algebra.polynomial.big_operators ⟷ Mathlib.Algebra.Polynomial.BigOperators

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 -/
 import Algebra.Order.WithZero
-import Data.Polynomial.Monic
+import Algebra.Polynomial.Monic
 
 #align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
 

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

@@ -81,7 +81,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).
     rw [← List.foldr_max_of_ne_nil]
     · congr
     contrapose! h
-    rw [List.map_eq_nil] at h 
+    rw [List.map_eq_nil] at h
     simp [h]
 #align polynomial.degree_list_sum_le Polynomial.degree_list_sum_le
 -/
@@ -319,9 +319,9 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
   nontriviality R
   convert multiset_prod_X_sub_C_next_coeff (by assumption)
   rw [next_coeff]; split_ifs
-  · rw [nat_degree_multiset_prod_of_monic] at h  <;> simp only [Multiset.mem_map] at *
+  · rw [nat_degree_multiset_prod_of_monic] at h <;> simp only [Multiset.mem_map] at *
     swap; · rintro _ ⟨_, _, rfl⟩; apply monic_X_sub_C
-    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h 
+    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h
     contrapose! h
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
     exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, nat_degree_X_sub_C _⟩, one_ne_zero⟩

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 -/
-import Mathbin.Algebra.Order.WithZero
-import Mathbin.Data.Polynomial.Monic
+import Algebra.Order.WithZero
+import Data.Polynomial.Monic
 
 #align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
-
-! This file was ported from Lean 3 source module algebra.polynomial.big_operators
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Order.WithZero
 import Mathbin.Data.Polynomial.Monic
 
+#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
 /-!
 # Lemmas for the interaction between polynomials and `∑` and `∏`.
 

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

@@ -51,20 +51,27 @@ section Semiring
 
 variable {S : Type _} [Semiring S]
 
+#print Polynomial.natDegree_list_sum_le /-
 theorem natDegree_list_sum_le (l : List S[X]) : natDegree l.Sum ≤ (l.map natDegree).foldr max 0 :=
   List.sum_le_foldr_max natDegree (by simp) natDegree_add_le _
 #align polynomial.nat_degree_list_sum_le Polynomial.natDegree_list_sum_le
+-/
 
+#print Polynomial.natDegree_multiset_sum_le /-
 theorem natDegree_multiset_sum_le (l : Multiset S[X]) :
     natDegree l.Sum ≤ (l.map natDegree).foldr max max_left_comm 0 :=
   Quotient.inductionOn l (by simpa using nat_degree_list_sum_le)
 #align polynomial.nat_degree_multiset_sum_le Polynomial.natDegree_multiset_sum_le
+-/
 
+#print Polynomial.natDegree_sum_le /-
 theorem natDegree_sum_le (f : ι → S[X]) :
     natDegree (∑ i in s, f i) ≤ s.fold max 0 (natDegree ∘ f) := by
   simpa using nat_degree_multiset_sum_le (s.val.map f)
 #align polynomial.nat_degree_sum_le Polynomial.natDegree_sum_le
+-/
 
+#print Polynomial.degree_list_sum_le /-
 theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).maximum :=
   by
   by_cases h : l.sum = 0
@@ -80,6 +87,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).
     rw [List.map_eq_nil] at h 
     simp [h]
 #align polynomial.degree_list_sum_le Polynomial.degree_list_sum_le
+-/
 
 #print Polynomial.natDegree_list_prod_le /-
 theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.Prod ≤ (l.map natDegree).Sum :=
@@ -99,6 +107,7 @@ theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).S
 #align polynomial.degree_list_prod_le Polynomial.degree_list_prod_le
 -/
 
+#print Polynomial.coeff_list_prod_of_natDegree_le /-
 theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p ∈ l, natDegree p ≤ n) :
     coeff (List.prod l) (l.length * n) = (l.map fun p => coeff p n).Prod :=
   by
@@ -124,6 +133,7 @@ theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p 
         MulZeroClass.zero_mul]
       exact nat_degree_mul_le.trans_lt (add_lt_add_of_lt_of_le hdn' h)
 #align polynomial.coeff_list_prod_of_nat_degree_le Polynomial.coeff_list_prod_of_natDegree_le
+-/
 
 end Semiring
 
@@ -158,6 +168,7 @@ theorem degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree :
 #align polynomial.degree_prod_le Polynomial.degree_prod_le
 -/
 
+#print Polynomial.leadingCoeff_multiset_prod' /-
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -171,7 +182,9 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).Prod ≠ 0) :
   simp only [Multiset.map_cons, Multiset.prod_cons] at h ⊢
   rw [Polynomial.leadingCoeff_mul'] <;> · rwa [ih]; apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
+-/
 
+#print Polynomial.leadingCoeff_prod' /-
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -182,7 +195,9 @@ theorem leadingCoeff_prod' (h : ∏ i in s, (f i).leadingCoeff ≠ 0) :
     (∏ i in s, f i).leadingCoeff = ∏ i in s, (f i).leadingCoeff := by
   simpa using leading_coeff_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.leading_coeff_prod' Polynomial.leadingCoeff_prod'
+-/
 
+#print Polynomial.natDegree_multiset_prod' /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -199,7 +214,9 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).Prod ≠ 0
   · apply right_ne_zero_of_mul ht
   · rwa [Polynomial.leadingCoeff_multiset_prod']; apply right_ne_zero_of_mul ht
 #align polynomial.nat_degree_multiset_prod' Polynomial.natDegree_multiset_prod'
+-/
 
+#print Polynomial.natDegree_prod' /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -210,6 +227,7 @@ theorem natDegree_prod' (h : ∏ i in s, (f i).leadingCoeff ≠ 0) :
     (∏ i in s, f i).natDegree = ∑ i in s, (f i).natDegree := by
   simpa using nat_degree_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.nat_degree_prod' Polynomial.natDegree_prod'
+-/
 
 #print Polynomial.natDegree_multiset_prod_of_monic /-
 theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
@@ -234,12 +252,14 @@ theorem natDegree_prod_of_monic (h : ∀ i ∈ s, (f i).Monic) :
 #align polynomial.nat_degree_prod_of_monic Polynomial.natDegree_prod_of_monic
 -/
 
+#print Polynomial.coeff_multiset_prod_of_natDegree_le /-
 theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegree p ≤ n) :
     coeff t.Prod (t.card * n) = (t.map fun p => coeff p n).Prod :=
   by
   induction t using Quotient.inductionOn
   simpa using coeff_list_prod_of_nat_degree_le _ _ hl
 #align polynomial.coeff_multiset_prod_of_nat_degree_le Polynomial.coeff_multiset_prod_of_natDegree_le
+-/
 
 #print Polynomial.coeff_prod_of_natDegree_le /-
 theorem coeff_prod_of_natDegree_le (f : ι → R[X]) (n : ℕ) (h : ∀ p ∈ s, natDegree (f p) ≤ n) :
@@ -275,6 +295,7 @@ variable [CommRing R]
 
 open Monic
 
+#print Polynomial.multiset_prod_X_sub_C_nextCoeff /-
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
 theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
@@ -285,12 +306,16 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
     exact t.sum_hom (-AddMonoidHom.id R)
   · intros; apply monic_X_sub_C
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
+-/
 
+#print Polynomial.prod_X_sub_C_nextCoeff /-
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, (X - C (f i))) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
+-/
 
+#print Polynomial.multiset_prod_X_sub_C_coeff_card_pred /-
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
   by
@@ -305,11 +330,14 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
     exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, nat_degree_X_sub_C _⟩, one_ne_zero⟩
   congr; rw [nat_degree_multiset_prod_of_monic] <;> · simp [nat_degree_X_sub_C, monic_X_sub_C]
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
+-/
 
+#print Polynomial.prod_X_sub_C_coeff_card_pred /-
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, (X - C (f i))).coeff (s.card - 1) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)
 #align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_pred
+-/
 
 end CommRing
 
@@ -319,6 +347,7 @@ section Semiring
 
 variable [Semiring R] [NoZeroDivisors R]
 
+#print Polynomial.degree_list_prod /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 `[nontrivial R]` is needed, otherwise for `l = []` we have `⊥` in the LHS and `0` in the RHS.
@@ -326,6 +355,7 @@ the sum of the degrees, where the degree of the zero polynomial is ⊥.
 theorem degree_list_prod [Nontrivial R] (l : List R[X]) : l.Prod.degree = (l.map degree).Sum :=
   map_list_prod (@degreeMonoidHom R _ _ _) l
 #align polynomial.degree_list_prod Polynomial.degree_list_prod
+-/
 
 end Semiring
 
@@ -333,6 +363,7 @@ section CommSemiring
 
 variable [CommSemiring R] [NoZeroDivisors R] (f : ι → R[X]) (t : Multiset R[X])
 
+#print Polynomial.natDegree_prod /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees.
 
@@ -347,7 +378,9 @@ theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
   rw [prod_ne_zero_iff]
   intro x hx; simp [h x hx]
 #align polynomial.nat_degree_prod Polynomial.natDegree_prod
+-/
 
+#print Polynomial.natDegree_multiset_prod /-
 theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.map natDegree).Sum :=
   by
   nontriviality R
@@ -356,21 +389,27 @@ theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.m
   rintro ⟨_, h, rfl⟩
   contradiction
 #align polynomial.nat_degree_multiset_prod Polynomial.natDegree_multiset_prod
+-/
 
+#print Polynomial.degree_multiset_prod /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
 theorem degree_multiset_prod [Nontrivial R] : t.Prod.degree = (t.map fun f => degree f).Sum :=
   map_multiset_prod (@degreeMonoidHom R _ _ _) _
 #align polynomial.degree_multiset_prod Polynomial.degree_multiset_prod
+-/
 
+#print Polynomial.degree_prod /-
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
 theorem degree_prod [Nontrivial R] : (∏ i in s, f i).degree = ∑ i in s, (f i).degree :=
   map_prod (@degreeMonoidHom R _ _ _) _ _
 #align polynomial.degree_prod Polynomial.degree_prod
+-/
 
+#print Polynomial.leadingCoeff_multiset_prod /-
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 
@@ -380,7 +419,9 @@ where additionally, the product of the leading coefficients must be nonzero.
 theorem leadingCoeff_multiset_prod : t.Prod.leadingCoeff = (t.map fun f => leadingCoeff f).Prod :=
   by rw [← leading_coeff_hom_apply, MonoidHom.map_multiset_prod]; rfl
 #align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prod
+-/
 
+#print Polynomial.leadingCoeff_prod /-
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 
@@ -390,6 +431,7 @@ where additionally, the product of the leading coefficients must be nonzero.
 theorem leadingCoeff_prod : (∏ i in s, f i).leadingCoeff = ∏ i in s, (f i).leadingCoeff := by
   simpa using leading_coeff_multiset_prod (s.1.map f)
 #align polynomial.leading_coeff_prod Polynomial.leadingCoeff_prod
+-/
 
 end CommSemiring
 

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

@@ -178,7 +178,7 @@ the product of the leading coefficients, provided that this product is nonzero.
 See `polynomial.leading_coeff_prod` (without the `'`) for a version for integral domains,
 where this condition is automatically satisfied.
 -/
-theorem leadingCoeff_prod' (h : (∏ i in s, (f i).leadingCoeff) ≠ 0) :
+theorem leadingCoeff_prod' (h : ∏ i in s, (f i).leadingCoeff ≠ 0) :
     (∏ i in s, f i).leadingCoeff = ∏ i in s, (f i).leadingCoeff := by
   simpa using leading_coeff_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.leading_coeff_prod' Polynomial.leadingCoeff_prod'
@@ -206,7 +206,7 @@ the sum of the degrees, provided that the product of leading coefficients is non
 See `polynomial.nat_degree_prod` (without the `'`) for a version for integral domains,
 where this condition is automatically satisfied.
 -/
-theorem natDegree_prod' (h : (∏ i in s, (f i).leadingCoeff) ≠ 0) :
+theorem natDegree_prod' (h : ∏ i in s, (f i).leadingCoeff ≠ 0) :
     (∏ i in s, f i).natDegree = ∑ i in s, (f i).natDegree := by
   simpa using nat_degree_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.nat_degree_prod' Polynomial.natDegree_prod'
@@ -287,7 +287,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
-    nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
+    nextCoeff (∏ i in s, (X - C (f i))) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
 
@@ -307,7 +307,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
 
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
-    (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by
+    (∏ i in s, (X - C (f i))).coeff (s.card - 1) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)
 #align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_pred
 

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

@@ -77,7 +77,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).
     rw [← List.foldr_max_of_ne_nil]
     · congr
     contrapose! h
-    rw [List.map_eq_nil] at h
+    rw [List.map_eq_nil] at h 
     simp [h]
 #align polynomial.degree_list_sum_le Polynomial.degree_list_sum_le
 
@@ -168,7 +168,7 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).Prod ≠ 0) :
     t.Prod.leadingCoeff = (t.map leadingCoeff).Prod :=
   by
   induction' t using Multiset.induction_on with a t ih; · simp
-  simp only [Multiset.map_cons, Multiset.prod_cons] at h⊢
+  simp only [Multiset.map_cons, Multiset.prod_cons] at h ⊢
   rw [Polynomial.leadingCoeff_mul'] <;> · rwa [ih]; apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
@@ -194,7 +194,7 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).Prod ≠ 0
   by
   revert h
   refine' Multiset.induction_on t _ fun a t ih ht => _; · simp
-  rw [Multiset.map_cons, Multiset.prod_cons] at ht⊢
+  rw [Multiset.map_cons, Multiset.prod_cons] at ht ⊢
   rw [Multiset.sum_cons, Polynomial.natDegree_mul', ih]
   · apply right_ne_zero_of_mul ht
   · rwa [Polynomial.leadingCoeff_multiset_prod']; apply right_ne_zero_of_mul ht
@@ -283,7 +283,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
   rw [next_coeff_multiset_prod]
   · simp only [next_coeff_X_sub_C]
     exact t.sum_hom (-AddMonoidHom.id R)
-  · intros ; apply monic_X_sub_C
+  · intros; apply monic_X_sub_C
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
@@ -297,13 +297,13 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
   nontriviality R
   convert multiset_prod_X_sub_C_next_coeff (by assumption)
   rw [next_coeff]; split_ifs
-  · rw [nat_degree_multiset_prod_of_monic] at h <;> simp only [Multiset.mem_map] at *
+  · rw [nat_degree_multiset_prod_of_monic] at h  <;> simp only [Multiset.mem_map] at *
     swap; · rintro _ ⟨_, _, rfl⟩; apply monic_X_sub_C
-    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h
+    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h 
     contrapose! h
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
     exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, nat_degree_X_sub_C _⟩, one_ne_zero⟩
-  congr ; rw [nat_degree_multiset_prod_of_monic] <;> · simp [nat_degree_X_sub_C, monic_X_sub_C]
+  congr; rw [nat_degree_multiset_prod_of_monic] <;> · simp [nat_degree_X_sub_C, monic_X_sub_C]
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
 
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :

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

@@ -37,7 +37,7 @@ open Finset
 
 open Multiset
 
-open BigOperators Polynomial
+open scoped BigOperators Polynomial
 
 universe u w
 
@@ -90,12 +90,14 @@ theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.Prod ≤ (l.map nat
 #align polynomial.nat_degree_list_prod_le Polynomial.natDegree_list_prod_le
 -/
 
+#print Polynomial.degree_list_prod_le /-
 theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).Sum :=
   by
   induction' l with hd tl IH
   · simp
   · simpa using (degree_mul_le _ _).trans (add_le_add_left IH _)
 #align polynomial.degree_list_prod_le Polynomial.degree_list_prod_le
+-/
 
 theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p ∈ l, natDegree p ≤ n) :
     coeff (List.prod l) (l.length * n) = (l.map fun p => coeff p n).Prod :=
@@ -141,16 +143,20 @@ theorem natDegree_prod_le : (∏ i in s, f i).natDegree ≤ ∑ i in s, (f i).na
 #align polynomial.nat_degree_prod_le Polynomial.natDegree_prod_le
 -/
 
+#print Polynomial.degree_multiset_prod_le /-
 /-- The degree of a product of polynomials is at most the sum of the degrees,
 where the degree of the zero polynomial is ⊥.
 -/
 theorem degree_multiset_prod_le : t.Prod.degree ≤ (t.map Polynomial.degree).Sum :=
   Quotient.inductionOn t (by simpa using degree_list_prod_le)
 #align polynomial.degree_multiset_prod_le Polynomial.degree_multiset_prod_le
+-/
 
+#print Polynomial.degree_prod_le /-
 theorem degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree := by
   simpa only [Multiset.map_map] using degree_multiset_prod_le (s.1.map f)
 #align polynomial.degree_prod_le Polynomial.degree_prod_le
+-/
 
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.

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

@@ -51,44 +51,20 @@ section Semiring
 
 variable {S : Type _} [Semiring S]
 
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 theorem natDegree_list_sum_le (l : List S[X]) : natDegree l.Sum ≤ (l.map natDegree).foldr max 0 :=
   List.sum_le_foldr_max natDegree (by simp) natDegree_add_le _
 #align polynomial.nat_degree_list_sum_le Polynomial.natDegree_list_sum_le
 
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 theorem natDegree_multiset_sum_le (l : Multiset S[X]) :
     natDegree l.Sum ≤ (l.map natDegree).foldr max max_left_comm 0 :=
   Quotient.inductionOn l (by simpa using nat_degree_list_sum_le)
 #align polynomial.nat_degree_multiset_sum_le Polynomial.natDegree_multiset_sum_le
 
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 theorem natDegree_sum_le (f : ι → S[X]) :
     natDegree (∑ i in s, f i) ≤ s.fold max 0 (natDegree ∘ f) := by
   simpa using nat_degree_multiset_sum_le (s.val.map f)
 #align polynomial.nat_degree_sum_le Polynomial.natDegree_sum_le
 
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 theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).maximum :=
   by
   by_cases h : l.sum = 0
@@ -114,12 +90,6 @@ theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.Prod ≤ (l.map nat
 #align polynomial.nat_degree_list_prod_le Polynomial.natDegree_list_prod_le
 -/
 
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 theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).Sum :=
   by
   induction' l with hd tl IH
@@ -127,12 +97,6 @@ theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).S
   · simpa using (degree_mul_le _ _).trans (add_le_add_left IH _)
 #align polynomial.degree_list_prod_le Polynomial.degree_list_prod_le
 
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 theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p ∈ l, natDegree p ≤ n) :
     coeff (List.prod l) (l.length * n) = (l.map fun p => coeff p n).Prod :=
   by
@@ -177,12 +141,6 @@ theorem natDegree_prod_le : (∏ i in s, f i).natDegree ≤ ∑ i in s, (f i).na
 #align polynomial.nat_degree_prod_le Polynomial.natDegree_prod_le
 -/
 
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 /-- The degree of a product of polynomials is at most the sum of the degrees,
 where the degree of the zero polynomial is ⊥.
 -/
@@ -190,22 +148,10 @@ theorem degree_multiset_prod_le : t.Prod.degree ≤ (t.map Polynomial.degree).Su
   Quotient.inductionOn t (by simpa using degree_list_prod_le)
 #align polynomial.degree_multiset_prod_le Polynomial.degree_multiset_prod_le
 
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 theorem degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree := by
   simpa only [Multiset.map_map] using degree_multiset_prod_le (s.1.map f)
 #align polynomial.degree_prod_le Polynomial.degree_prod_le
 
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 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -220,12 +166,6 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).Prod ≠ 0) :
   rw [Polynomial.leadingCoeff_mul'] <;> · rwa [ih]; apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
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 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -237,12 +177,6 @@ theorem leadingCoeff_prod' (h : (∏ i in s, (f i).leadingCoeff) ≠ 0) :
   simpa using leading_coeff_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.leading_coeff_prod' Polynomial.leadingCoeff_prod'
 
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 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -260,12 +194,6 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).Prod ≠ 0
   · rwa [Polynomial.leadingCoeff_multiset_prod']; apply right_ne_zero_of_mul ht
 #align polynomial.nat_degree_multiset_prod' Polynomial.natDegree_multiset_prod'
 
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 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -300,9 +228,6 @@ theorem natDegree_prod_of_monic (h : ∀ i ∈ s, (f i).Monic) :
 #align polynomial.nat_degree_prod_of_monic Polynomial.natDegree_prod_of_monic
 -/
 
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-<too large>
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 theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegree p ≤ n) :
     coeff t.Prod (t.card * n) = (t.map fun p => coeff p n).Prod :=
   by
@@ -344,12 +269,6 @@ variable [CommRing R]
 
 open Monic
 
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-Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
 theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
@@ -361,20 +280,11 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
   · intros ; apply monic_X_sub_C
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
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(Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeffₓ'. -/
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
 
-/- warning: polynomial.multiset_prod_X_sub_C_coeff_card_pred -> Polynomial.multiset_prod_X_sub_C_coeff_card_pred is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
   by
@@ -390,9 +300,6 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
   congr ; rw [nat_degree_multiset_prod_of_monic] <;> · simp [nat_degree_X_sub_C, monic_X_sub_C]
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
 
-/- warning: polynomial.prod_X_sub_C_coeff_card_pred -> Polynomial.prod_X_sub_C_coeff_card_pred is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)
@@ -406,12 +313,6 @@ section Semiring
 
 variable [Semiring R] [NoZeroDivisors R]
 
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-Case conversion may be inaccurate. Consider using '#align polynomial.degree_list_prod Polynomial.degree_list_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 `[nontrivial R]` is needed, otherwise for `l = []` we have `⊥` in the LHS and `0` in the RHS.
@@ -426,12 +327,6 @@ section CommSemiring
 
 variable [CommSemiring R] [NoZeroDivisors R] (f : ι → R[X]) (t : Multiset R[X])
 
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-Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_prod Polynomial.natDegree_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees.
 
@@ -447,12 +342,6 @@ theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
   intro x hx; simp [h x hx]
 #align polynomial.nat_degree_prod Polynomial.natDegree_prod
 
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 theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.map natDegree).Sum :=
   by
   nontriviality R
@@ -462,12 +351,6 @@ theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.m
   contradiction
 #align polynomial.nat_degree_multiset_prod Polynomial.natDegree_multiset_prod
 
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 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
@@ -475,12 +358,6 @@ theorem degree_multiset_prod [Nontrivial R] : t.Prod.degree = (t.map fun f => de
   map_multiset_prod (@degreeMonoidHom R _ _ _) _
 #align polynomial.degree_multiset_prod Polynomial.degree_multiset_prod
 
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 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
@@ -488,12 +365,6 @@ theorem degree_prod [Nontrivial R] : (∏ i in s, f i).degree = ∑ i in s, (f i
   map_prod (@degreeMonoidHom R _ _ _) _ _
 #align polynomial.degree_prod Polynomial.degree_prod
 
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 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 
@@ -504,12 +375,6 @@ theorem leadingCoeff_multiset_prod : t.Prod.leadingCoeff = (t.map fun f => leadi
   by rw [← leading_coeff_hom_apply, MonoidHom.map_multiset_prod]; rfl
 #align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prod
 
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 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 

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

@@ -217,9 +217,7 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).Prod ≠ 0) :
   by
   induction' t using Multiset.induction_on with a t ih; · simp
   simp only [Multiset.map_cons, Multiset.prod_cons] at h⊢
-  rw [Polynomial.leadingCoeff_mul'] <;>
-    · rwa [ih]
-      apply right_ne_zero_of_mul h
+  rw [Polynomial.leadingCoeff_mul'] <;> · rwa [ih]; apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
 /- warning: polynomial.leading_coeff_prod' -> Polynomial.leadingCoeff_prod' is a dubious translation:
@@ -259,8 +257,7 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).Prod ≠ 0
   rw [Multiset.map_cons, Multiset.prod_cons] at ht⊢
   rw [Multiset.sum_cons, Polynomial.natDegree_mul', ih]
   · apply right_ne_zero_of_mul ht
-  · rwa [Polynomial.leadingCoeff_multiset_prod']
-    apply right_ne_zero_of_mul ht
+  · rwa [Polynomial.leadingCoeff_multiset_prod']; apply right_ne_zero_of_mul ht
 #align polynomial.nat_degree_multiset_prod' Polynomial.natDegree_multiset_prod'
 
 /- warning: polynomial.nat_degree_prod' -> Polynomial.natDegree_prod' is a dubious translation:
@@ -286,16 +283,12 @@ theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
   by
   nontriviality R
   apply nat_degree_multiset_prod'
-  suffices (t.map fun f => leading_coeff f).Prod = 1
-    by
-    rw [this]
-    simp
+  suffices (t.map fun f => leading_coeff f).Prod = 1 by rw [this]; simp
   convert prod_replicate t.card (1 : R)
   · simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and_iff]
     rintro i hi
     obtain ⟨i, hi, rfl⟩ := multiset.mem_map.mp hi
-    apply h
-    assumption
+    apply h; assumption
   · simp
 #align polynomial.nat_degree_multiset_prod_of_monic Polynomial.natDegree_multiset_prod_of_monic
 -/
@@ -365,8 +358,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
   rw [next_coeff_multiset_prod]
   · simp only [next_coeff_X_sub_C]
     exact t.sum_hom (-AddMonoidHom.id R)
-  · intros
-    apply monic_X_sub_C
+  · intros ; apply monic_X_sub_C
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
 /- warning: polynomial.prod_X_sub_C_next_coeff -> Polynomial.prod_X_sub_C_nextCoeff is a dubious translation:
@@ -390,9 +382,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
   convert multiset_prod_X_sub_C_next_coeff (by assumption)
   rw [next_coeff]; split_ifs
   · rw [nat_degree_multiset_prod_of_monic] at h <;> simp only [Multiset.mem_map] at *
-    swap
-    · rintro _ ⟨_, _, rfl⟩
-      apply monic_X_sub_C
+    swap; · rintro _ ⟨_, _, rfl⟩; apply monic_X_sub_C
     simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h
     contrapose! h
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
@@ -511,9 +501,7 @@ See `polynomial.leading_coeff_multiset_prod'` (with a `'`) for a version for com
 where additionally, the product of the leading coefficients must be nonzero.
 -/
 theorem leadingCoeff_multiset_prod : t.Prod.leadingCoeff = (t.map fun f => leadingCoeff f).Prod :=
-  by
-  rw [← leading_coeff_hom_apply, MonoidHom.map_multiset_prod]
-  rfl
+  by rw [← leading_coeff_hom_apply, MonoidHom.map_multiset_prod]; rfl
 #align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prod
 
 /- warning: polynomial.leading_coeff_prod -> Polynomial.leadingCoeff_prod is a dubious translation:

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

@@ -308,10 +308,7 @@ theorem natDegree_prod_of_monic (h : ∀ i ∈ s, (f i).Monic) :
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align polynomial.coeff_multiset_prod_of_nat_degree_le Polynomial.coeff_multiset_prod_of_natDegree_leₓ'. -/
 theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegree p ≤ n) :
     coeff t.Prod (t.card * n) = (t.map fun p => coeff p n).Prod :=
@@ -384,10 +381,7 @@ theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
@@ -407,10 +401,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
 
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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by

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

@@ -358,7 +358,7 @@ open Monic
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
@@ -376,7 +376,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeffₓ'. -/
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
@@ -387,7 +387,7 @@ theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
@@ -410,7 +410,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by

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

@@ -85,7 +85,7 @@ theorem natDegree_sum_le (f : ι → S[X]) :
 
 /- warning: polynomial.degree_list_sum_le -> Polynomial.degree_list_sum_le is a dubious translation:
 lean 3 declaration is
-  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.maximum.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (fun (a : Nat) (b : Nat) => Nat.decidableLt a b) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toHasLe.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.maximum.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (fun (a : Nat) (b : Nat) => Nat.decidableLt a b) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
 but is expected to have type
   forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.maximum.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (fun (a : Nat) (b : Nat) => Nat.decLt a b) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
 Case conversion may be inaccurate. Consider using '#align polynomial.degree_list_sum_le Polynomial.degree_list_sum_leₓ'. -/
@@ -114,14 +114,18 @@ theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.Prod ≤ (l.map nat
 #align polynomial.nat_degree_list_prod_le Polynomial.natDegree_list_prod_le
 -/
 
-#print Polynomial.degree_list_prod_le /-
+/- warning: polynomial.degree_list_prod_le -> Polynomial.degree_list_prod_le is a dubious translation:
+lean 3 declaration is
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toHasLe.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} S _inst_1 (List.prod.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.mul'.{u1} S _inst_1) (Polynomial.hasOne.{u1} S _inst_1) l)) (List.sum.{0} (WithBot.{0} Nat) (WithBot.hasAdd.{0} Nat Nat.hasAdd) (WithBot.hasZero.{0} Nat Nat.hasZero) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) (WithBot.{0} Nat) (Polynomial.degree.{u1} S _inst_1) l))
+but is expected to have type
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} S _inst_1 (List.prod.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.mul'.{u1} S _inst_1) (Polynomial.one.{u1} S _inst_1) l)) (List.sum.{0} (WithBot.{0} Nat) (WithBot.add.{0} Nat instAddNat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) (WithBot.{0} Nat) (Polynomial.degree.{u1} S _inst_1) l))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_list_prod_le Polynomial.degree_list_prod_leₓ'. -/
 theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).Sum :=
   by
   induction' l with hd tl IH
   · simp
   · simpa using (degree_mul_le _ _).trans (add_le_add_left IH _)
 #align polynomial.degree_list_prod_le Polynomial.degree_list_prod_le
--/
 
 /- warning: polynomial.coeff_list_prod_of_nat_degree_le -> Polynomial.coeff_list_prod_of_natDegree_le is a dubious translation:
 lean 3 declaration is
@@ -173,20 +177,28 @@ theorem natDegree_prod_le : (∏ i in s, f i).natDegree ≤ ∑ i in s, (f i).na
 #align polynomial.nat_degree_prod_le Polynomial.natDegree_prod_le
 -/
 
-#print Polynomial.degree_multiset_prod_le /-
+/- warning: polynomial.degree_multiset_prod_le -> Polynomial.degree_multiset_prod_le is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), LE.le.{0} (WithBot.{0} Nat) (Preorder.toHasLe.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} (WithBot.{0} Nat) (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} (WithBot.{0} Nat) (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_multiset_prod_le Polynomial.degree_multiset_prod_leₓ'. -/
 /-- The degree of a product of polynomials is at most the sum of the degrees,
 where the degree of the zero polynomial is ⊥.
 -/
 theorem degree_multiset_prod_le : t.Prod.degree ≤ (t.map Polynomial.degree).Sum :=
   Quotient.inductionOn t (by simpa using degree_list_prod_le)
 #align polynomial.degree_multiset_prod_le Polynomial.degree_multiset_prod_le
--/
 
-#print Polynomial.degree_prod_le /-
+/- warning: polynomial.degree_prod_le -> Polynomial.degree_prod_le is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), LE.le.{0} (WithBot.{0} Nat) (Preorder.toHasLe.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} (WithBot.{0} Nat) ι (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) s (fun (i : ι) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} (WithBot.{0} Nat) ι (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) s (fun (i : ι) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_prod_le Polynomial.degree_prod_leₓ'. -/
 theorem degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree := by
   simpa only [Multiset.map_map] using degree_multiset_prod_le (s.1.map f)
 #align polynomial.degree_prod_le Polynomial.degree_prod_le
--/
 
 /- warning: polynomial.leading_coeff_multiset_prod' -> Polynomial.leadingCoeff_multiset_prod' is a dubious translation:
 lean 3 declaration is

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

@@ -346,7 +346,7 @@ open Monic
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
@@ -364,7 +364,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeffₓ'. -/
 theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
@@ -375,7 +375,7 @@ theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
@@ -398,7 +398,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} 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} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -360,16 +360,16 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
     apply monic_X_sub_C
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
-/- warning: polynomial.prod_X_sub_C_next_coeff -> Polynomial.prod_x_sub_c_nextCoeff is a dubious translation:
+/- warning: polynomial.prod_X_sub_C_next_coeff -> Polynomial.prod_X_sub_C_nextCoeff is a dubious translation:
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 but is expected to have type
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeffₓ'. -/
-theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
+Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeffₓ'. -/
+theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
-#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeff
+#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
 
 /- warning: polynomial.multiset_prod_X_sub_C_coeff_card_pred -> Polynomial.multiset_prod_X_sub_C_coeff_card_pred is a dubious translation:
 lean 3 declaration is

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

@@ -344,7 +344,7 @@ open Monic
 
 /- warning: polynomial.multiset_prod_X_sub_C_next_coeff -> Polynomial.multiset_prod_X_sub_C_nextCoeff is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
@@ -362,7 +362,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 
 /- warning: polynomial.prod_X_sub_C_next_coeff -> Polynomial.prod_x_sub_c_nextCoeff is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 but is expected to have type
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeffₓ'. -/
@@ -373,7 +373,7 @@ theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
 
 /- warning: polynomial.multiset_prod_X_sub_C_coeff_card_pred -> Polynomial.multiset_prod_X_sub_C_coeff_card_pred is a dubious translation:
 lean 3 declaration is
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(Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
@@ -396,7 +396,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 
 /- warning: polynomial.prod_X_sub_C_coeff_card_pred -> Polynomial.prod_X_sub_C_coeff_card_pred is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 but is expected to have type
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 
 ! This file was ported from Lean 3 source module algebra.polynomial.big_operators
-! leanprover-community/mathlib commit 47adfab39a11a072db552f47594bf8ed2cf8a722
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Data.Polynomial.Monic
 /-!
 # Lemmas for the interaction between polynomials and `∑` and `∏`.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Recall that `∑` and `∏` are notation for `finset.sum` and `finset.prod` respectively.
 
 ## Main results

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

@@ -296,7 +296,7 @@ theorem natDegree_prod_of_monic (h : ∀ i ∈ s, (f i).Monic) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (n : Nat), (forall (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Membership.Mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.hasMem.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p t) -> (LE.le.{0} Nat Nat.hasLe (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p) n)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.orderedCancelAddCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.orderedCancelAddCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.orderedCancelAddCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) t) n)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p n) t)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (n : Nat), (forall (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instMembershipMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p t) -> (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p) n)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t) (HMul.hMul.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) (instHMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) instMulNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (fun (_x : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) t) n)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p n) t)))
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (n : Nat), (forall (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instMembershipMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p t) -> (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p) n)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t) (HMul.hMul.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) (instHMul.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) t) instMulNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (fun (_x : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) t) n)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p n) t)))
 Case conversion may be inaccurate. Consider using '#align polynomial.coeff_multiset_prod_of_nat_degree_le Polynomial.coeff_multiset_prod_of_natDegree_leₓ'. -/
 theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegree p ≤ n) :
     coeff t.Prod (t.card * n) = (t.map fun p => coeff p n).Prod :=
@@ -343,7 +343,7 @@ open Monic
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
@@ -361,7 +361,7 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] {s : Finset.{u2} ι} (f : ι -> R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i))))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i)))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeffₓ'. -/
 theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
@@ -372,7 +372,7 @@ theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
 Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
@@ -395,7 +395,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 lean 3 declaration is
   forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 but is expected to have type
-  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
 Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
 theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by

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

@@ -144,9 +144,11 @@ theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p 
     rcases hdn.eq_or_lt with (rfl | hdn')
     · cases' h.eq_or_lt with h' h'
       · rw [← h', coeff_mul_degree_add_degree, leading_coeff, leading_coeff]
-      · rw [coeff_eq_zero_of_nat_degree_lt, coeff_eq_zero_of_nat_degree_lt h', mul_zero]
+      · rw [coeff_eq_zero_of_nat_degree_lt, coeff_eq_zero_of_nat_degree_lt h',
+          MulZeroClass.mul_zero]
         exact nat_degree_mul_le.trans_lt (add_lt_add_left h' _)
-    · rw [coeff_eq_zero_of_nat_degree_lt hdn', coeff_eq_zero_of_nat_degree_lt, zero_mul]
+    · rw [coeff_eq_zero_of_nat_degree_lt hdn', coeff_eq_zero_of_nat_degree_lt,
+        MulZeroClass.zero_mul]
       exact nat_degree_mul_le.trans_lt (add_lt_add_of_lt_of_le hdn' h)
 #align polynomial.coeff_list_prod_of_nat_degree_le Polynomial.coeff_list_prod_of_natDegree_le
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -48,20 +48,44 @@ section Semiring
 
 variable {S : Type _} [Semiring S]
 
+/- warning: polynomial.nat_degree_list_sum_le -> Polynomial.natDegree_list_sum_le is a dubious translation:
+lean 3 declaration is
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} Nat Nat.hasLe (Polynomial.natDegree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.foldr.{0, 0} Nat Nat (LinearOrder.max.{0} Nat Nat.linearOrder) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+but is expected to have type
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.foldr.{0, 0} Nat Nat (Max.max.{0} Nat Nat.instMaxNat) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_list_sum_le Polynomial.natDegree_list_sum_leₓ'. -/
 theorem natDegree_list_sum_le (l : List S[X]) : natDegree l.Sum ≤ (l.map natDegree).foldr max 0 :=
   List.sum_le_foldr_max natDegree (by simp) natDegree_add_le _
 #align polynomial.nat_degree_list_sum_le Polynomial.natDegree_list_sum_le
 
+/- warning: polynomial.nat_degree_multiset_sum_le -> Polynomial.natDegree_multiset_sum_le is a dubious translation:
+lean 3 declaration is
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : Multiset.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} Nat Nat.hasLe (Polynomial.natDegree.{u1} S _inst_1 (Multiset.sum.{u1} (Polynomial.{u1} S _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} S _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.semiring.{u1} S _inst_1)))) l)) (Multiset.foldr.{0, 0} Nat Nat (LinearOrder.max.{0} Nat Nat.linearOrder) (max_left_comm.{0} Nat Nat.linearOrder) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Multiset.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+but is expected to have type
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : Multiset.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} S _inst_1 (Multiset.sum.{u1} (Polynomial.{u1} S _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} S _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.semiring.{u1} S _inst_1)))) l)) (Multiset.foldr.{0, 0} Nat Nat (Max.max.{0} Nat (LinearOrder.toMax.{0} Nat Nat.linearOrder)) (max_left_comm.{0} Nat Nat.linearOrder) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Multiset.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_multiset_sum_le Polynomial.natDegree_multiset_sum_leₓ'. -/
 theorem natDegree_multiset_sum_le (l : Multiset S[X]) :
     natDegree l.Sum ≤ (l.map natDegree).foldr max max_left_comm 0 :=
   Quotient.inductionOn l (by simpa using nat_degree_list_sum_le)
 #align polynomial.nat_degree_multiset_sum_le Polynomial.natDegree_multiset_sum_le
 
+/- warning: polynomial.nat_degree_sum_le -> Polynomial.natDegree_sum_le is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} (s : Finset.{u1} ι) {S : Type.{u2}} [_inst_1 : Semiring.{u2} S] (f : ι -> (Polynomial.{u2} S _inst_1)), LE.le.{0} Nat Nat.hasLe (Polynomial.natDegree.{u2} S _inst_1 (Finset.sum.{u2, u1} (Polynomial.{u2} S _inst_1) ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} S _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} S _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} S _inst_1) (Polynomial.semiring.{u2} S _inst_1)))) s (fun (i : ι) => f i))) (Finset.fold.{u1, 0} ι Nat (LinearOrder.max.{0} Nat Nat.linearOrder) (sup_isCommutative.{0} Nat (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid)) (sup_isAssociative.{0} Nat (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid)) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Function.comp.{succ u1, succ u2, 1} ι (Polynomial.{u2} S _inst_1) Nat (Polynomial.natDegree.{u2} S _inst_1) f) s)
+but is expected to have type
+  forall {ι : Type.{u2}} (s : Finset.{u2} ι) {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (f : ι -> (Polynomial.{u1} S _inst_1)), LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} S _inst_1 (Finset.sum.{u1, u2} (Polynomial.{u1} S _inst_1) ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} S _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.semiring.{u1} S _inst_1)))) s (fun (i : ι) => f i))) (Finset.fold.{u2, 0} ι Nat (Max.max.{0} Nat (LinearOrder.toMax.{0} Nat Nat.linearOrder)) (instIsCommutativeMaxToMax.{0} Nat Nat.linearOrder) (instIsAssociativeMaxToMax.{0} Nat Nat.linearOrder) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Function.comp.{succ u2, succ u1, 1} ι (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) f) s)
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_sum_le Polynomial.natDegree_sum_leₓ'. -/
 theorem natDegree_sum_le (f : ι → S[X]) :
     natDegree (∑ i in s, f i) ≤ s.fold max 0 (natDegree ∘ f) := by
   simpa using nat_degree_multiset_sum_le (s.val.map f)
 #align polynomial.nat_degree_sum_le Polynomial.natDegree_sum_le
 
+/- warning: polynomial.degree_list_sum_le -> Polynomial.degree_list_sum_le is a dubious translation:
+lean 3 declaration is
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.maximum.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))) (fun (a : Nat) (b : Nat) => Nat.decidableLt a b) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+but is expected to have type
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)), LE.le.{0} (WithBot.{0} Nat) (Preorder.toLE.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} S _inst_1 (List.sum.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.add'.{u1} S _inst_1) (Polynomial.zero.{u1} S _inst_1) l)) (List.maximum.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)) (fun (a : Nat) (b : Nat) => Nat.decLt a b) (List.map.{u1, 0} (Polynomial.{u1} S _inst_1) Nat (Polynomial.natDegree.{u1} S _inst_1) l))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_list_sum_le Polynomial.degree_list_sum_leₓ'. -/
 theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).maximum :=
   by
   by_cases h : l.sum = 0
@@ -78,20 +102,30 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.Sum ≤ (l.map natDegree).
     simp [h]
 #align polynomial.degree_list_sum_le Polynomial.degree_list_sum_le
 
+#print Polynomial.natDegree_list_prod_le /-
 theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.Prod ≤ (l.map natDegree).Sum :=
   by
   induction' l with hd tl IH
   · simp
   · simpa using nat_degree_mul_le.trans (add_le_add_left IH _)
 #align polynomial.nat_degree_list_prod_le Polynomial.natDegree_list_prod_le
+-/
 
+#print Polynomial.degree_list_prod_le /-
 theorem degree_list_prod_le (l : List S[X]) : degree l.Prod ≤ (l.map degree).Sum :=
   by
   induction' l with hd tl IH
   · simp
   · simpa using (degree_mul_le _ _).trans (add_le_add_left IH _)
 #align polynomial.degree_list_prod_le Polynomial.degree_list_prod_le
+-/
 
+/- warning: polynomial.coeff_list_prod_of_nat_degree_le -> Polynomial.coeff_list_prod_of_natDegree_le is a dubious translation:
+lean 3 declaration is
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)) (n : Nat), (forall (p : Polynomial.{u1} S _inst_1), (Membership.Mem.{u1, u1} (Polynomial.{u1} S _inst_1) (List.{u1} (Polynomial.{u1} S _inst_1)) (List.hasMem.{u1} (Polynomial.{u1} S _inst_1)) p l) -> (LE.le.{0} Nat Nat.hasLe (Polynomial.natDegree.{u1} S _inst_1 p) n)) -> (Eq.{succ u1} S (Polynomial.coeff.{u1} S _inst_1 (List.prod.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.mul'.{u1} S _inst_1) (Polynomial.hasOne.{u1} S _inst_1) l) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) (List.length.{u1} (Polynomial.{u1} S _inst_1) l) n)) (List.prod.{u1} S (Distrib.toHasMul.{u1} S (NonUnitalNonAssocSemiring.toDistrib.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_1)))) (AddMonoidWithOne.toOne.{u1} S (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} S (NonAssocSemiring.toAddCommMonoidWithOne.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_1)))) (List.map.{u1, u1} (Polynomial.{u1} S _inst_1) S (fun (p : Polynomial.{u1} S _inst_1) => Polynomial.coeff.{u1} S _inst_1 p n) l)))
+but is expected to have type
+  forall {S : Type.{u1}} [_inst_1 : Semiring.{u1} S] (l : List.{u1} (Polynomial.{u1} S _inst_1)) (n : Nat), (forall (p : Polynomial.{u1} S _inst_1), (Membership.mem.{u1, u1} (Polynomial.{u1} S _inst_1) (List.{u1} (Polynomial.{u1} S _inst_1)) (List.instMembershipList.{u1} (Polynomial.{u1} S _inst_1)) p l) -> (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} S _inst_1 p) n)) -> (Eq.{succ u1} S (Polynomial.coeff.{u1} S _inst_1 (List.prod.{u1} (Polynomial.{u1} S _inst_1) (Polynomial.mul'.{u1} S _inst_1) (Polynomial.one.{u1} S _inst_1) l) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (List.length.{u1} (Polynomial.{u1} S _inst_1) l) n)) (List.prod.{u1} S (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_1))) (Semiring.toOne.{u1} S _inst_1) (List.map.{u1, u1} (Polynomial.{u1} S _inst_1) S (fun (p : Polynomial.{u1} S _inst_1) => Polynomial.coeff.{u1} S _inst_1 p n) l)))
+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_list_prod_of_nat_degree_le Polynomial.coeff_list_prod_of_natDegree_leₓ'. -/
 theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p ∈ l, natDegree p ≤ n) :
     coeff (List.prod l) (l.length * n) = (l.map fun p => coeff p n).Prod :=
   by
@@ -122,25 +156,39 @@ section CommSemiring
 
 variable [CommSemiring R] (f : ι → R[X]) (t : Multiset R[X])
 
+#print Polynomial.natDegree_multiset_prod_le /-
 theorem natDegree_multiset_prod_le : t.Prod.natDegree ≤ (t.map natDegree).Sum :=
   Quotient.inductionOn t (by simpa using nat_degree_list_prod_le)
 #align polynomial.nat_degree_multiset_prod_le Polynomial.natDegree_multiset_prod_le
+-/
 
+#print Polynomial.natDegree_prod_le /-
 theorem natDegree_prod_le : (∏ i in s, f i).natDegree ≤ ∑ i in s, (f i).natDegree := by
   simpa using nat_degree_multiset_prod_le (s.1.map f)
 #align polynomial.nat_degree_prod_le Polynomial.natDegree_prod_le
+-/
 
+#print Polynomial.degree_multiset_prod_le /-
 /-- The degree of a product of polynomials is at most the sum of the degrees,
 where the degree of the zero polynomial is ⊥.
 -/
 theorem degree_multiset_prod_le : t.Prod.degree ≤ (t.map Polynomial.degree).Sum :=
   Quotient.inductionOn t (by simpa using degree_list_prod_le)
 #align polynomial.degree_multiset_prod_le Polynomial.degree_multiset_prod_le
+-/
 
+#print Polynomial.degree_prod_le /-
 theorem degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree := by
   simpa only [Multiset.map_map] using degree_multiset_prod_le (s.1.map f)
 #align polynomial.degree_prod_le Polynomial.degree_prod_le
+-/
 
+/- warning: polynomial.leading_coeff_multiset_prod' -> Polynomial.leadingCoeff_multiset_prod' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Ne.{succ u1} R (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) -> (Eq.{succ u1} R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'ₓ'. -/
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -157,6 +205,12 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).Prod ≠ 0) :
       apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
+/- warning: polynomial.leading_coeff_prod' -> Polynomial.leadingCoeff_prod' is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.leading_coeff_prod' Polynomial.leadingCoeff_prod'ₓ'. -/
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients, provided that this product is nonzero.
 
@@ -168,6 +222,12 @@ theorem leadingCoeff_prod' (h : (∏ i in s, (f i).leadingCoeff) ≠ 0) :
   simpa using leading_coeff_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.leading_coeff_prod' Polynomial.leadingCoeff_prod'
 
+/- warning: polynomial.nat_degree_multiset_prod' -> Polynomial.natDegree_multiset_prod' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Ne.{succ u1} R (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t)) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} Nat Nat.addCommMonoid (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) Nat (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_multiset_prod' Polynomial.natDegree_multiset_prod'ₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -186,6 +246,12 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).Prod ≠ 0
     apply right_ne_zero_of_mul ht
 #align polynomial.nat_degree_multiset_prod' Polynomial.natDegree_multiset_prod'
 
+/- warning: polynomial.nat_degree_prod' -> Polynomial.natDegree_prod' is a dubious translation:
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+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Ne.{succ u1} R (Finset.prod.{u1, u2} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) s (fun (i : ι) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} Nat ι Nat.addCommMonoid s (fun (i : ι) => Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Ne.{succ u1} R (Finset.prod.{u1, u2} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) s (fun (i : ι) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} Nat ι Nat.addCommMonoid s (fun (i : ι) => Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_prod' Polynomial.natDegree_prod'ₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, provided that the product of leading coefficients is nonzero.
 
@@ -197,6 +263,7 @@ theorem natDegree_prod' (h : (∏ i in s, (f i).leadingCoeff) ≠ 0) :
   simpa using nat_degree_multiset_prod' (s.1.map f) (by simpa using h)
 #align polynomial.nat_degree_prod' Polynomial.natDegree_prod'
 
+#print Polynomial.natDegree_multiset_prod_of_monic /-
 theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
     t.Prod.natDegree = (t.map natDegree).Sum :=
   by
@@ -214,12 +281,21 @@ theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
     assumption
   · simp
 #align polynomial.nat_degree_multiset_prod_of_monic Polynomial.natDegree_multiset_prod_of_monic
+-/
 
+#print Polynomial.natDegree_prod_of_monic /-
 theorem natDegree_prod_of_monic (h : ∀ i ∈ s, (f i).Monic) :
     (∏ i in s, f i).natDegree = ∑ i in s, (f i).natDegree := by
   simpa using nat_degree_multiset_prod_of_monic (s.1.map f) (by simpa using h)
 #align polynomial.nat_degree_prod_of_monic Polynomial.natDegree_prod_of_monic
+-/
 
+/- warning: polynomial.coeff_multiset_prod_of_nat_degree_le -> Polynomial.coeff_multiset_prod_of_natDegree_le is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (n : Nat), (forall (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)), (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instMembershipMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p t) -> (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) p) n)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t) (HMul.hMul.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} 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+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_multiset_prod_of_nat_degree_le Polynomial.coeff_multiset_prod_of_natDegree_leₓ'. -/
 theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegree p ≤ n) :
     coeff t.Prod (t.card * n) = (t.map fun p => coeff p n).Prod :=
   by
@@ -227,6 +303,7 @@ theorem coeff_multiset_prod_of_natDegree_le (n : ℕ) (hl : ∀ p ∈ t, natDegr
   simpa using coeff_list_prod_of_nat_degree_le _ _ hl
 #align polynomial.coeff_multiset_prod_of_nat_degree_le Polynomial.coeff_multiset_prod_of_natDegree_le
 
+#print Polynomial.coeff_prod_of_natDegree_le /-
 theorem coeff_prod_of_natDegree_le (f : ι → R[X]) (n : ℕ) (h : ∀ p ∈ s, natDegree (f p) ≤ n) :
     coeff (∏ i in s, f i) (s.card * n) = ∏ i in s, coeff (f i) n :=
   by
@@ -236,16 +313,21 @@ theorem coeff_prod_of_natDegree_le (f : ι → R[X]) (n : ℕ) (h : ∀ p ∈ s,
   · simp
   · simpa using h
 #align polynomial.coeff_prod_of_nat_degree_le Polynomial.coeff_prod_of_natDegree_le
+-/
 
+#print Polynomial.coeff_zero_multiset_prod /-
 theorem coeff_zero_multiset_prod : t.Prod.coeff 0 = (t.map fun f => coeff f 0).Prod :=
   by
   refine' Multiset.induction_on t _ fun a t ht => _; · simp
   rw [Multiset.prod_cons, Multiset.map_cons, Multiset.prod_cons, Polynomial.mul_coeff_zero, ht]
 #align polynomial.coeff_zero_multiset_prod Polynomial.coeff_zero_multiset_prod
+-/
 
+#print Polynomial.coeff_zero_prod /-
 theorem coeff_zero_prod : (∏ i in s, f i).coeff 0 = ∏ i in s, (f i).coeff 0 := by
   simpa using coeff_zero_multiset_prod (s.1.map f)
 #align polynomial.coeff_zero_prod Polynomial.coeff_zero_prod
+-/
 
 end CommSemiring
 
@@ -255,9 +337,15 @@ variable [CommRing R]
 
 open Monic
 
+/- warning: polynomial.multiset_prod_X_sub_C_next_coeff -> Polynomial.multiset_prod_X_sub_C_nextCoeff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), Eq.{succ u1} R (Polynomial.nextCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R 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(Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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_inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t))
+Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeffₓ'. -/
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
-theorem multiset_prod_x_sub_c_nextCoeff (t : Multiset R) :
+theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
     nextCoeff (t.map fun x => X - C x).Prod = -t.Sum :=
   by
   rw [next_coeff_multiset_prod]
@@ -265,14 +353,26 @@ theorem multiset_prod_x_sub_c_nextCoeff (t : Multiset R) :
     exact t.sum_hom (-AddMonoidHom.id R)
   · intros
     apply monic_X_sub_C
-#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_x_sub_c_nextCoeff
-
+#align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
+
+/- warning: polynomial.prod_X_sub_C_next_coeff -> Polynomial.prod_x_sub_c_nextCoeff is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeffₓ'. -/
 theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeff
 
-theorem multiset_prod_x_sub_c_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
+/- warning: polynomial.multiset_prod_X_sub_C_coeff_card_pred -> Polynomial.multiset_prod_X_sub_C_coeff_card_pred is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (coeFn.{succ u1, succ u1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (fun (_x : AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) => (Multiset.{u1} R) -> Nat) (AddMonoidHom.hasCoeToFun.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.orderedCancelAddCommMonoid.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Multiset.sum.{u1} R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (t : Multiset.{u1} R), (LT.lt.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) instLTNat (OfNat.ofNat.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) 0 (instOfNatNat 0)) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) (Multiset.map.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x : R) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x)) t)) (HSub.hSub.{0, 0, 0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) Nat ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) (instHSub.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) t) instSubNat) (FunLike.coe.{succ u1, succ u1, 1} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) (fun (_x : Multiset.{u1} R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.398 : Multiset.{u1} R) => Nat) _x) (AddHomClass.toFunLike.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddZeroClass.toAdd.{u1} (Multiset.{u1} R) (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R))))))) (AddZeroClass.toAdd.{0} Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (AddMonoidHomClass.toAddHomClass.{u1, u1, 0} (AddMonoidHom.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)) (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid) (AddMonoidHom.addMonoidHomClass.{u1, 0} (Multiset.{u1} R) Nat (AddMonoid.toAddZeroClass.{u1} (Multiset.{u1} R) (AddRightCancelMonoid.toAddMonoid.{u1} (Multiset.{u1} R) (AddCancelMonoid.toAddRightCancelMonoid.{u1} (Multiset.{u1} R) (AddCancelCommMonoid.toAddCancelMonoid.{u1} (Multiset.{u1} R) (OrderedCancelAddCommMonoid.toCancelAddCommMonoid.{u1} (Multiset.{u1} R) (Multiset.instOrderedCancelAddCommMonoidMultiset.{u1} R)))))) (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid)))) (Multiset.card.{u1} R) t) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.sum.{u1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) t)))
+Case conversion may be inaccurate. Consider using '#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_predₓ'. -/
+theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
     (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
   by
   nontriviality R
@@ -287,12 +387,18 @@ theorem multiset_prod_x_sub_c_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
     exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, nat_degree_X_sub_C _⟩, one_ne_zero⟩
   congr ; rw [nat_degree_multiset_prod_of_monic] <;> · simp [nat_degree_X_sub_C, monic_X_sub_C]
-#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_x_sub_c_coeff_card_pred
-
-theorem prod_x_sub_c_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
+#align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
+
+/- warning: polynomial.prod_X_sub_C_coeff_card_pred -> Polynomial.prod_X_sub_C_coeff_card_pred is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Finset.sum.{u1, u2} R ι (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} [_inst_1 : CommRing.{u1} R] (s : Finset.{u2} ι) (f : ι -> R), (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Finset.card.{u2} ι s)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.prod.{u1, u2} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ι (CommRing.toCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.commRing.{u1} R _inst_1)) s (fun (i : ι) => HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.sub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.X.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (f i)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} ι s) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Neg.neg.{u1} R (Ring.toNeg.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Finset.sum.{u1, u2} R ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) s (fun (i : ι) => f i))))
+Case conversion may be inaccurate. Consider using '#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_predₓ'. -/
+theorem prod_X_sub_C_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
     (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)
-#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_x_sub_c_coeff_card_pred
+#align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_X_sub_C_coeff_card_pred
 
 end CommRing
 
@@ -302,6 +408,12 @@ section Semiring
 
 variable [Semiring R] [NoZeroDivisors R]
 
+/- warning: polynomial.degree_list_prod -> Polynomial.degree_list_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))] [_inst_3 : Nontrivial.{u1} R] (l : List.{u1} (Polynomial.{u1} R _inst_1)), Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R _inst_1 (List.prod.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1) (Polynomial.hasOne.{u1} R _inst_1) l)) (List.sum.{0} (WithBot.{0} Nat) (WithBot.hasAdd.{0} Nat Nat.hasAdd) (WithBot.hasZero.{0} Nat Nat.hasZero) (List.map.{u1, 0} (Polynomial.{u1} R _inst_1) (WithBot.{0} Nat) (Polynomial.degree.{u1} R _inst_1) l))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))] [_inst_3 : Nontrivial.{u1} R] (l : List.{u1} (Polynomial.{u1} R _inst_1)), Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R _inst_1 (List.prod.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1) (Polynomial.one.{u1} R _inst_1) l)) (List.sum.{0} (WithBot.{0} Nat) (WithBot.add.{0} Nat instAddNat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (List.map.{u1, 0} (Polynomial.{u1} R _inst_1) (WithBot.{0} Nat) (Polynomial.degree.{u1} R _inst_1) l))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_list_prod Polynomial.degree_list_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 `[nontrivial R]` is needed, otherwise for `l = []` we have `⊥` in the LHS and `0` in the RHS.
@@ -316,6 +428,12 @@ section CommSemiring
 
 variable [CommSemiring R] [NoZeroDivisors R] (f : ι → R[X]) (t : Multiset R[X])
 
+/- warning: polynomial.nat_degree_prod -> Polynomial.natDegree_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (forall (i : ι), (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) -> (Ne.{succ u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (f i) (OfNat.ofNat.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (OfNat.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (Zero.zero.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.zero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} Nat ι Nat.addCommMonoid s (fun (i : ι) => Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (forall (i : ι), (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i s) -> (Ne.{succ u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (f i) (OfNat.ofNat.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.zero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} Nat ι Nat.addCommMonoid s (fun (i : ι) => Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i))))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_prod Polynomial.natDegree_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees.
 
@@ -331,6 +449,12 @@ theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
   intro x hx; simp [h x hx]
 #align polynomial.nat_degree_prod Polynomial.natDegree_prod
 
+/- warning: polynomial.nat_degree_multiset_prod -> Polynomial.natDegree_multiset_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Not (Membership.Mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.hasMem.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (OfNat.mk.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (Zero.zero.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.zero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) t)) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} Nat Nat.addCommMonoid (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), (Not (Membership.mem.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Multiset.instMembershipMultiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (OfNat.ofNat.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.zero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) t)) -> (Eq.{1} Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} Nat Nat.addCommMonoid (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) Nat (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) t)))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_multiset_prod Polynomial.natDegree_multiset_prodₓ'. -/
 theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.map natDegree).Sum :=
   by
   nontriviality R
@@ -340,6 +464,12 @@ theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.Prod = (t.m
   contradiction
 #align polynomial.nat_degree_multiset_prod Polynomial.natDegree_multiset_prod
 
+/- warning: polynomial.degree_multiset_prod -> Polynomial.degree_multiset_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) [_inst_3 : Nontrivial.{u1} R], Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} (WithBot.{0} Nat) (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (WithBot.{0} Nat) (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) [_inst_3 : Nontrivial.{u1} R], Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.sum.{0} (WithBot.{0} Nat) (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) (Multiset.map.{u1, 0} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (WithBot.{0} Nat) (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_multiset_prod Polynomial.degree_multiset_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
@@ -347,6 +477,12 @@ theorem degree_multiset_prod [Nontrivial R] : t.Prod.degree = (t.map fun f => de
   map_multiset_prod (@degreeMonoidHom R _ _ _) _
 #align polynomial.degree_multiset_prod Polynomial.degree_multiset_prod
 
+/- warning: polynomial.degree_prod -> Polynomial.degree_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) [_inst_3 : Nontrivial.{u1} R], Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} (WithBot.{0} Nat) ι (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) s (fun (i : ι) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) [_inst_3 : Nontrivial.{u1} R], Eq.{1} (WithBot.{0} Nat) (Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.sum.{0, u2} (WithBot.{0} Nat) ι (WithBot.addCommMonoid.{0} Nat Nat.addCommMonoid) s (fun (i : ι) => Polynomial.degree.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_prod Polynomial.degree_prodₓ'. -/
 /-- The degree of a product of polynomials is equal to
 the sum of the degrees, where the degree of the zero polynomial is ⊥.
 -/
@@ -354,6 +490,12 @@ theorem degree_prod [Nontrivial R] : (∏ i in s, f i).degree = ∑ i in s, (f i
   map_prod (@degreeMonoidHom R _ _ _) _ _
 #align polynomial.degree_prod Polynomial.degree_prod
 
+/- warning: polynomial.leading_coeff_multiset_prod -> Polynomial.leadingCoeff_multiset_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), Eq.{succ u1} R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (t : Multiset.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), Eq.{succ u1} R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Multiset.prod.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) t)) (Multiset.prod.{u1} R (CommSemiring.toCommMonoid.{u1} R _inst_1) (Multiset.map.{u1, u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (fun (f : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) f) t))
+Case conversion may be inaccurate. Consider using '#align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prodₓ'. -/
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 
@@ -366,6 +508,12 @@ theorem leadingCoeff_multiset_prod : t.Prod.leadingCoeff = (t.map fun f => leadi
   rfl
 #align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prod
 
+/- warning: polynomial.leading_coeff_prod -> Polynomial.leadingCoeff_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), Eq.{succ u1} R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.prod.{u1, u2} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) s (fun (i : ι) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+but is expected to have type
+  forall {R : Type.{u1}} {ι : Type.{u2}} (s : Finset.{u2} ι) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] (f : ι -> (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))), Eq.{succ u1} R (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (Finset.prod.{u1, u2} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) ι (CommSemiring.toCommMonoid.{u1} (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Polynomial.commSemiring.{u1} R _inst_1)) s (fun (i : ι) => f i))) (Finset.prod.{u1, u2} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) s (fun (i : ι) => Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) (f i)))
+Case conversion may be inaccurate. Consider using '#align polynomial.leading_coeff_prod Polynomial.leadingCoeff_prodₓ'. -/
 /-- The leading coefficient of a product of polynomials is equal to
 the product of the leading coefficients.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -258,7 +258,7 @@ open Monic
 -- Eventually this can be generalized with Vieta's formulas
 -- plus the connection between roots and factorization.
 theorem multiset_prod_x_sub_c_nextCoeff (t : Multiset R) :
-    nextCoeff (t.map fun x => x - c x).Prod = -t.Sum :=
+    nextCoeff (t.map fun x => X - C x).Prod = -t.Sum :=
   by
   rw [next_coeff_multiset_prod]
   · simp only [next_coeff_X_sub_C]
@@ -268,12 +268,12 @@ theorem multiset_prod_x_sub_c_nextCoeff (t : Multiset R) :
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_x_sub_c_nextCoeff
 
 theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
-    nextCoeff (∏ i in s, x - c (f i)) = -∑ i in s, f i := by
+    nextCoeff (∏ i in s, X - C (f i)) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f)
 #align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeff
 
 theorem multiset_prod_x_sub_c_coeff_card_pred (t : Multiset R) (ht : 0 < t.card) :
-    (t.map fun x => x - c x).Prod.coeff (t.card - 1) = -t.Sum :=
+    (t.map fun x => X - C x).Prod.coeff (t.card - 1) = -t.Sum :=
   by
   nontriviality R
   convert multiset_prod_X_sub_C_next_coeff (by assumption)
@@ -290,7 +290,7 @@ theorem multiset_prod_x_sub_c_coeff_card_pred (t : Multiset R) (ht : 0 < t.card)
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_x_sub_c_coeff_card_pred
 
 theorem prod_x_sub_c_coeff_card_pred (s : Finset ι) (f : ι → R) (hs : 0 < s.card) :
-    (∏ i in s, x - c (f i)).coeff (s.card - 1) = -∑ i in s, f i := by
+    (∏ i in s, X - C (f i)).coeff (s.card - 1) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs)
 #align polynomial.prod_X_sub_C_coeff_card_pred Polynomial.prod_x_sub_c_coeff_card_pred
 

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

A PR analogous to #12338 and #12361: reformatting proofs following the multiple goals linter of #12339.

@@ -150,7 +150,7 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod ≠ 0) :
     simp only [ne_eq]
     apply right_ne_zero_of_mul h
   · rw [ih]
-    exact h
+    · exact h
     simp only [ne_eq, not_false_eq_true]
     apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'

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

@@ -3,7 +3,7 @@ Copyright (c) 2020 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 -/
-import Mathlib.Data.Polynomial.Monic
+import Mathlib.Algebra.Polynomial.Monic
 
 #align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
 

@@ -329,7 +329,7 @@ theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) :
     natDegree t.prod = (t.map natDegree).sum := by
   nontriviality R
   rw [natDegree_multiset_prod']
-  simp_rw [Ne.def, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]
+  simp_rw [Ne, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]
   rintro ⟨_, h, rfl⟩
   contradiction
 #align polynomial.nat_degree_multiset_prod Polynomial.natDegree_multiset_prod

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

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

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

Before

After

@@ -3,7 +3,6 @@ Copyright (c) 2020 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 -/
-import Mathlib.Algebra.Order.WithZero
 import Mathlib.Data.Polynomial.Monic
 
 #align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"

These will be caught by the linter in a future lean version.

@@ -270,16 +270,16 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multise
     (t.map fun x => X - C x).prod.coeff ((Multiset.card t) - 1) = -t.sum := by
   nontriviality R
   convert multiset_prod_X_sub_C_nextCoeff (by assumption)
-  rw [nextCoeff]; split_ifs with h
-  · rw [natDegree_multiset_prod_of_monic] at h
+  rw [nextCoeff, if_neg]
+  swap
+  · rw [natDegree_multiset_prod_of_monic]
     swap
     · simp only [Multiset.mem_map]
       rintro _ ⟨_, _, rfl⟩
       apply monic_X_sub_C
-    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h
-    contrapose! h
+    simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map]
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
-    exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, natDegree_X_sub_C _⟩, one_ne_zero⟩
+    exact fun h => one_ne_zero <| h 1 ⟨_, ⟨x, hx, rfl⟩, natDegree_X_sub_C _⟩
   congr; rw [natDegree_multiset_prod_of_monic] <;> · simp [natDegree_X_sub_C, monic_X_sub_C]
 set_option linter.uppercaseLean3 false in
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred

Also add two missing simp attributes

@@ -59,6 +59,10 @@ theorem natDegree_sum_le (f : ι → S[X]) :
   simpa using natDegree_multiset_sum_le (s.val.map f)
 #align polynomial.nat_degree_sum_le Polynomial.natDegree_sum_le
 
+lemma natDegree_sum_le_of_forall_le {n : ℕ} (f : ι → S[X]) (h : ∀ i ∈ s, natDegree (f i) ≤ n) :
+    natDegree (∑ i in s, f i) ≤ n :=
+  le_trans (natDegree_sum_le s f) <| (Finset.fold_max_le n).mpr <| by simpa
+
 theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).maximum := by
   by_cases h : l.sum = 0
   · simp [h]

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -99,7 +99,7 @@ theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p 
       simpa using hl'
     have hdn : natDegree hd ≤ n := hl _ (List.mem_cons_self _ _)
     rcases hdn.eq_or_lt with (rfl | hdn')
-    · cases' h.eq_or_lt with h' h'
+    · rcases h.eq_or_lt with h' | h'
       · rw [← h', coeff_mul_degree_add_degree, leadingCoeff, leadingCoeff]
       · rw [coeff_eq_zero_of_natDegree_lt, coeff_eq_zero_of_natDegree_lt h', mul_zero]
         exact natDegree_mul_le.trans_lt (add_lt_add_left h' _)

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

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

@@ -267,9 +267,10 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multise
   nontriviality R
   convert multiset_prod_X_sub_C_nextCoeff (by assumption)
   rw [nextCoeff]; split_ifs with h
-  · rw [natDegree_multiset_prod_of_monic] at h <;> simp only [Multiset.mem_map] at *
+  · rw [natDegree_multiset_prod_of_monic] at h
     swap
-    · rintro _ ⟨_, _, rfl⟩
+    · simp only [Multiset.mem_map]
+      rintro _ ⟨_, _, rfl⟩
       apply monic_X_sub_C
     simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h
     contrapose! h

Removes nonterminal simps on lines looking like simp [...]

@@ -142,10 +142,14 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod ≠ 0) :
     t.prod.leadingCoeff = (t.map leadingCoeff).prod := by
   induction' t using Multiset.induction_on with a t ih; · simp
   simp only [Multiset.map_cons, Multiset.prod_cons] at h ⊢
-  rw [Polynomial.leadingCoeff_mul'] <;>
-    · rw [ih]
-      simp [*]
-      apply right_ne_zero_of_mul h
+  rw [Polynomial.leadingCoeff_mul']
+  · rw [ih]
+    simp only [ne_eq]
+    apply right_ne_zero_of_mul h
+  · rw [ih]
+    exact h
+    simp only [ne_eq, not_false_eq_true]
+    apply right_ne_zero_of_mul h
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
 /-- The leading coefficient of a product of polynomials is equal to

Also WithBot.charZero

@@ -65,7 +65,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).
   · rw [degree_eq_natDegree h]
     suffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : ℕ) by
       rw [this]
-      simpa [this, Nat.cast_withBot] using natDegree_list_sum_le l
+      simpa using natDegree_list_sum_le l
     rw [← List.foldr_max_of_ne_nil]
     · congr
     contrapose! h

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

This has nice performance benefits.

@@ -43,7 +43,7 @@ variable (s : Finset ι)
 
 section Semiring
 
-variable {S : Type _} [Semiring S]
+variable {S : Type*} [Semiring S]
 
 theorem natDegree_list_sum_le (l : List S[X]) : natDegree l.sum ≤ (l.map natDegree).foldr max 0 :=
   List.sum_le_foldr_max natDegree (by simp) natDegree_add_le _

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
-
-! This file was ported from Lean 3 source module algebra.polynomial.big_operators
-! leanprover-community/mathlib commit 47adfab39a11a072db552f47594bf8ed2cf8a722
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Order.WithZero
 import Mathlib.Data.Polynomial.Monic
 
+#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
+
 /-!
 # Lemmas for the interaction between polynomials and `∑` and `∏`.
 

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.

@@ -274,7 +274,7 @@ theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multise
     contrapose! h
     obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht
     exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, natDegree_X_sub_C _⟩, one_ne_zero⟩
-  congr ; rw [natDegree_multiset_prod_of_monic] <;> · simp [natDegree_X_sub_C, monic_X_sub_C]
+  congr; rw [natDegree_multiset_prod_of_monic] <;> · simp [natDegree_X_sub_C, monic_X_sub_C]
 set_option linter.uppercaseLean3 false in
 #align polynomial.multiset_prod_X_sub_C_coeff_card_pred Polynomial.multiset_prod_X_sub_C_coeff_card_pred
 

Changes are of the form

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

@@ -172,7 +172,7 @@ theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).prod ≠ 0
     t.prod.natDegree = (t.map fun f => natDegree f).sum := by
   revert h
   refine' Multiset.induction_on t _ fun a t ih ht => _; · simp
-  rw [Multiset.map_cons, Multiset.prod_cons] at ht⊢
+  rw [Multiset.map_cons, Multiset.prod_cons] at ht ⊢
   rw [Multiset.sum_cons, Polynomial.natDegree_mul', ih]
   · apply right_ne_zero_of_mul ht
   · rwa [Polynomial.leadingCoeff_multiset_prod']

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -66,8 +66,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).
   by_cases h : l.sum = 0
   · simp [h]
   · rw [degree_eq_natDegree h]
-    suffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : ℕ)
-      by
+    suffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : ℕ) by
       rw [this]
       simpa [this, Nat.cast_withBot] using natDegree_list_sum_le l
     rw [← List.foldr_max_of_ne_nil]
@@ -96,8 +95,7 @@ theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p 
   · have hl' : ∀ p ∈ tl, natDegree p ≤ n := fun p hp => hl p (List.mem_cons_of_mem _ hp)
     simp only [List.prod_cons, List.map, List.length]
     rw [add_mul, one_mul, add_comm, ← IH hl', mul_comm tl.length]
-    have h : natDegree tl.prod ≤ n * tl.length :=
-      by
+    have h : natDegree tl.prod ≤ n * tl.length := by
       refine' (natDegree_list_prod_le _).trans _
       rw [← tl.length_map natDegree, mul_comm]
       refine' List.sum_le_card_nsmul _ _ _
@@ -196,8 +194,7 @@ theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
     t.prod.natDegree = (t.map natDegree).sum := by
   nontriviality R
   apply natDegree_multiset_prod'
-  suffices (t.map fun f => leadingCoeff f).prod = 1
-    by
+  suffices (t.map fun f => leadingCoeff f).prod = 1 by
     rw [this]
     simp
   convert prod_replicate (Multiset.card t) (1 : R)
@@ -323,8 +320,8 @@ theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
   intro x hx; simp [h x hx]
 #align polynomial.nat_degree_prod Polynomial.natDegree_prod
 
-theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) : natDegree t.prod = (t.map natDegree).sum :=
-  by
+theorem natDegree_multiset_prod (h : (0 : R[X]) ∉ t) :
+    natDegree t.prod = (t.map natDegree).sum := by
   nontriviality R
   rw [natDegree_multiset_prod']
   simp_rw [Ne.def, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]
@@ -352,8 +349,8 @@ the product of the leading coefficients.
 See `Polynomial.leadingCoeff_multiset_prod'` (with a `'`) for a version for commutative semirings,
 where additionally, the product of the leading coefficients must be nonzero.
 -/
-theorem leadingCoeff_multiset_prod : t.prod.leadingCoeff = (t.map fun f => leadingCoeff f).prod :=
-  by
+theorem leadingCoeff_multiset_prod :
+    t.prod.leadingCoeff = (t.map fun f => leadingCoeff f).prod := by
   rw [← leadingCoeffHom_apply, MonoidHom.map_multiset_prod]
   rfl
 #align polynomial.leading_coeff_multiset_prod Polynomial.leadingCoeff_multiset_prod

This PR fixes two things:

• Most align statements for definitions and theorems and instances that are separated by two newlines from the relevant declaration (s/\n\n#align/\n#align). This is often seen in the mathport output after ending calc blocks.
• All remaining more-than-one-line #align statements. (This was needed for a script I wrote for #3630.)

@@ -151,7 +151,6 @@ theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod ≠ 0) :
     · rw [ih]
       simp [*]
       apply right_ne_zero_of_mul h
-
 #align polynomial.leading_coeff_multiset_prod' Polynomial.leadingCoeff_multiset_prod'
 
 /-- The leading coefficient of a product of polynomials is equal to

@@ -259,11 +259,11 @@ theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
 set_option linter.uppercaseLean3 false in
 #align polynomial.multiset_prod_X_sub_C_next_coeff Polynomial.multiset_prod_X_sub_C_nextCoeff
 
-theorem prod_x_sub_c_nextCoeff {s : Finset ι} (f : ι → R) :
+theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
     nextCoeff (∏ i in s, (X - C (f i))) = -∑ i in s, f i := by
   simpa using multiset_prod_X_sub_C_nextCoeff (s.1.map f)
 set_option linter.uppercaseLean3 false in
-#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_x_sub_c_nextCoeff
+#align polynomial.prod_X_sub_C_next_coeff Polynomial.prod_X_sub_C_nextCoeff
 
 theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multiset.card t) :
     (t.map fun x => X - C x).prod.coeff ((Multiset.card t) - 1) = -t.sum := by