_sum_c_mul_x_pow /-
-theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
+#print wittPolynomial_eq_sum_C_mul_X_pow /-
+theorem wittPolynomial_eq_sum_C_mul_X_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
   by
   apply sum_congr rfl
   rintro i -
   rw [monomial_eq, Finsupp.prod_single_index]
   rw [pow_zero]
-#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_pow
+#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_C_mul_X_pow
 -/
 
 /-! We set up notation locally to this file, to keep statements short and comprehensible.
@@ -157,7 +157,7 @@ theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
 #print wittPolynomial_one /-
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
-  simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
+  simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,
     one_mul, pow_one, C_1, pow_zero]
 #align witt_polynomial_one wittPolynomial_one
 -/
@@ -177,7 +177,7 @@ by expanding the `n`th Witt polynomial by `p`.
 theorem wittPolynomial_zMod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) :=
   by
-  simp only [wittPolynomial_eq_sum_c_mul_x_pow]
+  simp only [wittPolynomial_eq_sum_C_mul_X_pow]
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
     MulZeroClass.zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
@@ -335,7 +335,7 @@ theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     bind₁ (xInTermsOfW p R) (W_ R k) = X k :=
   by
-  rw [wittPolynomial_eq_sum_c_mul_x_pow, AlgHom.map_sum]
+  rw [wittPolynomial_eq_sum_C_mul_X_pow, AlgHom.map_sum]
   simp only [AlgHom.map_pow, C_pow, AlgHom.map_mul, alg_hom_C]
   rw [sum_range_succ_comm, tsub_self, pow_zero, pow_one, bind₁_X_right, mul_comm, ← C_pow,
     xInTermsOfW_aux]
@@ -352,7 +352,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
   clear n; intro n H
   rw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind₁_X_right, alg_hom_C, AlgHom.map_sum]
   have : W_ R n - ∑ i in range n, C (p ^ i : R) * X i ^ p ^ (n - i) = C (p ^ n : R) * X n := by
-    simp only [wittPolynomial_eq_sum_c_mul_x_pow, tsub_self, sum_range_succ_comm, pow_one,
+    simp only [wittPolynomial_eq_sum_C_mul_X_pow, tsub_self, sum_range_succ_comm, pow_one,
       add_sub_cancel, pow_zero]
   rw [sum_congr rfl, this]
   ·-- this is really slow for some reason

38df578a

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -89,6 +89,7 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
 #align witt_polynomial wittPolynomial
 -/
 
+#print wittPolynomial_eq_sum_c_mul_x_pow /-
 theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
   by
@@ -97,16 +98,15 @@ theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
   rw [monomial_eq, Finsupp.prod_single_index]
   rw [pow_zero]
 #align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_pow
+-/
 
 /-! We set up notation locally to this file, to keep statements short and comprehensible.
 This allows us to simply write `W n` or `W_ ℤ n`. -/
 
 
--- mathport name: witt_polynomial
 -- Notation with ring of coefficients explicit
 scoped[Witt] notation "W_" => wittPolynomial p
 
--- mathport name: witt_polynomial.infer
 -- Notation with ring of coefficients implicit
 scoped[Witt] notation "W" => wittPolynomial p hole!
 
@@ -121,6 +121,7 @@ section
 
 variable {R} {S : Type _} [CommRing S]
 
+#print map_wittPolynomial /-
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
   by
@@ -128,9 +129,11 @@ theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
   intro i hi
   rw [map_monomial, RingHom.map_pow, map_natCast]
 #align map_witt_polynomial map_wittPolynomial
+-/
 
 variable (R)
 
+#print constantCoeff_wittPolynomial /-
 @[simp]
 theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
     constantCoeff (wittPolynomial p R n) = 0 :=
@@ -142,6 +145,7 @@ theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
   rw [Finsupp.single_eq_zero]
   exact ne_of_gt (pow_pos hp.1.Pos _)
 #align constant_coeff_witt_polynomial constantCoeff_wittPolynomial
+-/
 
 #print wittPolynomial_zero /-
 @[simp]
@@ -150,17 +154,22 @@ theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
 #align witt_polynomial_zero wittPolynomial_zero
 -/
 
+#print wittPolynomial_one /-
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
   simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
     one_mul, pow_one, C_1, pow_zero]
 #align witt_polynomial_one wittPolynomial_one
+-/
 
+#print aeval_wittPolynomial /-
 theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
     aeval f (W_ R n) = ∑ i in range (n + 1), p ^ i * f i ^ p ^ (n - i) := by
   simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
 #align aeval_witt_polynomial aeval_wittPolynomial
+-/
 
+#print wittPolynomial_zMod_self /-
 /-- Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
 by expanding the `n`th Witt polynomial by `p`.
 -/
@@ -177,13 +186,13 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   rw [mem_range] at hk 
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
 #align witt_polynomial_zmod_self wittPolynomial_zMod_self
+-/
 
 section PPrime
 
 variable [hp : NeZero p]
 
-include hp
-
+#print wittPolynomial_vars /-
 theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars = range (n + 1) :=
   by
   have : ∀ i, (monomial (Finsupp.single i (p ^ (n - i))) (p ^ i : R)).vars = {i} :=
@@ -199,6 +208,7 @@ theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars
     apply disjoint_singleton_left.mpr
     rwa [mem_singleton]
 #align witt_polynomial_vars wittPolynomial_vars
+-/
 
 #print wittPolynomial_vars_subset /-
 theorem wittPolynomial_vars_subset (n : ℕ) : (wittPolynomial p R n).vars ⊆ range (n + 1) :=
@@ -222,6 +232,7 @@ The polynomials `X_in_terms_of_W` give the coordinate transformation in the back
 -/
 
 
+#print xInTermsOfW /-
 /-- The `X_in_terms_of_W p R n` is the polynomial on the basis of Witt polynomials
 that corresponds to the ordinary `X n`. -/
 noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
@@ -232,13 +243,17 @@ noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
           C (p ^ (i : ℕ) : R) * xInTermsOfW i ^ p ^ (n - i)) *
       C (⅟ p ^ n : R)
 #align X_in_terms_of_W xInTermsOfW
+-/
 
+#print xInTermsOfW_eq /-
 theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} :
     xInTermsOfW p R n =
       (X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i)) * C (⅟ p ^ n : R) :=
   by rw [xInTermsOfW, ← Fin.sum_univ_eq_sum_range]
 #align X_in_terms_of_W_eq xInTermsOfW_eq
+-/
 
+#print constantCoeff_xInTermsOfW /-
 @[simp]
 theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
     constantCoeff (xInTermsOfW p R n) = 0 :=
@@ -254,18 +269,19 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
     rw [zero_pow, MulZeroClass.mul_zero]
     apply pow_pos hp.1.Pos
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW
+-/
 
+#print xInTermsOfW_zero /-
 @[simp]
 theorem xInTermsOfW_zero [Invertible (p : R)] : xInTermsOfW p R 0 = X 0 := by
   rw [xInTermsOfW_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]
 #align X_in_terms_of_W_zero xInTermsOfW_zero
+-/
 
 section PPrime
 
 variable [hp : Fact p.Prime]
 
-include hp
-
 #print xInTermsOfW_vars_aux /-
 theorem xInTermsOfW_vars_aux (n : ℕ) :
     n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) :=
@@ -306,12 +322,15 @@ theorem xInTermsOfW_vars_subset (n : ℕ) : (xInTermsOfW p ℚ n).vars ⊆ range
 
 end PPrime
 
+#print xInTermsOfW_aux /-
 theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
     xInTermsOfW p R n * C (p ^ n : R) =
       X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i) :=
   by rw [xInTermsOfW_eq, mul_assoc, ← C_mul, ← mul_pow, invOf_mul_self, one_pow, C_1, mul_one]
 #align X_in_terms_of_W_aux xInTermsOfW_aux
+-/
 
+#print bind₁_xInTermsOfW_wittPolynomial /-
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     bind₁ (xInTermsOfW p R) (W_ R k) = X k :=
@@ -322,7 +341,9 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     xInTermsOfW_aux]
   simp only [C_pow, bind₁_X_right, sub_add_cancel]
 #align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomial
+-/
 
+#print bind₁_wittPolynomial_xInTermsOfW /-
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     bind₁ (W_ R) (xInTermsOfW p R n) = X n :=
@@ -340,4 +361,5 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     rw [mem_range] at h 
     simp only [AlgHom.map_mul, AlgHom.map_pow, alg_hom_C, H i h]
 #align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfW
+-/

38df578a

bump to nightly-2023-06-10-07

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

3fdc57bc

Diff

@@ -330,7 +330,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
   apply Nat.strong_induction_on n
   clear n; intro n H
   rw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind₁_X_right, alg_hom_C, AlgHom.map_sum]
-  have : (W_ R n - ∑ i in range n, C (p ^ i : R) * X i ^ p ^ (n - i)) = C (p ^ n : R) * X n := by
+  have : W_ R n - ∑ i in range n, C (p ^ i : R) * X i ^ p ^ (n - i) = C (p ^ n : R) * X n := by
     simp only [wittPolynomial_eq_sum_c_mul_x_pow, tsub_self, sum_range_succ_comm, pow_one,
       add_sub_cancel, pow_zero]
   rw [sum_congr rfl, this]

38df578a

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -174,7 +174,7 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   intro k hk
   rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ]
   congr
-  rw [mem_range] at hk
+  rw [mem_range] at hk 
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
 #align witt_polynomial_zmod_self wittPolynomial_zMod_self
 
@@ -249,7 +249,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
     sum_eq_zero]
   · simp only [constant_coeff_X, sub_zero, MulZeroClass.mul_zero]
   · intro m H
-    rw [mem_range] at H
+    rw [mem_range] at H 
     simp only [RingHom.map_mul, RingHom.map_pow, constant_coeff_C, IH m H]
     rw [zero_pow, MulZeroClass.mul_zero]
     apply pow_pos hp.1.Pos
@@ -284,14 +284,14 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
   all_goals
     intro H
     replace H := vars_sum_subset _ _ H
-    rw [mem_bUnion] at H
+    rw [mem_bUnion] at H 
     rcases H with ⟨j, hj, H⟩
-    rw [vars_C_mul] at H
+    rw [vars_C_mul] at H 
     swap
     · apply pow_ne_zero; exact_mod_cast hp.1.NeZero
-    rw [mem_range] at hj
+    rw [mem_range] at hj 
     replace H := (ih j hj).2 (vars_pow _ _ H)
-    rw [mem_range] at H
+    rw [mem_range] at H 
   · rw [mem_range]
     exact lt_of_lt_of_le H hj
   · exact lt_irrefl n (lt_of_lt_of_le H hj)
@@ -337,7 +337,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
   ·-- this is really slow for some reason
     rw [mul_right_comm, ← C_mul, ← mul_pow, mul_invOf_self, one_pow, C_1, one_mul]
   · intro i h
-    rw [mem_range] at h
+    rw [mem_range] at h 
     simp only [AlgHom.map_mul, AlgHom.map_pow, alg_hom_C, H i h]
 #align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfW

38df578a

bump to nightly-2023-06-01-04

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

4d9eaa85

Diff

@@ -156,14 +156,11 @@ theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
     one_mul, pow_one, C_1, pow_zero]
 #align witt_polynomial_one wittPolynomial_one
 
-#print aeval_wittPolynomial /-
 theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
     aeval f (W_ R n) = ∑ i in range (n + 1), p ^ i * f i ^ p ^ (n - i) := by
   simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
 #align aeval_witt_polynomial aeval_wittPolynomial
--/
 
-#print wittPolynomial_zMod_self /-
 /-- Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
 by expanding the `n`th Witt polynomial by `p`.
 -/
@@ -180,7 +177,6 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
 #align witt_polynomial_zmod_self wittPolynomial_zMod_self
--/
 
 section PPrime

38df578a

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -70,7 +70,7 @@ open Finset hiding map
 
 open Finsupp (single)
 
-open BigOperators
+open scoped BigOperators
 
 attribute [-simp] coe_eval₂_hom
 
@@ -110,7 +110,7 @@ scoped[Witt] notation "W_" => wittPolynomial p
 -- Notation with ring of coefficients implicit
 scoped[Witt] notation "W" => wittPolynomial p hole!
 
-open Witt
+open scoped Witt
 
 open MvPolynomial

38df578a

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -89,9 +89,6 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
 #align witt_polynomial wittPolynomial
 -/
 
-/- warning: witt_polynomial_eq_sum_C_mul_X_pow -> wittPolynomial_eq_sum_c_mul_x_pow is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_powₓ'. -/
 theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
   by
@@ -124,9 +121,6 @@ section
 
 variable {R} {S : Type _} [CommRing S]
 
-/- warning: map_witt_polynomial -> map_wittPolynomial is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align map_witt_polynomial map_wittPolynomialₓ'. -/
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
   by
@@ -137,12 +131,6 @@ theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
 
 variable (R)
 
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-Case conversion may be inaccurate. Consider using '#align constant_coeff_witt_polynomial constantCoeff_wittPolynomialₓ'. -/
 @[simp]
 theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
     constantCoeff (wittPolynomial p R n) = 0 :=
@@ -162,9 +150,6 @@ theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
 #align witt_polynomial_zero wittPolynomial_zero
 -/
 
-/- warning: witt_polynomial_one -> wittPolynomial_one is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align witt_polynomial_one wittPolynomial_oneₓ'. -/
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
   simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
@@ -203,12 +188,6 @@ variable [hp : NeZero p]
 
 include hp
 
-/- warning: witt_polynomial_vars -> wittPolynomial_vars is a dubious translation:
-lean 3 declaration is
-  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : NeZero.{0} Nat Nat.hasZero p] [_inst_3 : CharZero.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))] (n : Nat), Eq.{1} (Finset.{0} Nat) (MvPolynomial.vars.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1 n)) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))
-but is expected to have type
-  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : NeZero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) p] [_inst_3 : CharZero.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))] (n : Nat), Eq.{1} (Finset.{0} Nat) (MvPolynomial.vars.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1 n)) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))
-Case conversion may be inaccurate. Consider using '#align witt_polynomial_vars wittPolynomial_varsₓ'. -/
 theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars = range (n + 1) :=
   by
   have : ∀ i, (monomial (Finsupp.single i (p ^ (n - i))) (p ^ i : R)).vars = {i} :=
@@ -247,12 +226,6 @@ The polynomials `X_in_terms_of_W` give the coordinate transformation in the back
 -/
 
 
-/- warning: X_in_terms_of_W -> xInTermsOfW is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W xInTermsOfWₓ'. -/
 /-- The `X_in_terms_of_W p R n` is the polynomial on the basis of Witt polynomials
 that corresponds to the ordinary `X n`. -/
 noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
@@ -264,21 +237,12 @@ noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
       C (⅟ p ^ n : R)
 #align X_in_terms_of_W xInTermsOfW
 
-/- warning: X_in_terms_of_W_eq -> xInTermsOfW_eq is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_eq xInTermsOfW_eqₓ'. -/
 theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} :
     xInTermsOfW p R n =
       (X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i)) * C (⅟ p ^ n : R) :=
   by rw [xInTermsOfW, ← Fin.sum_univ_eq_sum_range]
 #align X_in_terms_of_W_eq xInTermsOfW_eq
 
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-Case conversion may be inaccurate. Consider using '#align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfWₓ'. -/
 @[simp]
 theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
     constantCoeff (xInTermsOfW p R n) = 0 :=
@@ -295,12 +259,6 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
     apply pow_pos hp.1.Pos
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW
 
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-Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_zero xInTermsOfW_zeroₓ'. -/
 @[simp]
 theorem xInTermsOfW_zero [Invertible (p : R)] : xInTermsOfW p R 0 = X 0 := by
   rw [xInTermsOfW_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]
@@ -352,18 +310,12 @@ theorem xInTermsOfW_vars_subset (n : ℕ) : (xInTermsOfW p ℚ n).vars ⊆ range
 
 end PPrime
 
-/- warning: X_in_terms_of_W_aux -> xInTermsOfW_aux is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_aux xInTermsOfW_auxₓ'. -/
 theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
     xInTermsOfW p R n * C (p ^ n : R) =
       X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i) :=
   by rw [xInTermsOfW_eq, mul_assoc, ← C_mul, ← mul_pow, invOf_mul_self, one_pow, C_1, mul_one]
 #align X_in_terms_of_W_aux xInTermsOfW_aux
 
-/- warning: bind₁_X_in_terms_of_W_witt_polynomial -> bind₁_xInTermsOfW_wittPolynomial is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomialₓ'. -/
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     bind₁ (xInTermsOfW p R) (W_ R k) = X k :=
@@ -375,9 +327,6 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
   simp only [C_pow, bind₁_X_right, sub_add_cancel]
 #align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomial
 
-/- warning: bind₁_witt_polynomial_X_in_terms_of_W -> bind₁_wittPolynomial_xInTermsOfW is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfWₓ'. -/
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     bind₁ (W_ R) (xInTermsOfW p R n) = X n :=

38df578a

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -334,8 +334,7 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
     rcases H with ⟨j, hj, H⟩
     rw [vars_C_mul] at H
     swap
-    · apply pow_ne_zero
-      exact_mod_cast hp.1.NeZero
+    · apply pow_ne_zero; exact_mod_cast hp.1.NeZero
     rw [mem_range] at hj
     replace H := (ih j hj).2 (vars_pow _ _ H)
     rw [mem_range] at H
@@ -384,8 +383,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     bind₁ (W_ R) (xInTermsOfW p R n) = X n :=
   by
   apply Nat.strong_induction_on n
-  clear n
-  intro n H
+  clear n; intro n H
   rw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind₁_X_right, alg_hom_C, AlgHom.map_sum]
   have : (W_ R n - ∑ i in range n, C (p ^ i : R) * X i ^ p ^ (n - i)) = C (p ^ n : R) * X n := by
     simp only [wittPolynomial_eq_sum_c_mul_x_pow, tsub_self, sum_range_succ_comm, pow_one,

38df578a

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -90,10 +90,7 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
 -/
 
 /- warning: witt_polynomial_eq_sum_C_mul_X_pow -> wittPolynomial_eq_sum_c_mul_x_pow is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_powₓ'. -/
 theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
@@ -128,10 +125,7 @@ section
 variable {R} {S : Type _} [CommRing S]
 
 /- warning: map_witt_polynomial -> map_wittPolynomial is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align map_witt_polynomial map_wittPolynomialₓ'. -/
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
@@ -169,10 +163,7 @@ theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align witt_polynomial_one wittPolynomial_oneₓ'. -/
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
@@ -274,10 +265,7 @@ noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
 #align X_in_terms_of_W xInTermsOfW
 
 /- warning: X_in_terms_of_W_eq -> xInTermsOfW_eq is a dubious translation:
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(Invertible.invOf.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) _inst_2) n)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_eq xInTermsOfW_eqₓ'. -/
 theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} :
     xInTermsOfW p R n =
@@ -366,10 +354,7 @@ theorem xInTermsOfW_vars_subset (n : ℕ) : (xInTermsOfW p ℚ n).vars ⊆ range
 end PPrime
 
 /- warning: X_in_terms_of_W_aux -> xInTermsOfW_aux is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_aux xInTermsOfW_auxₓ'. -/
 theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
     xInTermsOfW p R n * C (p ^ n : R) =
@@ -378,10 +363,7 @@ theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
 #align X_in_terms_of_W_aux xInTermsOfW_aux
 
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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R 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+<too large>
 Case conversion may be inaccurate. Consider using '#align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomialₓ'. -/
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
@@ -395,10 +377,7 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
 #align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomial
 
 /- warning: bind₁_witt_polynomial_X_in_terms_of_W -> bind₁_wittPolynomial_xInTermsOfW is a dubious translation:
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(CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n)
+<too large>
 Case conversion may be inaccurate. Consider using '#align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfWₓ'. -/
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :

38df578a

bump to nightly-2023-05-24-06

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

fbc7b476

Diff

@@ -381,7 +381,7 @@ theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
 lean 3 declaration is
   forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)] (k : Nat), Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (fun (_x : AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) => (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) ([anonymous].{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (xInTermsOfW.{u1} p R _inst_1 _inst_2)) (wittPolynomial.{u1} p R _inst_1 k)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) k)
 but is expected to have type
-  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (k : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (wittPolynomial.{u1} p R _inst_1 k)) (FunLike.coe.{succ u1, succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (fun (_x : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (SMulHomClass.toFunLike.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (xInTermsOfW.{u1} p R _inst_1 _inst_2)) (wittPolynomial.{u1} p R _inst_1 k)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) k)
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (k : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (wittPolynomial.{u1} p R _inst_1 k)) (FunLike.coe.{succ u1, succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (fun (_x : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (SMulHomClass.toFunLike.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (xInTermsOfW.{u1} p R _inst_1 _inst_2)) (wittPolynomial.{u1} p R _inst_1 k)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) k)
 Case conversion may be inaccurate. Consider using '#align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomialₓ'. -/
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
@@ -398,7 +398,7 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
 lean 3 declaration is
   forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)] (n : Nat), Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (fun (_x : AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) => (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) ([anonymous].{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n)
 but is expected to have type
-  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (n : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (FunLike.coe.{succ u1, succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (fun (_x : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (SMulHomClass.toFunLike.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n)
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (n : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (FunLike.coe.{succ u1, succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (fun (_x : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (SMulHomClass.toFunLike.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n)
 Case conversion may be inaccurate. Consider using '#align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfWₓ'. -/
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :

38df578a

bump to nightly-2023-05-24-03

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

2c55cf2c

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Robert Y. Lewis
 
 ! This file was ported from Lean 3 source module ring_theory.witt_vector.witt_polynomial
-! leanprover-community/mathlib commit c3019c79074b0619edb4b27553a91b2e82242395
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Data.Zmod.Basic
 /-!
 # Witt polynomials
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 To endow `witt_vector p R` with a ring structure,
 we need to study the so-called Witt polynomials.

c3019c79

bump to nightly-2023-05-23-03

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

ed98987c

Diff

@@ -75,6 +75,7 @@ variable (p : ℕ)
 
 variable (R : Type _) [CommRing R]
 
+#print wittPolynomial /-
 /-- `witt_polynomial p R n` is the `n`-th Witt polynomial
 with respect to a prime `p` with coefficients in a commutative ring `R`.
 It is defined as:
@@ -83,7 +84,14 @@ It is defined as:
 noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
   ∑ i in range (n + 1), monomial (single i (p ^ (n - i))) (p ^ i : R)
 #align witt_polynomial wittPolynomial
+-/
 
+/- warning: witt_polynomial_eq_sum_C_mul_X_pow -> wittPolynomial_eq_sum_c_mul_x_pow is a dubious translation:
+lean 3 declaration is
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(MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (HPow.hPow.{u1, 0, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHPow.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (Monoid.Pow.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toMonoidWithZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) i) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat instPowNat) p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n i)))))
+Case conversion may be inaccurate. Consider using '#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_powₓ'. -/
 theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
   by
@@ -116,6 +124,12 @@ section
 
 variable {R} {S : Type _} [CommRing S]
 
+/- warning: map_witt_polynomial -> map_wittPolynomial is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (n : Nat), Eq.{succ u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (CommSemiring.toSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (MvPolynomial.commSemiring.{u2, 0} S Nat (CommRing.toCommSemiring.{u2} S _inst_2))))) (fun (_x : RingHom.{u1, u2} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (CommSemiring.toSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (MvPolynomial.commSemiring.{u2, 0} S Nat (CommRing.toCommSemiring.{u2} S _inst_2))))) => (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) -> (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.hasCoeToFun.{u1, u2} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (CommSemiring.toSemiring.{u2} (MvPolynomial.{0, u2} Nat S (CommRing.toCommSemiring.{u2} S _inst_2)) (MvPolynomial.commSemiring.{u2, 0} S Nat (CommRing.toCommSemiring.{u2} S _inst_2))))) (MvPolynomial.map.{u1, u2, 0} R S Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} S _inst_2) f) (wittPolynomial.{u1} p R _inst_1 n)) (wittPolynomial.{u2} p S _inst_2 n)
+but is expected to have type
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(CommRing.toCommSemiring.{u1} S _inst_2)))) (RingHom.instRingHomClassRingHom.{u2, u1} (MvPolynomial.{0, u2} Nat R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.{0, u1} Nat S (CommRing.toCommSemiring.{u1} S _inst_2)) (Semiring.toNonAssocSemiring.{u2} (MvPolynomial.{0, u2} Nat R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} (MvPolynomial.{0, u2} Nat R (CommRing.toCommSemiring.{u2} R _inst_1)) (MvPolynomial.commSemiring.{u2, 0} R Nat (CommRing.toCommSemiring.{u2} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat S (CommRing.toCommSemiring.{u1} S _inst_2)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat S (CommRing.toCommSemiring.{u1} S _inst_2)) (MvPolynomial.commSemiring.{u1, 0} S Nat (CommRing.toCommSemiring.{u1} S _inst_2)))))))) (MvPolynomial.map.{u2, u1, 0} R S Nat (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} S _inst_2) f) (wittPolynomial.{u2} p R _inst_1 n)) (wittPolynomial.{u1} p S _inst_2 n)
+Case conversion may be inaccurate. Consider using '#align map_witt_polynomial map_wittPolynomialₓ'. -/
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
   by
@@ -126,6 +140,12 @@ theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n :=
 
 variable (R)
 
+/- warning: constant_coeff_witt_polynomial -> constantCoeff_wittPolynomial is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align constant_coeff_witt_polynomial constantCoeff_wittPolynomialₓ'. -/
 @[simp]
 theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
     constantCoeff (wittPolynomial p R n) = 0 :=
@@ -138,22 +158,33 @@ theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
   exact ne_of_gt (pow_pos hp.1.Pos _)
 #align constant_coeff_witt_polynomial constantCoeff_wittPolynomial
 
+#print wittPolynomial_zero /-
 @[simp]
 theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
   simp only [wittPolynomial, X, sum_singleton, range_one, pow_zero]
 #align witt_polynomial_zero wittPolynomial_zero
+-/
 
+/- warning: witt_polynomial_one -> wittPolynomial_one is a dubious translation:
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(CommRing.toCommSemiring.{u1} R _inst_1)) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)) (NonUnitalNonAssocRing.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R 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(Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (HPow.hPow.{u1, 0, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHPow.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (Monoid.Pow.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toMonoidWithZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) p))
+Case conversion may be inaccurate. Consider using '#align witt_polynomial_one wittPolynomial_oneₓ'. -/
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
   simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
     one_mul, pow_one, C_1, pow_zero]
 #align witt_polynomial_one wittPolynomial_one
 
+#print aeval_wittPolynomial /-
 theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
     aeval f (W_ R n) = ∑ i in range (n + 1), p ^ i * f i ^ p ^ (n - i) := by
   simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
 #align aeval_witt_polynomial aeval_wittPolynomial
+-/
 
+#print wittPolynomial_zMod_self /-
 /-- Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
 by expanding the `n`th Witt polynomial by `p`.
 -/
@@ -170,6 +201,7 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (nat.lt_succ_iff.mp hk), ← add_comm]
 #align witt_polynomial_zmod_self wittPolynomial_zMod_self
+-/
 
 section PPrime
 
@@ -177,6 +209,12 @@ variable [hp : NeZero p]
 
 include hp
 
+/- warning: witt_polynomial_vars -> wittPolynomial_vars is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : NeZero.{0} Nat Nat.hasZero p] [_inst_3 : CharZero.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))] (n : Nat), Eq.{1} (Finset.{0} Nat) (MvPolynomial.vars.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1 n)) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : NeZero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) p] [_inst_3 : CharZero.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (Ring.toAddGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))] (n : Nat), Eq.{1} (Finset.{0} Nat) (MvPolynomial.vars.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1 n)) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))
+Case conversion may be inaccurate. Consider using '#align witt_polynomial_vars wittPolynomial_varsₓ'. -/
 theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars = range (n + 1) :=
   by
   have : ∀ i, (monomial (Finsupp.single i (p ^ (n - i))) (p ^ i : R)).vars = {i} :=
@@ -193,11 +231,13 @@ theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars
     rwa [mem_singleton]
 #align witt_polynomial_vars wittPolynomial_vars
 
+#print wittPolynomial_vars_subset /-
 theorem wittPolynomial_vars_subset (n : ℕ) : (wittPolynomial p R n).vars ⊆ range (n + 1) :=
   by
   rw [← map_wittPolynomial p (Int.castRingHom R), ← wittPolynomial_vars p ℤ]
   apply vars_map
 #align witt_polynomial_vars_subset wittPolynomial_vars_subset
+-/
 
 end PPrime
 
@@ -213,6 +253,12 @@ The polynomials `X_in_terms_of_W` give the coordinate transformation in the back
 -/
 
 
+/- warning: X_in_terms_of_W -> xInTermsOfW is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)], Nat -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)], Nat -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))
+Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W xInTermsOfWₓ'. -/
 /-- The `X_in_terms_of_W p R n` is the polynomial on the basis of Witt polynomials
 that corresponds to the ordinary `X n`. -/
 noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
@@ -224,12 +270,24 @@ noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
       C (⅟ p ^ n : R)
 #align X_in_terms_of_W xInTermsOfW
 
+/- warning: X_in_terms_of_W_eq -> xInTermsOfW_eq is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)] {n : Nat}, Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n) (HMul.hMul.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Distrib.toHasMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toDistrib.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1))))) (HSub.hSub.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SubNegMonoid.toHasSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddGroup.toSubNegMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddGroupWithOne.toAddGroup.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroupWithOne.toAddGroupWithOne.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toAddCommGroupWithOne.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1)))))))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n) (Finset.sum.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (AddCommGroup.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toAddCommGroup.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1)))))) (Finset.range n) (fun (i : Nat) => HMul.hMul.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Distrib.toHasMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toDistrib.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1))))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) => R -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.hasCoeToFun.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p) i)) (HPow.hPow.{u1, 0, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHPow.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (Monoid.Pow.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1))))) (xInTermsOfW.{u1} p R _inst_1 _inst_2 i) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.Pow.{0} Nat Nat.monoid)) p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) n i)))))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) => R -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.hasCoeToFun.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Invertible.invOf.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p) _inst_2) n)))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] {n : Nat}, Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n) (HMul.hMul.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Invertible.invOf.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) _inst_2) n)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))))) (HSub.hSub.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n) (Finset.sum.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))))) (Finset.range n) (fun (i : Nat) => HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (NonUnitalNonAssocRing.toMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (Ring.toNonAssocRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) i)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R 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(Invertible.invOf.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) _inst_2) n)))
+Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_eq xInTermsOfW_eqₓ'. -/
 theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} :
     xInTermsOfW p R n =
       (X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i)) * C (⅟ p ^ n : R) :=
   by rw [xInTermsOfW, ← Fin.sum_univ_eq_sum_range]
 #align X_in_terms_of_W_eq xInTermsOfW_eq
 
+/- warning: constant_coeff_X_in_terms_of_W -> constantCoeff_xInTermsOfW is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : Fact (Nat.Prime p)] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)] (n : Nat), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (fun (_x : RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) => (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) -> R) (RingHom.hasCoeToFun.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.constantCoeff.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [hp : Fact (Nat.Prime p)] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (n : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (fun (_x : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (NonUnitalNonAssocSemiring.toMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.instRingHomClassRingHom.{u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) R (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (MvPolynomial.constantCoeff.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (CommSemiring.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => R) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfWₓ'. -/
 @[simp]
 theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
     constantCoeff (xInTermsOfW p R n) = 0 :=
@@ -246,6 +304,12 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
     apply pow_pos hp.1.Pos
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW
 
+/- warning: X_in_terms_of_W_zero -> xInTermsOfW_zero is a dubious translation:
+lean 3 declaration is
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p)], Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)], Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))
+Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_zero xInTermsOfW_zeroₓ'. -/
 @[simp]
 theorem xInTermsOfW_zero [Invertible (p : R)] : xInTermsOfW p R 0 = X 0 := by
   rw [xInTermsOfW_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]
@@ -257,6 +321,7 @@ variable [hp : Fact p.Prime]
 
 include hp
 
+#print xInTermsOfW_vars_aux /-
 theorem xInTermsOfW_vars_aux (n : ℕ) :
     n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) :=
   by
@@ -287,19 +352,34 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
     exact lt_of_lt_of_le H hj
   · exact lt_irrefl n (lt_of_lt_of_le H hj)
 #align X_in_terms_of_W_vars_aux xInTermsOfW_vars_aux
+-/
 
+#print xInTermsOfW_vars_subset /-
 theorem xInTermsOfW_vars_subset (n : ℕ) : (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) :=
   (xInTermsOfW_vars_aux p n).2
 #align X_in_terms_of_W_vars_subset xInTermsOfW_vars_subset
+-/
 
 end PPrime
 
+/- warning: X_in_terms_of_W_aux -> xInTermsOfW_aux is a dubious translation:
+lean 3 declaration is
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(CommRing.toCommSemiring.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) => R -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.hasCoeToFun.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p) n))) (HSub.hSub.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SubNegMonoid.toHasSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddGroup.toSubNegMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddGroupWithOne.toAddGroup.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroupWithOne.toAddGroupWithOne.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toAddCommGroupWithOne.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1)))))))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n) (Finset.sum.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (AddCommGroup.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toAddCommGroup.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1)))))) (Finset.range n) (fun (i : Nat) => HMul.hMul.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Distrib.toHasMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toDistrib.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1))))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) => R -> (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1))) (RingHom.hasCoeToFun.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTCₓ.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) p) i)) (HPow.hPow.{u1, 0, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHPow.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (Monoid.Pow.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commRing.{u1, 0} R Nat _inst_1))))) (xInTermsOfW.{u1} p R _inst_1 _inst_2 i) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.Pow.{0} Nat Nat.monoid)) p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) n i))))))
+but is expected to have type
+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (n : Nat), Eq.{succ u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HMul.hMul.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) n)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toNonAssocRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))))) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.C.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p) n))) (HSub.hSub.{u1, u1, u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (instHSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSub.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toRing.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.instCommRingMvPolynomialToCommSemiring.{u1, 0} R Nat _inst_1)))) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n) (Finset.sum.{u1, 0} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R 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+Case conversion may be inaccurate. Consider using '#align X_in_terms_of_W_aux xInTermsOfW_auxₓ'. -/
 theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
     xInTermsOfW p R n * C (p ^ n : R) =
       X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i) :=
   by rw [xInTermsOfW_eq, mul_assoc, ← C_mul, ← mul_pow, invOf_mul_self, one_pow, C_1, mul_one]
 #align X_in_terms_of_W_aux xInTermsOfW_aux
 
+/- warning: bind₁_X_in_terms_of_W_witt_polynomial -> bind₁_xInTermsOfW_wittPolynomial is a dubious translation:
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+  forall (p : Nat) (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] [_inst_2 : Invertible.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)] (k : Nat), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) => MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (wittPolynomial.{u1} p R _inst_1 k)) (FunLike.coe.{succ u1, succ u1, succ u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} 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MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) _x) (SMulHomClass.toFunLike.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (SMulZeroClass.toSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddMonoid.toAddZeroClass.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) 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+Case conversion may be inaccurate. Consider using '#align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomialₓ'. -/
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     bind₁ (xInTermsOfW p R) (W_ R k) = X k :=
@@ -311,6 +391,12 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
   simp only [C_pow, bind₁_X_right, sub_add_cancel]
 #align bind₁_X_in_terms_of_W_witt_polynomial bind₁_xInTermsOfW_wittPolynomial
 
+/- warning: bind₁_witt_polynomial_X_in_terms_of_W -> bind₁_wittPolynomial_xInTermsOfW is a dubious translation:
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(AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (DistribMulActionHomClass.toSMulHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R 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(CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u1, u1, u1, u1} (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Module.toDistribMulAction.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Algebra.toModule.{u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AlgHom.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AlgHom.algHomClass.{u1, u1, u1} R (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} (MvPolynomial.{0, u1} Nat R (CommRing.toCommSemiring.{u1} R _inst_1)) (MvPolynomial.commSemiring.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (MvPolynomial.algebra.{u1, u1, 0} R R Nat (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u1} R _inst_1) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (MvPolynomial.bind₁.{0, 0, u1} Nat Nat R (CommRing.toCommSemiring.{u1} R _inst_1) (wittPolynomial.{u1} p R _inst_1)) (xInTermsOfW.{u1} p R _inst_1 _inst_2 n)) (MvPolynomial.X.{u1, 0} R Nat (CommRing.toCommSemiring.{u1} R _inst_1) n)
+Case conversion may be inaccurate. Consider using '#align bind₁_witt_polynomial_X_in_terms_of_W bind₁_wittPolynomial_xInTermsOfWₓ'. -/
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     bind₁ (W_ R) (xInTermsOfW p R n) = X n :=

c3019c79

bump to nightly-2023-03-15-10

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd

f73428b0

Diff

@@ -85,7 +85,7 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
 #align witt_polynomial wittPolynomial
 
 theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
-    wittPolynomial p R n = ∑ i in range (n + 1), c (p ^ i : R) * x i ^ p ^ (n - i) :=
+    wittPolynomial p R n = ∑ i in range (n + 1), C (p ^ i : R) * X i ^ p ^ (n - i) :=
   by
   apply sum_congr rfl
   rintro i -
@@ -139,12 +139,12 @@ theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
 #align constant_coeff_witt_polynomial constantCoeff_wittPolynomial
 
 @[simp]
-theorem wittPolynomial_zero : wittPolynomial p R 0 = x 0 := by
+theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
   simp only [wittPolynomial, X, sum_singleton, range_one, pow_zero]
 #align witt_polynomial_zero wittPolynomial_zero
 
 @[simp]
-theorem wittPolynomial_one : wittPolynomial p R 1 = c ↑p * x 1 + x 0 ^ p := by
+theorem wittPolynomial_one : wittPolynomial p R 1 = C ↑p * X 1 + X 0 ^ p := by
   simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
     one_mul, pow_one, C_1, pow_zero]
 #align witt_polynomial_one wittPolynomial_one
@@ -217,16 +217,16 @@ The polynomials `X_in_terms_of_W` give the coordinate transformation in the back
 that corresponds to the ordinary `X n`. -/
 noncomputable def xInTermsOfW [Invertible (p : R)] : ℕ → MvPolynomial ℕ R
   | n =>
-    (x n -
+    (X n -
         ∑ i : Fin n,
           have := i.2
-          c (p ^ (i : ℕ) : R) * xInTermsOfW i ^ p ^ (n - i)) *
-      c (⅟ p ^ n : R)
+          C (p ^ (i : ℕ) : R) * xInTermsOfW i ^ p ^ (n - i)) *
+      C (⅟ p ^ n : R)
 #align X_in_terms_of_W xInTermsOfW
 
 theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} :
     xInTermsOfW p R n =
-      (x n - ∑ i in range n, c (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i)) * c (⅟ p ^ n : R) :=
+      (X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i)) * C (⅟ p ^ n : R) :=
   by rw [xInTermsOfW, ← Fin.sum_univ_eq_sum_range]
 #align X_in_terms_of_W_eq xInTermsOfW_eq
 
@@ -247,7 +247,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW
 
 @[simp]
-theorem xInTermsOfW_zero [Invertible (p : R)] : xInTermsOfW p R 0 = x 0 := by
+theorem xInTermsOfW_zero [Invertible (p : R)] : xInTermsOfW p R 0 = X 0 := by
   rw [xInTermsOfW_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]
 #align X_in_terms_of_W_zero xInTermsOfW_zero
 
@@ -295,14 +295,14 @@ theorem xInTermsOfW_vars_subset (n : ℕ) : (xInTermsOfW p ℚ n).vars ⊆ range
 end PPrime
 
 theorem xInTermsOfW_aux [Invertible (p : R)] (n : ℕ) :
-    xInTermsOfW p R n * c (p ^ n : R) =
-      x n - ∑ i in range n, c (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i) :=
+    xInTermsOfW p R n * C (p ^ n : R) =
+      X n - ∑ i in range n, C (p ^ i : R) * xInTermsOfW p R i ^ p ^ (n - i) :=
   by rw [xInTermsOfW_eq, mul_assoc, ← C_mul, ← mul_pow, invOf_mul_self, one_pow, C_1, mul_one]
 #align X_in_terms_of_W_aux xInTermsOfW_aux
 
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
-    bind₁ (xInTermsOfW p R) (W_ R k) = x k :=
+    bind₁ (xInTermsOfW p R) (W_ R k) = X k :=
   by
   rw [wittPolynomial_eq_sum_c_mul_x_pow, AlgHom.map_sum]
   simp only [AlgHom.map_pow, C_pow, AlgHom.map_mul, alg_hom_C]
@@ -313,7 +313,7 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
 
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
-    bind₁ (W_ R) (xInTermsOfW p R n) = x n :=
+    bind₁ (W_ R) (xInTermsOfW p R n) = X n :=
   by
   apply Nat.strong_induction_on n
   clear n

c3019c79

bump to nightly-2023-03-11-16

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

cd44fb92

Diff

@@ -163,7 +163,7 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   by
   simp only [wittPolynomial_eq_sum_c_mul_x_pow]
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
-    zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
+    MulZeroClass.zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
   rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, alg_hom_C, ← pow_mul, ← pow_succ]
   congr
@@ -238,11 +238,11 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
   intro n IH
   rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, RingHom.map_sum, constant_coeff_C,
     sum_eq_zero]
-  · simp only [constant_coeff_X, sub_zero, mul_zero]
+  · simp only [constant_coeff_X, sub_zero, MulZeroClass.mul_zero]
   · intro m H
     rw [mem_range] at H
     simp only [RingHom.map_mul, RingHom.map_pow, constant_coeff_C, IH m H]
-    rw [zero_pow, mul_zero]
+    rw [zero_pow, MulZeroClass.mul_zero]
     apply pow_pos hp.1.Pos
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW

c3019c79

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

c3019c79

doc: convert many comments into doc comments (#11940)

All of these changes appear to be oversights to me.

678a5c79

Diff

@@ -107,7 +107,7 @@ open Witt
 
 open MvPolynomial
 
-/- The first observation is that the Witt polynomial doesn't really depend on the coefficient ring.
+/-! The first observation is that the Witt polynomial doesn't really depend on the coefficient ring.
 If we map the coefficients through a ring homomorphism, we obtain the corresponding Witt polynomial
 over the target ring. -/
 section
@@ -151,8 +151,7 @@ theorem aeval_wittPolynomial {A : Type*} [CommRing A] [Algebra R A] (f : ℕ →
 #align aeval_witt_polynomial aeval_wittPolynomial
 
 /-- Over the ring `ZMod (p^(n+1))`, we produce the `n+1`st Witt polynomial
-by expanding the `n`th Witt polynomial by `p`.
--/
+by expanding the `n`th Witt polynomial by `p`. -/
 @[simp]
 theorem wittPolynomial_zmod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by

c3019c79

chore: remove mathport name: <expression> lines (#11928)

Quoting [@digama0](https://github.com/digama0):

These were actually never meant to go in the file, they are basically debugging information and only useful on significantly broken mathport files. You can safely remove all of them.

11431c65

Diff

@@ -93,13 +93,11 @@ set_option linter.uppercaseLean3 false in
 This allows us to simply write `W n` or `W_ ℤ n`. -/
 
 
--- mathport name: witt_polynomial
 -- Notation with ring of coefficients explicit
 set_option quotPrecheck false in
 @[inherit_doc]
 scoped[Witt] notation "W_" => wittPolynomial p
 
--- mathport name: witt_polynomial.infer
 -- Notation with ring of coefficients implicit
 set_option quotPrecheck false in
 @[inherit_doc]

c3019c79

move(Polynomial): Move out of Data (#11751)

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

56e439ec

Diff

@@ -4,10 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Robert Y. Lewis
 -/
 import Mathlib.Algebra.CharP.Invertible
+import Mathlib.Algebra.MvPolynomial.Variables
+import Mathlib.Algebra.MvPolynomial.CommRing
+import Mathlib.Algebra.MvPolynomial.Expand
 import Mathlib.Data.Fintype.BigOperators
-import Mathlib.Data.MvPolynomial.Variables
-import Mathlib.Data.MvPolynomial.CommRing
-import Mathlib.Data.MvPolynomial.Expand
 import Mathlib.Data.ZMod.Basic
 
 #align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"

c3019c79

change the order of operation in zsmulRec and nsmulRec (#11451)

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.

9a3feeae

Diff

@@ -162,7 +162,7 @@ theorem wittPolynomial_zmod_self (n : ℕ) :
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
     zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
-  rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, ← pow_mul, ← pow_succ]
+  rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, ← pow_mul, ← pow_succ']
   congr
   rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (Nat.lt_succ_iff.mp hk), ← add_comm]

c3019c79

chore(*): remove empty lines between variable statements (#11418)

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)

75c86f04

Diff

@@ -69,7 +69,6 @@ open BigOperators
 --attribute [-simp] coe_eval₂_hom
 
 variable (p : ℕ)
-
 variable (R : Type*) [CommRing R] [DecidableEq R]
 
 /-- `wittPolynomial p R n` is the `n`-th Witt polynomial

c3019c79

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -229,7 +229,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
   intro n IH
   rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
     constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]
-  -- porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that
+  -- Porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that
   -- is created is not what we expect, and the sum is not replaced by zero...
   -- is it a bug in `rw` tactic?
   refine' Eq.trans (_ : _ = ((⅟↑p : R) ^ n)* 0) (mul_zero _)

c3019c79

refactor: optimize proofs with omega (#11093)

I ran tryAtEachStep on all files under Mathlib to find all locations where omega succeeds. For each that was a linarith without an only, I tried replacing it with omega, and I verified that elaboration time got smaller. (In almost all cases, there was a noticeable speedup.) I also replaced some slow aesops along the way.

37bc4aba

Diff

@@ -278,8 +278,8 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
     replace H := (ih j hj).2 (vars_pow _ _ H)
     rw [mem_range] at H
   · rw [mem_range]
-    linarith
-  · linarith
+    omega
+  · omega
 set_option linter.uppercaseLean3 false in
 #align X_in_terms_of_W_vars_aux xInTermsOfW_vars_aux

c3019c79

feat: The support of f ^ n (#9617)

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

From LeanAPAP

9d1a7d26

Diff

@@ -239,7 +239,7 @@ theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n :
   rw [mem_range] at H
   simp only [RingHom.map_mul, RingHom.map_pow, map_natCast, IH m H]
   rw [zero_pow, mul_zero]
-  apply pow_pos hp.1.pos
+  exact pow_ne_zero _ hp.1.ne_zero
 set_option linter.uppercaseLean3 false in
 #align constant_coeff_X_in_terms_of_W constantCoeff_xInTermsOfW

c3019c79

doc: @[inherit_doc] on notations (#9942)

Make all the notations that unambiguously should inherit the docstring of their definition actually inherit it.

Also write a few docstrings by hand. I only wrote the ones I was competent to write and which I was sure of. Some docstrings come from mathlib3 as they were lost during the early port.

This PR is only intended as a first pass There are many more docstrings to add.

08f7e99d

Diff

@@ -97,11 +97,13 @@ This allows us to simply write `W n` or `W_ ℤ n`. -/
 -- mathport name: witt_polynomial
 -- Notation with ring of coefficients explicit
 set_option quotPrecheck false in
+@[inherit_doc]
 scoped[Witt] notation "W_" => wittPolynomial p
 
 -- mathport name: witt_polynomial.infer
 -- Notation with ring of coefficients implicit
 set_option quotPrecheck false in
+@[inherit_doc]
 scoped[Witt] notation "W" => wittPolynomial p _
 
 open Witt

c3019c79

chore: replace exact_mod_cast tactic with mod_cast elaborator where possible (#8404)

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>

693fd795

Diff

@@ -271,7 +271,7 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
     rw [vars_C_mul] at H
     swap
     · apply pow_ne_zero
-      exact_mod_cast hp.1.ne_zero
+      exact mod_cast hp.1.ne_zero
     rw [mem_range] at hj
     replace H := (ih j hj).2 (vars_pow _ _ H)
     rw [mem_range] at H

c3019c79

chore: drop MulZeroClass. in mul_zero/zero_mul (#6682)

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).

277dea95

Diff

@@ -159,7 +159,7 @@ theorem wittPolynomial_zmod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by
   simp only [wittPolynomial_eq_sum_C_mul_X_pow]
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
-    MulZeroClass.zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
+    zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
   rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, ← pow_mul, ← pow_succ]
   congr

c3019c79

chore: banish Type _ and Sort _ (#6499)

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

This has nice performance benefits.

32de3f67

Diff

@@ -70,7 +70,7 @@ open BigOperators
 
 variable (p : ℕ)
 
-variable (R : Type _) [CommRing R] [DecidableEq R]
+variable (R : Type*) [CommRing R] [DecidableEq R]
 
 /-- `wittPolynomial p R n` is the `n`-th Witt polynomial
 with respect to a prime `p` with coefficients in a commutative ring `R`.
@@ -113,7 +113,7 @@ If we map the coefficients through a ring homomorphism, we obtain the correspond
 over the target ring. -/
 section
 
-variable {R} {S : Type _} [CommRing S]
+variable {R} {S : Type*} [CommRing S]
 
 @[simp]
 theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n := by
@@ -146,7 +146,7 @@ theorem wittPolynomial_one : wittPolynomial p R 1 = C (p : R) * X 1 + X 0 ^ p :=
     one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]
 #align witt_polynomial_one wittPolynomial_one
 
-theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
+theorem aeval_wittPolynomial {A : Type*} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
     aeval f (W_ R n) = ∑ i in range (n + 1), (p : A) ^ i * f i ^ p ^ (n - i) := by
   simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
 #align aeval_witt_polynomial aeval_wittPolynomial

c3019c79

chore: tidy various files (#6291)

c4679d7b

Diff

@@ -155,7 +155,7 @@ theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ 
 by expanding the `n`th Witt polynomial by `p`.
 -/
 @[simp]
-theorem wittPolynomial_zMod_self (n : ℕ) :
+theorem wittPolynomial_zmod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by
   simp only [wittPolynomial_eq_sum_C_mul_X_pow]
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
@@ -165,7 +165,7 @@ theorem wittPolynomial_zMod_self (n : ℕ) :
   congr
   rw [mem_range] at hk
   rw [add_comm, add_tsub_assoc_of_le (Nat.lt_succ_iff.mp hk), ← add_comm]
-#align witt_polynomial_zmod_self wittPolynomial_zMod_self
+#align witt_polynomial_zmod_self wittPolynomial_zmod_self
 
 section PPrime

c3019c79

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

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

89aa3a3b

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Robert Y. Lewis
-
-! This file was ported from Lean 3 source module ring_theory.witt_vector.witt_polynomial
-! leanprover-community/mathlib commit c3019c79074b0619edb4b27553a91b2e82242395
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.CharP.Invertible
 import Mathlib.Data.Fintype.BigOperators
@@ -15,6 +10,8 @@ import Mathlib.Data.MvPolynomial.CommRing
 import Mathlib.Data.MvPolynomial.Expand
 import Mathlib.Data.ZMod.Basic
 
+#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
+
 /-!
 # Witt polynomials

c3019c79

chore: remove occurrences of semicolon after space (#5713)

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

c795bd35

Diff

@@ -226,7 +226,7 @@ set_option linter.uppercaseLean3 false in
 @[simp]
 theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
     constantCoeff (xInTermsOfW p R n) = 0 := by
-  apply Nat.strongInductionOn n ; clear n
+  apply Nat.strongInductionOn n; clear n
   intro n IH
   rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
     constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]

c3019c79

chore: fix focusing dots (#5708)

This PR is the result of running

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

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

cb02d09e

Diff

@@ -273,14 +273,14 @@ theorem xInTermsOfW_vars_aux (n : ℕ) :
     rcases H with ⟨j, hj, H⟩
     rw [vars_C_mul] at H
     swap
-    . apply pow_ne_zero
+    · apply pow_ne_zero
       exact_mod_cast hp.1.ne_zero
     rw [mem_range] at hj
     replace H := (ih j hj).2 (vars_pow _ _ H)
     rw [mem_range] at H
-  . rw [mem_range]
+  · rw [mem_range]
     linarith
-  . linarith
+  · linarith
 set_option linter.uppercaseLean3 false in
 #align X_in_terms_of_W_vars_aux xInTermsOfW_vars_aux

c3019c79

feat: port RingTheory.WittVector.MulCoeff (#5554)

b47e7d28

Diff

@@ -84,14 +84,14 @@ noncomputable def wittPolynomial (n : ℕ) : MvPolynomial ℕ R :=
   ∑ i in range (n + 1), monomial (single i (p ^ (n - i))) ((p : R) ^ i)
 #align witt_polynomial wittPolynomial
 
-theorem wittPolynomial_eq_sum_c_mul_x_pow (n : ℕ) :
+theorem wittPolynomial_eq_sum_C_mul_X_pow (n : ℕ) :
     wittPolynomial p R n = ∑ i in range (n + 1), C ((p : R) ^ i) * X i ^ p ^ (n - i) := by
   apply sum_congr rfl
   rintro i -
   rw [monomial_eq, Finsupp.prod_single_index]
   rw [pow_zero]
 set_option linter.uppercaseLean3 false in
-#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_c_mul_x_pow
+#align witt_polynomial_eq_sum_C_mul_X_pow wittPolynomial_eq_sum_C_mul_X_pow
 
 /-! We set up notation locally to this file, to keep statements short and comprehensible.
 This allows us to simply write `W n` or `W_ ℤ n`. -/
@@ -145,7 +145,7 @@ theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by
 
 @[simp]
 theorem wittPolynomial_one : wittPolynomial p R 1 = C (p : R) * X 1 + X 0 ^ p := by
-  simp only [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm, range_one, sum_singleton,
+  simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,
     one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]
 #align witt_polynomial_one wittPolynomial_one
 
@@ -160,7 +160,7 @@ by expanding the `n`th Witt polynomial by `p`.
 @[simp]
 theorem wittPolynomial_zMod_self (n : ℕ) :
     W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by
-  simp only [wittPolynomial_eq_sum_c_mul_x_pow]
+  simp only [wittPolynomial_eq_sum_C_mul_X_pow]
   rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
     MulZeroClass.zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
   intro k hk
@@ -302,7 +302,7 @@ set_option linter.uppercaseLean3 false in
 @[simp]
 theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
     bind₁ (xInTermsOfW p R) (W_ R k) = X k := by
-  rw [wittPolynomial_eq_sum_c_mul_x_pow, AlgHom.map_sum]
+  rw [wittPolynomial_eq_sum_C_mul_X_pow, AlgHom.map_sum]
   simp only [Nat.cast_pow, AlgHom.map_pow, C_pow, AlgHom.map_mul, algHom_C]
   rw [sum_range_succ_comm, tsub_self, pow_zero, pow_one, bind₁_X_right, mul_comm, ← C_pow,
     xInTermsOfW_aux]
@@ -320,7 +320,7 @@ theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     show X n = (X n * C ((p : R) ^ n)) * C ((⅟p : R) ^ n) by
       rw [mul_assoc, ← C_mul, ← mul_pow, mul_invOf_self, one_pow, map_one, mul_one]]
   congr 1
-  rw [wittPolynomial_eq_sum_c_mul_x_pow, sum_range_succ_comm,
+  rw [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm,
     tsub_self, pow_zero, pow_one, mul_comm (X n), add_sub_assoc, add_right_eq_self, sub_eq_zero]
   apply sum_congr rfl
   intro i h

c3019c79

chore: fix upper/lowercase in comments (#4360)

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

27d257fc

Diff

@@ -154,7 +154,7 @@ theorem aeval_wittPolynomial {A : Type _} [CommRing A] [Algebra R A] (f : ℕ 
   simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
 #align aeval_witt_polynomial aeval_wittPolynomial
 
-/-- Over the ring `zmod (p^(n+1))`, we produce the `n+1`st Witt polynomial
+/-- Over the ring `ZMod (p^(n+1))`, we produce the `n+1`st Witt polynomial
 by expanding the `n`th Witt polynomial by `p`.
 -/
 @[simp]

c3019c79

feat: port RingTheory.WittVector.WittPolynomial (#3355)

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

06326c9c

`ring_theory.witt_vector.witt_polynomial` ⟷ `Mathlib.RingTheory.WittVector.WittPolynomial`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 499

ring_theory.witt_vector.witt_polynomial ⟷ Mathlib.RingTheory.WittVector.WittPolynomial

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 499

`ring_theory.witt_vector.witt_polynomial` ⟷ `Mathlib.RingTheory.WittVector.WittPolynomial`