data.polynomial.degree.trailing_degree ⟷ Mathlib.Data.Polynomial.Degree.TrailingDegree

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

@@ -291,11 +291,11 @@ theorem natTrailingDegree_one : natTrailingDegree (1 : R[X]) = 0 :=
 #align polynomial.nat_trailing_degree_one Polynomial.natTrailingDegree_one
 -/
 
-#print Polynomial.natTrailingDegree_nat_cast /-
+#print Polynomial.natTrailingDegree_natCast /-
 @[simp]
-theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0 := by
+theorem natTrailingDegree_natCast (n : ℕ) : natTrailingDegree (n : R[X]) = 0 := by
   simp only [← C_eq_nat_cast, nat_trailing_degree_C]
-#align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
+#align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_natCast
 -/
 
 #print Polynomial.trailingDegree_C_mul_X_pow /-
@@ -557,11 +557,11 @@ theorem natTrailingDegree_neg (p : R[X]) : natTrailingDegree (-p) = natTrailingD
 #align polynomial.nat_trailing_degree_neg Polynomial.natTrailingDegree_neg
 -/
 
-#print Polynomial.natTrailingDegree_int_cast /-
+#print Polynomial.natTrailingDegree_intCast /-
 @[simp]
-theorem natTrailingDegree_int_cast (n : ℤ) : natTrailingDegree (n : R[X]) = 0 := by
+theorem natTrailingDegree_intCast (n : ℤ) : natTrailingDegree (n : R[X]) = 0 := by
   simp only [← C_eq_int_cast, nat_trailing_degree_C]
-#align polynomial.nat_trailing_degree_int_cast Polynomial.natTrailingDegree_int_cast
+#align polynomial.nat_trailing_degree_int_cast Polynomial.natTrailingDegree_intCast
 -/
 
 end Ring

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

@@ -80,10 +80,10 @@ def TrailingMonic (p : R[X]) :=
 #align polynomial.trailing_monic Polynomial.TrailingMonic
 -/
 
-#print Polynomial.TrailingMonic.definition /-
-theorem TrailingMonic.definition : TrailingMonic p ↔ trailingCoeff p = 1 :=
+#print Polynomial.TrailingMonic.def /-
+theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
   Iff.rfl
-#align polynomial.trailing_monic.def Polynomial.TrailingMonic.definition
+#align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
 -/
 
 #print Polynomial.TrailingMonic.decidable /-

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: Damiano Testa
 -/
 import Data.ENat.Basic
-import Data.Polynomial.Degree.Definitions
+import Algebra.Polynomial.Degree.Definitions
 
 #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
 

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

@@ -3,7 +3,7 @@ Copyright (c) 2020 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Data.Enat.Basic
+import Data.ENat.Basic
 import Data.Polynomial.Degree.Definitions
 
 #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
@@ -80,10 +80,10 @@ def TrailingMonic (p : R[X]) :=
 #align polynomial.trailing_monic Polynomial.TrailingMonic
 -/
 
-#print Polynomial.TrailingMonic.def /-
-theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
+#print Polynomial.TrailingMonic.definition /-
+theorem TrailingMonic.definition : TrailingMonic p ↔ trailingCoeff p = 1 :=
   Iff.rfl
-#align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
+#align polynomial.trailing_monic.def Polynomial.TrailingMonic.definition
 -/
 
 #print Polynomial.TrailingMonic.decidable /-
@@ -182,10 +182,10 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
 -/
 
-#print Polynomial.le_trailingDegree_of_ne_zero /-
-theorem le_trailingDegree_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
+#print Polynomial.trailingDegree_le_of_ne_zero /-
+theorem trailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
   show @LE.le ℕ∞ _ p.support.min n from min_le (mem_support_iff.2 h)
-#align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zero
+#align polynomial.le_trailing_degree_of_ne_zero Polynomial.trailingDegree_le_of_ne_zero
 -/
 
 #print Polynomial.natTrailingDegree_le_of_ne_zero /-
@@ -311,10 +311,10 @@ theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trail
 #align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
 -/
 
-#print Polynomial.coeff_eq_zero_of_trailingDegree_lt /-
-theorem coeff_eq_zero_of_trailingDegree_lt (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
-  Classical.not_not.1 (mt le_trailingDegree_of_ne_zero (not_le_of_gt h))
-#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_lt
+#print Polynomial.coeff_eq_zero_of_lt_trailingDegree /-
+theorem coeff_eq_zero_of_lt_trailingDegree (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
+  Classical.not_not.1 (mt trailingDegree_le_of_ne_zero (not_le_of_gt h))
+#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_lt_trailingDegree
 -/
 
 #print Polynomial.coeff_eq_zero_of_lt_natTrailingDegree /-
@@ -583,11 +583,11 @@ theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by rw [next_coe
 #align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
 -/
 
-#print Polynomial.nextCoeffUp_of_pos_natTrailingDegree /-
-theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingDegree) :
+#print Polynomial.nextCoeffUp_of_constantCoeff_eq_zero /-
+theorem nextCoeffUp_of_constantCoeff_eq_zero (p : R[X]) (hp : 0 < p.natTrailingDegree) :
     nextCoeffUp p = p.coeff (p.natTrailingDegree + 1) := by rw [next_coeff_up, if_neg];
   contrapose! hp; simpa
-#align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_pos_natTrailingDegree
+#align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_constantCoeff_eq_zero
 -/
 
 end Semiring
@@ -599,7 +599,7 @@ variable [Semiring R] {p q : R[X]} {ι : Type _}
 #print Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt /-
 theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
     (h : trailingDegree p < trailingDegree q) : coeff q (natTrailingDegree p) = 0 :=
-  coeff_eq_zero_of_trailingDegree_lt <| natTrailingDegree_le_trailingDegree.trans_lt h
+  coeff_eq_zero_of_lt_trailingDegree <| natTrailingDegree_le_trailingDegree.trans_lt h
 #align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
 -/
 

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

@@ -150,16 +150,16 @@ theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (
   by
   constructor
   · intro H; rwa [← trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
-    rw [trailing_degree_zero] at H ; exact Option.noConfusion H
+    rw [trailing_degree_zero] at H; exact Option.noConfusion H
   · intro H; rwa [trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
-    rw [nat_trailing_degree_zero] at H ; rw [H] at hn ; exact lt_irrefl _ hn
+    rw [nat_trailing_degree_zero] at H; rw [H] at hn; exact lt_irrefl _ hn
 #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
 -/
 
 #print Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some /-
 theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
     (h : trailingDegree p = n) : natTrailingDegree p = n :=
-  have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h  <;> exact Option.noConfusion h
+  have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h <;> exact Option.noConfusion h
   Option.some_inj.1 <|
     show (natTrailingDegree p : ℕ∞) = n by rwa [← trailing_degree_eq_nat_trailing_degree hp0]
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
@@ -218,7 +218,7 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p :=
   by
-  rw [trailing_degree_eq_nat_trailing_degree hp] at H 
+  rw [trailing_degree_eq_nat_trailing_degree hp] at H
   exact with_top.coe_le_coe.mp H
 #align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_le
 -/
@@ -229,7 +229,7 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
   by
   by_cases hp : p = 0; · rw [hp, nat_trailing_degree_zero]; exact zero_le _
   rwa [trailing_degree_eq_nat_trailing_degree hp, trailing_degree_eq_nat_trailing_degree hq,
-    WithTop.coe_le_coe] at hpq 
+    WithTop.coe_le_coe] at hpq
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
 -/
 
@@ -423,9 +423,9 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
   · rw [nat_trailing_degree_eq_support_min' fun h => hp (mul_X_pow_eq_zero h), Finset.le_min'_iff]
     intro y hy
     have key : n ≤ y := by
-      rw [mem_support_iff, coeff_mul_X_pow'] at hy 
+      rw [mem_support_iff, coeff_mul_X_pow'] at hy
       exact by_contra fun h => hy (if_neg h)
-    rw [mem_support_iff, coeff_mul_X_pow', if_pos key] at hy 
+    rw [mem_support_iff, coeff_mul_X_pow', if_pos key] at hy
     exact (le_tsub_iff_right key).mp (nat_trailing_degree_le_of_ne_zero hy)
 #align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_X_pow
 -/
@@ -434,7 +434,7 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
 theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree :=
   by
   refine' Finset.le_min fun n hn => _
-  rw [mem_support_iff, coeff_mul] at hn 
+  rw [mem_support_iff, coeff_mul] at hn
   obtain ⟨⟨i, j⟩, hij, hpq⟩ := exists_ne_zero_of_sum_ne_zero hn
   refine'
     (add_le_add (min_le (mem_support_iff.mpr (left_ne_zero_of_mul hpq)))
@@ -465,12 +465,12 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
     Finset.sum_eq_single (p.nat_trailing_degree, q.nat_trailing_degree) _ fun h =>
       (h (nat.mem_antidiagonal.mpr rfl)).elim
   rintro ⟨i, j⟩ h₁ h₂
-  rw [nat.mem_antidiagonal] at h₁ 
+  rw [nat.mem_antidiagonal] at h₁
   by_cases hi : i < p.nat_trailing_degree
   · rw [coeff_eq_zero_of_lt_nat_trailing_degree hi, MulZeroClass.zero_mul]
   by_cases hj : j < q.nat_trailing_degree
   · rw [coeff_eq_zero_of_lt_nat_trailing_degree hj, MulZeroClass.mul_zero]
-  rw [not_lt] at hi hj 
+  rw [not_lt] at hi hj
   refine' (h₂ (prod.ext_iff.mpr _).symm).elim
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
 #align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree

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

@@ -54,7 +54,7 @@ def trailingDegree (p : R[X]) : ℕ∞ :=
 
 #print Polynomial.trailingDegree_lt_wf /-
 theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q :=
-  InvImage.wf trailingDegree (WithTop.wellFounded_lt Nat.lt_wfRel)
+  InvImage.wf trailingDegree (WithTop.instWellFoundedLT Nat.lt_wfRel)
 #align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf
 -/
 

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

@@ -131,7 +131,7 @@ theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) :
     trailingDegree p = (natTrailingDegree p : ℕ∞) :=
   by
   let ⟨n, hn⟩ :=
-    not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt trailingDegree_eq_top.1 hp))
+    Classical.not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt trailingDegree_eq_top.1 hp))
   have hn : trailingDegree p = n := Classical.not_not.1 hn
   rw [nat_trailing_degree, hn] <;> rfl
 #align polynomial.trailing_degree_eq_nat_trailing_degree Polynomial.trailingDegree_eq_natTrailingDegree

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathbin.Data.Enat.Basic
-import Mathbin.Data.Polynomial.Degree.Definitions
+import Data.Enat.Basic
+import Data.Polynomial.Degree.Definitions
 
 #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
 

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

@@ -210,7 +210,7 @@ theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠
 #print Polynomial.trailingDegree_ne_of_natTrailingDegree_ne /-
 theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
     p.natTrailingDegree ≠ n → trailingDegree p ≠ n :=
-  mt fun h => by rw [nat_trailing_degree, h, Option.getD_coe]
+  mt fun h => by rw [nat_trailing_degree, h, Option.getD_some]
 #align polynomial.trailing_degree_ne_of_nat_trailing_degree_ne Polynomial.trailingDegree_ne_of_natTrailingDegree_ne
 -/
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
-
-! This file was ported from Lean 3 source module data.polynomial.degree.trailing_degree
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Enat.Basic
 import Mathbin.Data.Polynomial.Degree.Definitions
 
+#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
 /-!
 # Trailing degree of univariate polynomials
 

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

@@ -55,9 +55,11 @@ def trailingDegree (p : R[X]) : ℕ∞ :=
 #align polynomial.trailing_degree Polynomial.trailingDegree
 -/
 
+#print Polynomial.trailingDegree_lt_wf /-
 theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q :=
   InvImage.wf trailingDegree (WithTop.wellFounded_lt Nat.lt_wfRel)
 #align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf
+-/
 
 #print Polynomial.natTrailingDegree /-
 /-- `nat_trailing_degree p` forces `trailing_degree p` to `ℕ`, by defining
@@ -87,9 +89,11 @@ theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
 #align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
 -/
 
+#print Polynomial.TrailingMonic.decidable /-
 instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) := by
   unfold trailing_monic <;> infer_instance
 #align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidable
+-/
 
 #print Polynomial.TrailingMonic.trailingCoeff /-
 @[simp]
@@ -105,10 +109,12 @@ theorem trailingDegree_zero : trailingDegree (0 : R[X]) = ⊤ :=
 #align polynomial.trailing_degree_zero Polynomial.trailingDegree_zero
 -/
 
+#print Polynomial.trailingCoeff_zero /-
 @[simp]
 theorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 :=
   rfl
 #align polynomial.trailing_coeff_zero Polynomial.trailingCoeff_zero
+-/
 
 #print Polynomial.natTrailingDegree_zero /-
 @[simp]
@@ -162,6 +168,7 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
 -/
 
+#print Polynomial.natTrailingDegree_le_trailingDegree /-
 @[simp]
 theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p :=
   by
@@ -169,6 +176,7 @@ theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trail
   rw [trailing_degree_eq_nat_trailing_degree hp]
   exact le_rfl
 #align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegree
+-/
 
 #print Polynomial.natTrailingDegree_eq_of_trailingDegree_eq /-
 theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
@@ -177,17 +185,22 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
 -/
 
+#print Polynomial.le_trailingDegree_of_ne_zero /-
 theorem le_trailingDegree_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
   show @LE.le ℕ∞ _ p.support.min n from min_le (mem_support_iff.2 h)
 #align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zero
+-/
 
+#print Polynomial.natTrailingDegree_le_of_ne_zero /-
 theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n :=
   by
   rw [← WithTop.coe_le_coe, ← trailing_degree_eq_nat_trailing_degree]
   · exact le_trailing_degree_of_ne_zero h
   · intro h; subst h; exact h rfl
 #align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero
+-/
 
+#print Polynomial.trailingDegree_le_trailingDegree /-
 theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
     trailingDegree q ≤ trailingDegree p :=
   by
@@ -195,6 +208,7 @@ theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠
   · rw [hp]; exact le_top
   · rw [trailing_degree_eq_nat_trailing_degree hp]; exact le_trailing_degree_of_ne_zero h
 #align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegree
+-/
 
 #print Polynomial.trailingDegree_ne_of_natTrailingDegree_ne /-
 theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
@@ -203,13 +217,16 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
 #align polynomial.trailing_degree_ne_of_nat_trailing_degree_ne Polynomial.trailingDegree_ne_of_natTrailingDegree_ne
 -/
 
+#print Polynomial.natTrailingDegree_le_of_trailingDegree_le /-
 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p :=
   by
   rw [trailing_degree_eq_nat_trailing_degree hp] at H 
   exact with_top.coe_le_coe.mp H
 #align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_le
+-/
 
+#print Polynomial.natTrailingDegree_le_natTrailingDegree /-
 theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     (hpq : p.trailingDegree ≤ q.trailingDegree) : p.natTrailingDegree ≤ q.natTrailingDegree :=
   by
@@ -217,41 +234,58 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
   rwa [trailing_degree_eq_nat_trailing_degree hp, trailing_degree_eq_nat_trailing_degree hq,
     WithTop.coe_le_coe] at hpq 
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
+-/
 
+#print Polynomial.trailingDegree_monomial /-
 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
   rw [trailing_degree, support_monomial n ha, min_singleton]
 #align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomial
+-/
 
+#print Polynomial.natTrailingDegree_monomial /-
 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
 #align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomial
+-/
 
+#print Polynomial.natTrailingDegree_monomial_le /-
 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
 #align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_le
+-/
 
+#print Polynomial.le_trailingDegree_monomial /-
 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge
 #align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomial
+-/
 
+#print Polynomial.trailingDegree_C /-
 @[simp]
 theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
   trailingDegree_monomial ha
 #align polynomial.trailing_degree_C Polynomial.trailingDegree_C
+-/
 
+#print Polynomial.le_trailingDegree_C /-
 theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_C
+-/
 
+#print Polynomial.trailingDegree_one_le /-
 theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
   rw [← C_1] <;> exact le_trailing_degree_C
 #align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_le
+-/
 
+#print Polynomial.natTrailingDegree_C /-
 @[simp]
 theorem natTrailingDegree_C (a : R) : natTrailingDegree (C a) = 0 :=
   nonpos_iff_eq_zero.1 natTrailingDegree_monomial_le
 #align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_C
+-/
 
 #print Polynomial.natTrailingDegree_one /-
 @[simp]
@@ -267,19 +301,26 @@ theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0
 #align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
 -/
 
+#print Polynomial.trailingDegree_C_mul_X_pow /-
 @[simp]
 theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
   rw [C_mul_X_pow_eq_monomial, trailing_degree_monomial ha]
 #align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_pow
+-/
 
+#print Polynomial.le_trailingDegree_C_mul_X_pow /-
 theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by rw [C_mul_X_pow_eq_monomial]; exact le_trailing_degree_monomial
 #align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
+-/
 
+#print Polynomial.coeff_eq_zero_of_trailingDegree_lt /-
 theorem coeff_eq_zero_of_trailingDegree_lt (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
   Classical.not_not.1 (mt le_trailingDegree_of_ne_zero (not_le_of_gt h))
 #align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_lt
+-/
 
+#print Polynomial.coeff_eq_zero_of_lt_natTrailingDegree /-
 theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
     p.coeff n = 0 := by
   apply coeff_eq_zero_of_trailing_degree_lt
@@ -287,21 +328,28 @@ theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.na
   · rw [hp, trailing_degree_zero]; exact WithTop.coe_lt_top n
   · rwa [trailing_degree_eq_nat_trailing_degree hp, WithTop.coe_lt_coe]
 #align polynomial.coeff_eq_zero_of_lt_nat_trailing_degree Polynomial.coeff_eq_zero_of_lt_natTrailingDegree
+-/
 
+#print Polynomial.coeff_natTrailingDegree_pred_eq_zero /-
 @[simp]
 theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < natTrailingDegree p} :
     p.coeff (p.natTrailingDegree - 1) = 0 :=
   coeff_eq_zero_of_lt_natTrailingDegree <|
     Nat.sub_lt ((WithTop.zero_lt_coe (natTrailingDegree p)).mp hp) Nat.one_pos
 #align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zero
+-/
 
+#print Polynomial.le_trailingDegree_X_pow /-
 theorem le_trailingDegree_X_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by
   simpa only [C_1, one_mul] using le_trailing_degree_C_mul_X_pow n (1 : R)
 #align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_X_pow
+-/
 
+#print Polynomial.le_trailingDegree_X /-
 theorem le_trailingDegree_X : (1 : ℕ∞) ≤ trailingDegree (X : R[X]) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_X
+-/
 
 #print Polynomial.natTrailingDegree_X_le /-
 theorem natTrailingDegree_X_le : (X : R[X]).natTrailingDegree ≤ 1 :=
@@ -309,6 +357,7 @@ theorem natTrailingDegree_X_le : (X : R[X]).natTrailingDegree ≤ 1 :=
 #align polynomial.nat_trailing_degree_X_le Polynomial.natTrailingDegree_X_le
 -/
 
+#print Polynomial.trailingCoeff_eq_zero /-
 @[simp]
 theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
   ⟨fun h =>
@@ -317,10 +366,13 @@ theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
         (mem_of_min (trailingDegree_eq_natTrailingDegree hp)),
     fun h => h.symm ▸ leadingCoeff_zero⟩
 #align polynomial.trailing_coeff_eq_zero Polynomial.trailingCoeff_eq_zero
+-/
 
+#print Polynomial.trailingCoeff_nonzero_iff_nonzero /-
 theorem trailingCoeff_nonzero_iff_nonzero : trailingCoeff p ≠ 0 ↔ p ≠ 0 :=
   not_congr trailingCoeff_eq_zero
 #align polynomial.trailing_coeff_nonzero_iff_nonzero Polynomial.trailingCoeff_nonzero_iff_nonzero
+-/
 
 #print Polynomial.natTrailingDegree_mem_support_of_nonzero /-
 theorem natTrailingDegree_mem_support_of_nonzero : p ≠ 0 → natTrailingDegree p ∈ p.support :=
@@ -347,11 +399,13 @@ theorem natTrailingDegree_eq_support_min' (h : p ≠ 0) :
 #align polynomial.nat_trailing_degree_eq_support_min' Polynomial.natTrailingDegree_eq_support_min'
 -/
 
+#print Polynomial.le_natTrailingDegree /-
 theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) : n ≤ p.natTrailingDegree :=
   by
   rw [nat_trailing_degree_eq_support_min' hp]
   exact Finset.le_min' _ _ _ fun m hm => not_lt.1 fun hmn => mem_support_iff.1 hm <| hn _ hmn
 #align polynomial.le_nat_trailing_degree Polynomial.le_natTrailingDegree
+-/
 
 #print Polynomial.natTrailingDegree_le_natDegree /-
 theorem natTrailingDegree_le_natDegree (p : R[X]) : p.natTrailingDegree ≤ p.natDegree :=
@@ -379,6 +433,7 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
 #align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_X_pow
 -/
 
+#print Polynomial.le_trailingDegree_mul /-
 theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree :=
   by
   refine' Finset.le_min fun n hn => _
@@ -390,6 +445,7 @@ theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).
       (le_of_eq _)
   rwa [← WithTop.coe_add, WithTop.coe_eq_coe, ← nat.mem_antidiagonal]
 #align polynomial.le_trailing_degree_mul Polynomial.le_trailingDegree_mul
+-/
 
 #print Polynomial.le_natTrailingDegree_mul /-
 theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
@@ -403,6 +459,7 @@ theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
 #align polynomial.le_nat_trailing_degree_mul Polynomial.le_natTrailingDegree_mul
 -/
 
+#print Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree /-
 theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
     (p * q).coeff (p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff :=
   by
@@ -420,7 +477,9 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
   refine' (h₂ (prod.ext_iff.mpr _).symm).elim
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
 #align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree
+-/
 
+#print Polynomial.trailingDegree_mul' /-
 theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).trailingDegree = p.trailingDegree + q.trailingDegree :=
   by
@@ -432,7 +491,9 @@ theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
   apply le_trailing_degree_of_ne_zero
   rwa [coeff_mul_nat_trailing_degree_add_nat_trailing_degree]
 #align polynomial.trailing_degree_mul' Polynomial.trailingDegree_mul'
+-/
 
+#print Polynomial.natTrailingDegree_mul' /-
 theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   by
@@ -442,12 +503,15 @@ theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
   rw [trailing_degree_mul' h, WithTop.coe_add, ← trailing_degree_eq_nat_trailing_degree hp, ←
     trailing_degree_eq_nat_trailing_degree hq]
 #align polynomial.nat_trailing_degree_mul' Polynomial.natTrailingDegree_mul'
+-/
 
+#print Polynomial.natTrailingDegree_mul /-
 theorem natTrailingDegree_mul [NoZeroDivisors R] (hp : p ≠ 0) (hq : q ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   natTrailingDegree_mul'
     (mul_ne_zero (mt trailingCoeff_eq_zero.mp hp) (mt trailingCoeff_eq_zero.mp hq))
 #align polynomial.nat_trailing_degree_mul Polynomial.natTrailingDegree_mul
+-/
 
 end Semiring
 
@@ -516,9 +580,11 @@ def nextCoeffUp (p : R[X]) : R :=
 #align polynomial.next_coeff_up Polynomial.nextCoeffUp
 -/
 
+#print Polynomial.nextCoeffUp_C_eq_zero /-
 @[simp]
 theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by rw [next_coeff_up]; simp
 #align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
+-/
 
 #print Polynomial.nextCoeffUp_of_pos_natTrailingDegree /-
 theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingDegree) :
@@ -533,14 +599,18 @@ section Semiring
 
 variable [Semiring R] {p q : R[X]} {ι : Type _}
 
+#print Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt /-
 theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
     (h : trailingDegree p < trailingDegree q) : coeff q (natTrailingDegree p) = 0 :=
   coeff_eq_zero_of_trailingDegree_lt <| natTrailingDegree_le_trailingDegree.trans_lt h
 #align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
+-/
 
+#print Polynomial.ne_zero_of_trailingDegree_lt /-
 theorem ne_zero_of_trailingDegree_lt {n : ℕ∞} (h : trailingDegree p < n) : p ≠ 0 := fun h₀ =>
   h.not_le (by simp [h₀])
 #align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_lt
+-/
 
 end Semiring
 

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

@@ -147,16 +147,16 @@ theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (
   by
   constructor
   · intro H; rwa [← trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
-    rw [trailing_degree_zero] at H; exact Option.noConfusion H
+    rw [trailing_degree_zero] at H ; exact Option.noConfusion H
   · intro H; rwa [trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
-    rw [nat_trailing_degree_zero] at H; rw [H] at hn; exact lt_irrefl _ hn
+    rw [nat_trailing_degree_zero] at H ; rw [H] at hn ; exact lt_irrefl _ hn
 #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
 -/
 
 #print Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some /-
 theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
     (h : trailingDegree p = n) : natTrailingDegree p = n :=
-  have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h <;> exact Option.noConfusion h
+  have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h  <;> exact Option.noConfusion h
   Option.some_inj.1 <|
     show (natTrailingDegree p : ℕ∞) = n by rwa [← trailing_degree_eq_nat_trailing_degree hp0]
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
@@ -206,7 +206,7 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p :=
   by
-  rw [trailing_degree_eq_nat_trailing_degree hp] at H
+  rw [trailing_degree_eq_nat_trailing_degree hp] at H 
   exact with_top.coe_le_coe.mp H
 #align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_le
 
@@ -215,7 +215,7 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
   by
   by_cases hp : p = 0; · rw [hp, nat_trailing_degree_zero]; exact zero_le _
   rwa [trailing_degree_eq_nat_trailing_degree hp, trailing_degree_eq_nat_trailing_degree hq,
-    WithTop.coe_le_coe] at hpq
+    WithTop.coe_le_coe] at hpq 
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
 
 @[simp]
@@ -372,9 +372,9 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
   · rw [nat_trailing_degree_eq_support_min' fun h => hp (mul_X_pow_eq_zero h), Finset.le_min'_iff]
     intro y hy
     have key : n ≤ y := by
-      rw [mem_support_iff, coeff_mul_X_pow'] at hy
+      rw [mem_support_iff, coeff_mul_X_pow'] at hy 
       exact by_contra fun h => hy (if_neg h)
-    rw [mem_support_iff, coeff_mul_X_pow', if_pos key] at hy
+    rw [mem_support_iff, coeff_mul_X_pow', if_pos key] at hy 
     exact (le_tsub_iff_right key).mp (nat_trailing_degree_le_of_ne_zero hy)
 #align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_X_pow
 -/
@@ -382,7 +382,7 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
 theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree :=
   by
   refine' Finset.le_min fun n hn => _
-  rw [mem_support_iff, coeff_mul] at hn
+  rw [mem_support_iff, coeff_mul] at hn 
   obtain ⟨⟨i, j⟩, hij, hpq⟩ := exists_ne_zero_of_sum_ne_zero hn
   refine'
     (add_le_add (min_le (mem_support_iff.mpr (left_ne_zero_of_mul hpq)))
@@ -411,12 +411,12 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
     Finset.sum_eq_single (p.nat_trailing_degree, q.nat_trailing_degree) _ fun h =>
       (h (nat.mem_antidiagonal.mpr rfl)).elim
   rintro ⟨i, j⟩ h₁ h₂
-  rw [nat.mem_antidiagonal] at h₁
+  rw [nat.mem_antidiagonal] at h₁ 
   by_cases hi : i < p.nat_trailing_degree
   · rw [coeff_eq_zero_of_lt_nat_trailing_degree hi, MulZeroClass.zero_mul]
   by_cases hj : j < q.nat_trailing_degree
   · rw [coeff_eq_zero_of_lt_nat_trailing_degree hj, MulZeroClass.mul_zero]
-  rw [not_lt] at hi hj
+  rw [not_lt] at hi hj 
   refine' (h₂ (prod.ext_iff.mpr _).symm).elim
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
 #align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree

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

@@ -32,7 +32,7 @@ noncomputable section
 
 open Function Polynomial Finsupp Finset
 
-open BigOperators Classical Polynomial
+open scoped BigOperators Classical Polynomial
 
 namespace Polynomial
 

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

@@ -55,12 +55,6 @@ def trailingDegree (p : R[X]) : ℕ∞ :=
 #align polynomial.trailing_degree Polynomial.trailingDegree
 -/
 
-/- warning: polynomial.trailing_degree_lt_wf -> Polynomial.trailingDegree_lt_wf is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
-Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wfₓ'. -/
 theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q :=
   InvImage.wf trailingDegree (WithTop.wellFounded_lt Nat.lt_wfRel)
 #align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf
@@ -93,12 +87,6 @@ theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
 #align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
 -/
 
-/- warning: polynomial.trailing_monic.decidable -> Polynomial.TrailingMonic.decidable is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} [_inst_2 : DecidableEq.{succ u1} R], Decidable (Polynomial.TrailingMonic.{u1} R _inst_1 p)
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Decidable (Polynomial.TrailingMonic.{u1} R _inst_1 p)
-Case conversion may be inaccurate. Consider using '#align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidableₓ'. -/
 instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) := by
   unfold trailing_monic <;> infer_instance
 #align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidable
@@ -117,12 +105,6 @@ theorem trailingDegree_zero : trailingDegree (0 : R[X]) = ⊤ :=
 #align polynomial.trailing_degree_zero Polynomial.trailingDegree_zero
 -/
 
-/- warning: polynomial.trailing_coeff_zero -> Polynomial.trailingCoeff_zero is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) (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 _inst_1)))))))
-but is expected to have type
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 @[simp]
 theorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 :=
   rfl
@@ -180,12 +162,6 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
 -/
 
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 @[simp]
 theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p :=
   by
@@ -201,22 +177,10 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
 -/
 
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 theorem le_trailingDegree_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
   show @LE.le ℕ∞ _ p.support.min n from min_le (mem_support_iff.2 h)
 #align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zero
 
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 theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n :=
   by
   rw [← WithTop.coe_le_coe, ← trailing_degree_eq_nat_trailing_degree]
@@ -224,12 +188,6 @@ theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegre
   · intro h; subst h; exact h rfl
 #align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero
 
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 theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
     trailingDegree q ≤ trailingDegree p :=
   by
@@ -245,12 +203,6 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
 #align polynomial.trailing_degree_ne_of_nat_trailing_degree_ne Polynomial.trailingDegree_ne_of_natTrailingDegree_ne
 -/
 
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 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p :=
   by
@@ -258,12 +210,6 @@ theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
   exact with_top.coe_le_coe.mp H
 #align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_le
 
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 theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     (hpq : p.trailingDegree ≤ q.trailingDegree) : p.natTrailingDegree ≤ q.natTrailingDegree :=
   by
@@ -272,84 +218,36 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     WithTop.coe_le_coe] at hpq
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
 
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 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
   rw [trailing_degree, support_monomial n ha, min_singleton]
 #align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomial
 
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 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
 #align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomial
 
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 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
 #align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_le
 
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 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge
 #align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomial
 
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 @[simp]
 theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
   trailingDegree_monomial ha
 #align polynomial.trailing_degree_C Polynomial.trailingDegree_C
 
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 theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_C
 
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 theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
   rw [← C_1] <;> exact le_trailing_degree_C
 #align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_le
 
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 @[simp]
 theorem natTrailingDegree_C (a : R) : natTrailingDegree (C a) = 0 :=
   nonpos_iff_eq_zero.1 natTrailingDegree_monomial_le
@@ -369,43 +267,19 @@ theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0
 #align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
 -/
 
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 @[simp]
 theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
   rw [C_mul_X_pow_eq_monomial, trailing_degree_monomial ha]
 #align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_pow
 
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 theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by rw [C_mul_X_pow_eq_monomial]; exact le_trailing_degree_monomial
 #align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
 
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 theorem coeff_eq_zero_of_trailingDegree_lt (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
   Classical.not_not.1 (mt le_trailingDegree_of_ne_zero (not_le_of_gt h))
 #align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_lt
 
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 theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
     p.coeff n = 0 := by
   apply coeff_eq_zero_of_trailing_degree_lt
@@ -414,12 +288,6 @@ theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.na
   · rwa [trailing_degree_eq_nat_trailing_degree hp, WithTop.coe_lt_coe]
 #align polynomial.coeff_eq_zero_of_lt_nat_trailing_degree Polynomial.coeff_eq_zero_of_lt_natTrailingDegree
 
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 @[simp]
 theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < natTrailingDegree p} :
     p.coeff (p.natTrailingDegree - 1) = 0 :=
@@ -427,22 +295,10 @@ theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < nat
     Nat.sub_lt ((WithTop.zero_lt_coe (natTrailingDegree p)).mp hp) Nat.one_pos
 #align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zero
 
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 theorem le_trailingDegree_X_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by
   simpa only [C_1, one_mul] using le_trailing_degree_C_mul_X_pow n (1 : R)
 #align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_X_pow
 
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 theorem le_trailingDegree_X : (1 : ℕ∞) ≤ trailingDegree (X : R[X]) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_X
@@ -453,12 +309,6 @@ theorem natTrailingDegree_X_le : (X : R[X]).natTrailingDegree ≤ 1 :=
 #align polynomial.nat_trailing_degree_X_le Polynomial.natTrailingDegree_X_le
 -/
 
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 @[simp]
 theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
   ⟨fun h =>
@@ -468,12 +318,6 @@ theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
     fun h => h.symm ▸ leadingCoeff_zero⟩
 #align polynomial.trailing_coeff_eq_zero Polynomial.trailingCoeff_eq_zero
 
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 theorem trailingCoeff_nonzero_iff_nonzero : trailingCoeff p ≠ 0 ↔ p ≠ 0 :=
   not_congr trailingCoeff_eq_zero
 #align polynomial.trailing_coeff_nonzero_iff_nonzero Polynomial.trailingCoeff_nonzero_iff_nonzero
@@ -503,12 +347,6 @@ theorem natTrailingDegree_eq_support_min' (h : p ≠ 0) :
 #align polynomial.nat_trailing_degree_eq_support_min' Polynomial.natTrailingDegree_eq_support_min'
 -/
 
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 theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) : n ≤ p.natTrailingDegree :=
   by
   rw [nat_trailing_degree_eq_support_min' hp]
@@ -541,12 +379,6 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
 #align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_X_pow
 -/
 
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 theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree :=
   by
   refine' Finset.le_min fun n hn => _
@@ -571,12 +403,6 @@ theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
 #align polynomial.le_nat_trailing_degree_mul Polynomial.le_natTrailingDegree_mul
 -/
 
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 theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
     (p * q).coeff (p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff :=
   by
@@ -595,12 +421,6 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
 #align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree
 
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 theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).trailingDegree = p.trailingDegree + q.trailingDegree :=
   by
@@ -613,12 +433,6 @@ theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
   rwa [coeff_mul_nat_trailing_degree_add_nat_trailing_degree]
 #align polynomial.trailing_degree_mul' Polynomial.trailingDegree_mul'
 
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 theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   by
@@ -629,12 +443,6 @@ theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     trailing_degree_eq_nat_trailing_degree hq]
 #align polynomial.nat_trailing_degree_mul' Polynomial.natTrailingDegree_mul'
 
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 theorem natTrailingDegree_mul [NoZeroDivisors R] (hp : p ≠ 0) (hq : q ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   natTrailingDegree_mul'
@@ -708,12 +516,6 @@ def nextCoeffUp (p : R[X]) : R :=
 #align polynomial.next_coeff_up Polynomial.nextCoeffUp
 -/
 
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 @[simp]
 theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by rw [next_coeff_up]; simp
 #align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
@@ -731,23 +533,11 @@ section Semiring
 
 variable [Semiring R] {p q : R[X]} {ι : Type _}
 
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-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
-Case conversion may be inaccurate. Consider using '#align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_ltₓ'. -/
 theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
     (h : trailingDegree p < trailingDegree q) : coeff q (natTrailingDegree p) = 0 :=
   coeff_eq_zero_of_trailingDegree_lt <| natTrailingDegree_le_trailingDegree.trans_lt h
 #align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
 
-/- warning: polynomial.ne_zero_of_trailing_degree_lt -> Polynomial.ne_zero_of_trailingDegree_lt is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))
-Case conversion may be inaccurate. Consider using '#align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_ltₓ'. -/
 theorem ne_zero_of_trailingDegree_lt {n : ℕ∞} (h : trailingDegree p < n) : p ≠ 0 := fun h₀ =>
   h.not_le (by simp [h₀])
 #align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_lt

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

@@ -164,17 +164,10 @@ theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (
     p.trailingDegree = n ↔ p.natTrailingDegree = n :=
   by
   constructor
-  · intro H
-    rwa [← trailing_degree_eq_iff_nat_trailing_degree_eq]
-    rintro rfl
-    rw [trailing_degree_zero] at H
-    exact Option.noConfusion H
-  · intro H
-    rwa [trailing_degree_eq_iff_nat_trailing_degree_eq]
-    rintro rfl
-    rw [nat_trailing_degree_zero] at H
-    rw [H] at hn
-    exact lt_irrefl _ hn
+  · intro H; rwa [← trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
+    rw [trailing_degree_zero] at H; exact Option.noConfusion H
+  · intro H; rwa [trailing_degree_eq_iff_nat_trailing_degree_eq]; rintro rfl
+    rw [nat_trailing_degree_zero] at H; rw [H] at hn; exact lt_irrefl _ hn
 #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
 -/
 
@@ -196,9 +189,7 @@ Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailin
 @[simp]
 theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p :=
   by
-  by_cases hp : p = 0;
-  · rw [hp, trailing_degree_zero]
-    exact le_top
+  by_cases hp : p = 0; · rw [hp, trailing_degree_zero]; exact le_top
   rw [trailing_degree_eq_nat_trailing_degree hp]
   exact le_rfl
 #align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegree
@@ -230,9 +221,7 @@ theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegre
   by
   rw [← WithTop.coe_le_coe, ← trailing_degree_eq_nat_trailing_degree]
   · exact le_trailing_degree_of_ne_zero h
-  · intro h
-    subst h
-    exact h rfl
+  · intro h; subst h; exact h rfl
 #align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero
 
 /- warning: polynomial.trailing_degree_le_trailing_degree -> Polynomial.trailingDegree_le_trailingDegree is a dubious translation:
@@ -245,10 +234,8 @@ theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠
     trailingDegree q ≤ trailingDegree p :=
   by
   by_cases hp : p = 0
-  · rw [hp]
-    exact le_top
-  · rw [trailing_degree_eq_nat_trailing_degree hp]
-    exact le_trailing_degree_of_ne_zero h
+  · rw [hp]; exact le_top
+  · rw [trailing_degree_eq_nat_trailing_degree hp]; exact le_trailing_degree_of_ne_zero h
 #align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegree
 
 #print Polynomial.trailingDegree_ne_of_natTrailingDegree_ne /-
@@ -280,9 +267,7 @@ Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailin
 theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     (hpq : p.trailingDegree ≤ q.trailingDegree) : p.natTrailingDegree ≤ q.natTrailingDegree :=
   by
-  by_cases hp : p = 0;
-  · rw [hp, nat_trailing_degree_zero]
-    exact zero_le _
+  by_cases hp : p = 0; · rw [hp, nat_trailing_degree_zero]; exact zero_le _
   rwa [trailing_degree_eq_nat_trailing_degree hp, trailing_degree_eq_nat_trailing_degree hq,
     WithTop.coe_le_coe] at hpq
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
@@ -402,9 +387,7 @@ but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_powₓ'. -/
 theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
-  by
-  rw [C_mul_X_pow_eq_monomial]
-  exact le_trailing_degree_monomial
+  by rw [C_mul_X_pow_eq_monomial]; exact le_trailing_degree_monomial
 #align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
 
 /- warning: polynomial.coeff_eq_zero_of_trailing_degree_lt -> Polynomial.coeff_eq_zero_of_trailingDegree_lt is a dubious translation:
@@ -427,8 +410,7 @@ theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.na
     p.coeff n = 0 := by
   apply coeff_eq_zero_of_trailing_degree_lt
   by_cases hp : p = 0
-  · rw [hp, trailing_degree_zero]
-    exact WithTop.coe_lt_top n
+  · rw [hp, trailing_degree_zero]; exact WithTop.coe_lt_top n
   · rwa [trailing_degree_eq_nat_trailing_degree hp, WithTop.coe_lt_coe]
 #align polynomial.coeff_eq_zero_of_lt_nat_trailing_degree Polynomial.coeff_eq_zero_of_lt_natTrailingDegree
 
@@ -733,19 +715,13 @@ but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) c)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zeroₓ'. -/
 @[simp]
-theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=
-  by
-  rw [next_coeff_up]
-  simp
+theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by rw [next_coeff_up]; simp
 #align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
 
 #print Polynomial.nextCoeffUp_of_pos_natTrailingDegree /-
 theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingDegree) :
-    nextCoeffUp p = p.coeff (p.natTrailingDegree + 1) :=
-  by
-  rw [next_coeff_up, if_neg]
-  contrapose! hp
-  simpa
+    nextCoeffUp p = p.coeff (p.natTrailingDegree + 1) := by rw [next_coeff_up, if_neg];
+  contrapose! hp; simpa
 #align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_pos_natTrailingDegree
 -/
 

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

@@ -291,7 +291,7 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomialₓ'. -/
 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
@@ -302,7 +302,7 @@ theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) =
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomialₓ'. -/
 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
@@ -312,7 +312,7 @@ theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_leₓ'. -/
 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
@@ -322,7 +322,7 @@ theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomialₓ'. -/
 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge

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

@@ -332,7 +332,7 @@ theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C Polynomial.trailingDegree_Cₓ'. -/
 @[simp]
 theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
@@ -343,7 +343,7 @@ theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_Cₓ'. -/
 theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
@@ -363,7 +363,7 @@ theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_Cₓ'. -/
 @[simp]
 theorem natTrailingDegree_C (a : R) : natTrailingDegree (C a) = 0 :=
@@ -388,7 +388,7 @@ theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_powₓ'. -/
 @[simp]
 theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
@@ -399,7 +399,7 @@ theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_powₓ'. -/
 theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by
@@ -730,7 +730,7 @@ def nextCoeffUp (p : R[X]) : R :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) c)) (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 _inst_1)))))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) c)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) c)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zeroₓ'. -/
 @[simp]
 theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=

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

@@ -291,7 +291,7 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomialₓ'. -/
 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
@@ -302,7 +302,7 @@ theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) =
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomialₓ'. -/
 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
@@ -312,7 +312,7 @@ theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_leₓ'. -/
 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
@@ -322,7 +322,7 @@ theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomialₓ'. -/
 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge

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

@@ -57,7 +57,7 @@ def trailingDegree (p : R[X]) : ℕ∞ :=
 
 /- warning: polynomial.trailing_degree_lt_wf -> Polynomial.trailingDegree_lt_wf is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wfₓ'. -/
@@ -189,7 +189,7 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
 
 /- warning: polynomial.nat_trailing_degree_le_trailing_degree -> Polynomial.natTrailingDegree_le_trailingDegree is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (Polynomial.trailingDegree.{u1} R _inst_1 p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (Polynomial.trailingDegree.{u1} R _inst_1 p)
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (Polynomial.trailingDegree.{u1} R _inst_1 p)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegreeₓ'. -/
@@ -212,7 +212,7 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
 
 /- warning: polynomial.le_trailing_degree_of_ne_zero -> Polynomial.le_trailingDegree_of_ne_zero is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
   forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zeroₓ'. -/
@@ -237,7 +237,7 @@ theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegre
 
 /- warning: polynomial.trailing_degree_le_trailing_degree -> Polynomial.trailingDegree_le_trailingDegree is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 q) (Polynomial.trailingDegree.{u1} R _inst_1 p))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 q) (Polynomial.trailingDegree.{u1} R _inst_1 p))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 q) (Polynomial.trailingDegree.{u1} R _inst_1 p))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegreeₓ'. -/
@@ -260,7 +260,7 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
 
 /- warning: polynomial.nat_trailing_degree_le_of_trailing_degree_le -> Polynomial.natTrailingDegree_le_of_trailingDegree_le is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat} {hp : Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (LE.le.{0} Nat Nat.hasLe n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat} {hp : Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))}, (LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (LE.le.{0} Nat Nat.hasLe n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat} {hp : Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (LE.le.{0} Nat instLENat n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_leₓ'. -/
@@ -273,7 +273,7 @@ theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
 
 /- warning: polynomial.nat_trailing_degree_le_nat_trailing_degree -> Polynomial.natTrailingDegree_le_natTrailingDegree is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} {hq : Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} {hq : Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))}, (LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} {hq : Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q))
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegreeₓ'. -/
@@ -320,7 +320,7 @@ theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :
 
 /- warning: polynomial.le_trailing_degree_monomial -> Polynomial.le_trailingDegree_monomial is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 but is expected to have type
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomialₓ'. -/
@@ -341,7 +341,7 @@ theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
 
 /- warning: polynomial.le_trailing_degree_C -> Polynomial.le_trailingDegree_C is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a))
 but is expected to have type
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_Cₓ'. -/
@@ -351,7 +351,7 @@ theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
 
 /- warning: polynomial.trailing_degree_one_le -> Polynomial.trailingDegree_one_le is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 1 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 1 (One.one.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.hasOne.{u1} R _inst_1)))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 1 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 1 (One.one.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.hasOne.{u1} R _inst_1)))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 1 (One.toOfNat1.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.one.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_leₓ'. -/
@@ -397,7 +397,7 @@ theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C
 
 /- warning: polynomial.le_trailing_degree_C_mul_X_pow -> Polynomial.le_trailingDegree_C_mul_X_pow is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_powₓ'. -/
@@ -409,7 +409,7 @@ theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trail
 
 /- warning: polynomial.coeff_eq_zero_of_trailing_degree_lt -> Polynomial.coeff_eq_zero_of_trailingDegree_lt is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1))))))))
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1))))))))
 but is expected to have type
   forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
 Case conversion may be inaccurate. Consider using '#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_ltₓ'. -/
@@ -434,7 +434,7 @@ theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.na
 
 /- warning: polynomial.coeff_nat_trailing_degree_pred_eq_zero -> Polynomial.coeff_natTrailingDegree_pred_eq_zero is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {hp : LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) (Polynomial.natTrailingDegree.{u1} R _inst_1 p))}, Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (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 _inst_1)))))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {hp : LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) (Polynomial.natTrailingDegree.{u1} R _inst_1 p))}, Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (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 _inst_1)))))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {hp : LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) (Polynomial.natTrailingDegree.{u1} R _inst_1 p))}, Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zeroₓ'. -/
@@ -447,7 +447,7 @@ theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < nat
 
 /- warning: polynomial.le_trailing_degree_X_pow -> Polynomial.le_trailingDegree_X_pow is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_X_powₓ'. -/
@@ -457,7 +457,7 @@ theorem le_trailingDegree_X_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^
 
 /- warning: polynomial.le_trailing_degree_X -> Polynomial.le_trailingDegree_X is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 1 (OfNat.mk.{0} ENat 1 (One.one.{0} ENat (AddMonoidWithOne.toOne.{0} ENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENat ENat.addCommMonoidWithOne))))) (Polynomial.trailingDegree.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 1 (OfNat.mk.{0} ENat 1 (One.one.{0} ENat (AddMonoidWithOne.toOne.{0} ENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENat ENat.addCommMonoidWithOne))))) (Polynomial.trailingDegree.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 1 (One.toOfNat1.{0} ENat (CanonicallyOrderedCommSemiring.toOne.{0} ENat instENatCanonicallyOrderedCommSemiring))) (Polynomial.trailingDegree.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_Xₓ'. -/
@@ -561,7 +561,7 @@ theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
 
 /- warning: polynomial.le_trailing_degree_mul -> Polynomial.le_trailingDegree_mul is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (HAdd.hAdd.{0, 0, 0} ENat ENat ENat (instHAdd.{0} ENat (Distrib.toHasAdd.{0} ENat (NonUnitalNonAssocSemiring.toDistrib.{0} ENat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENat (Semiring.toNonAssocSemiring.{0} ENat (OrderedSemiring.toSemiring.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toHasLe.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (HAdd.hAdd.{0, 0, 0} ENat ENat ENat (instHAdd.{0} ENat (Distrib.toHasAdd.{0} ENat (NonUnitalNonAssocSemiring.toDistrib.{0} ENat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENat (Semiring.toNonAssocSemiring.{0} ENat (OrderedSemiring.toSemiring.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (HAdd.hAdd.{0, 0, 0} ENat ENat ENat (instHAdd.{0} ENat (Distrib.toAdd.{0} ENat (NonUnitalNonAssocSemiring.toDistrib.{0} ENat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENat (Semiring.toNonAssocSemiring.{0} ENat (OrderedSemiring.toSemiring.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring)))))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_mul Polynomial.le_trailingDegree_mulₓ'. -/
@@ -757,7 +757,7 @@ variable [Semiring R] {p q : R[X]} {ι : Type _}
 
 /- warning: polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt -> Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1))))))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1))))))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
 Case conversion may be inaccurate. Consider using '#align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_ltₓ'. -/
@@ -768,7 +768,7 @@ theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
 
 /- warning: polynomial.ne_zero_of_trailing_degree_lt -> Polynomial.ne_zero_of_trailingDegree_lt is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toHasLt.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_ltₓ'. -/

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: Damiano Testa
 
 ! This file was ported from Lean 3 source module data.polynomial.degree.trailing_degree
-! leanprover-community/mathlib commit 302eab4f46abb63de520828de78c04cb0f9b5836
+! 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.Degree.Definitions
 /-!
 # Trailing degree of univariate polynomials
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main definitions
 
 * `trailing_degree p`: the multiplicity of `X` in the polynomial `p`

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

@@ -288,7 +288,7 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6178 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomialₓ'. -/
 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
@@ -299,7 +299,7 @@ theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) =
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6178 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomialₓ'. -/
 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
@@ -309,7 +309,7 @@ theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6178 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a)) n
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_leₓ'. -/
 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
@@ -319,7 +319,7 @@ theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6178 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
+  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomialₓ'. -/
 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge
@@ -329,7 +329,7 @@ theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C Polynomial.trailingDegree_Cₓ'. -/
 @[simp]
 theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
@@ -340,7 +340,7 @@ theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_Cₓ'. -/
 theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
@@ -360,7 +360,7 @@ theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_Cₓ'. -/
 @[simp]
 theorem natTrailingDegree_C (a : R) : natTrailingDegree (C a) = 0 :=
@@ -385,7 +385,7 @@ theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0
 lean 3 declaration is
   forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (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 _inst_1)))))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
 but is expected to have type
-  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R] (n : Nat), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} ENat (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
 Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_powₓ'. -/
 @[simp]
 theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
@@ -396,7 +396,7 @@ theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat) (a : R), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)))
 Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_powₓ'. -/
 theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by
@@ -727,7 +727,7 @@ def nextCoeffUp (p : R[X]) : R :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) c)) (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 _inst_1)))))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) c)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (c : R), Eq.{succ u1} R (Polynomial.nextCoeffUp.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) c)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zeroₓ'. -/
 @[simp]
 theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=

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

@@ -578,8 +578,8 @@ theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).
 theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
     p.natTrailingDegree + q.natTrailingDegree ≤ (p * q).natTrailingDegree :=
   by
-  have hp : p ≠ 0 := fun hp => h (by rw [hp, zero_mul])
-  have hq : q ≠ 0 := fun hq => h (by rw [hq, mul_zero])
+  have hp : p ≠ 0 := fun hp => h (by rw [hp, MulZeroClass.zero_mul])
+  have hq : q ≠ 0 := fun hq => h (by rw [hq, MulZeroClass.mul_zero])
   rw [← WithTop.coe_le_coe, WithTop.coe_add, ← trailing_degree_eq_nat_trailing_degree hp, ←
     trailing_degree_eq_nat_trailing_degree hq, ← trailing_degree_eq_nat_trailing_degree h]
   exact le_trailing_degree_mul
@@ -602,9 +602,9 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
   rintro ⟨i, j⟩ h₁ h₂
   rw [nat.mem_antidiagonal] at h₁
   by_cases hi : i < p.nat_trailing_degree
-  · rw [coeff_eq_zero_of_lt_nat_trailing_degree hi, zero_mul]
+  · rw [coeff_eq_zero_of_lt_nat_trailing_degree hi, MulZeroClass.zero_mul]
   by_cases hj : j < q.nat_trailing_degree
-  · rw [coeff_eq_zero_of_lt_nat_trailing_degree hj, mul_zero]
+  · rw [coeff_eq_zero_of_lt_nat_trailing_degree hj, MulZeroClass.mul_zero]
   rw [not_lt] at hi hj
   refine' (h₂ (prod.ext_iff.mpr _).symm).elim
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
@@ -619,8 +619,8 @@ Case conversion may be inaccurate. Consider using '#align polynomial.trailing_de
 theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).trailingDegree = p.trailingDegree + q.trailingDegree :=
   by
-  have hp : p ≠ 0 := fun hp => h (by rw [hp, trailing_coeff_zero, zero_mul])
-  have hq : q ≠ 0 := fun hq => h (by rw [hq, trailing_coeff_zero, mul_zero])
+  have hp : p ≠ 0 := fun hp => h (by rw [hp, trailing_coeff_zero, MulZeroClass.zero_mul])
+  have hq : q ≠ 0 := fun hq => h (by rw [hq, trailing_coeff_zero, MulZeroClass.mul_zero])
   refine' le_antisymm _ le_trailing_degree_mul
   rw [trailing_degree_eq_nat_trailing_degree hp, trailing_degree_eq_nat_trailing_degree hq, ←
     ENat.coe_add]
@@ -637,8 +637,8 @@ Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailin
 theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   by
-  have hp : p ≠ 0 := fun hp => h (by rw [hp, trailing_coeff_zero, zero_mul])
-  have hq : q ≠ 0 := fun hq => h (by rw [hq, trailing_coeff_zero, mul_zero])
+  have hp : p ≠ 0 := fun hp => h (by rw [hp, trailing_coeff_zero, MulZeroClass.zero_mul])
+  have hq : q ≠ 0 := fun hq => h (by rw [hq, trailing_coeff_zero, MulZeroClass.mul_zero])
   apply nat_trailing_degree_eq_of_trailing_degree_eq_some
   rw [trailing_degree_mul' h, WithTop.coe_add, ← trailing_degree_eq_nat_trailing_degree hp, ←
     trailing_degree_eq_nat_trailing_degree hq]

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

@@ -41,6 +41,7 @@ section Semiring
 
 variable [Semiring R] {p q r : R[X]}
 
+#print Polynomial.trailingDegree /-
 /-- `trailing_degree p` is the multiplicity of `x` in the polynomial `p`, i.e. the smallest
 `X`-exponent in `p`.
 `trailing_degree p = some n` when `p ≠ 0` and `n` is the smallest power of `X` that appears
@@ -49,59 +50,95 @@ in `p`, otherwise
 def trailingDegree (p : R[X]) : ℕ∞ :=
   p.support.min
 #align polynomial.trailing_degree Polynomial.trailingDegree
+-/
 
+/- warning: polynomial.trailing_degree_lt_wf -> Polynomial.trailingDegree_lt_wf is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], WellFounded.{succ u1} (Polynomial.{u1} R _inst_1) (fun (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) => LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wfₓ'. -/
 theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q :=
   InvImage.wf trailingDegree (WithTop.wellFounded_lt Nat.lt_wfRel)
 #align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf
 
+#print Polynomial.natTrailingDegree /-
 /-- `nat_trailing_degree p` forces `trailing_degree p` to `ℕ`, by defining
 `nat_trailing_degree ⊤ = 0`. -/
 def natTrailingDegree (p : R[X]) : ℕ :=
   (trailingDegree p).getD 0
 #align polynomial.nat_trailing_degree Polynomial.natTrailingDegree
+-/
 
+#print Polynomial.trailingCoeff /-
 /-- `trailing_coeff p` gives the coefficient of the smallest power of `X` in `p`-/
 def trailingCoeff (p : R[X]) : R :=
   coeff p (natTrailingDegree p)
 #align polynomial.trailing_coeff Polynomial.trailingCoeff
+-/
 
+#print Polynomial.TrailingMonic /-
 /-- a polynomial is `monic_at` if its trailing coefficient is 1 -/
 def TrailingMonic (p : R[X]) :=
   trailingCoeff p = (1 : R)
 #align polynomial.trailing_monic Polynomial.TrailingMonic
+-/
 
+#print Polynomial.TrailingMonic.def /-
 theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
   Iff.rfl
 #align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
+-/
 
+/- warning: polynomial.trailing_monic.decidable -> Polynomial.TrailingMonic.decidable is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} [_inst_2 : DecidableEq.{succ u1} R], Decidable (Polynomial.TrailingMonic.{u1} R _inst_1 p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Decidable (Polynomial.TrailingMonic.{u1} R _inst_1 p)
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidableₓ'. -/
 instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) := by
   unfold trailing_monic <;> infer_instance
 #align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidable
 
+#print Polynomial.TrailingMonic.trailingCoeff /-
 @[simp]
 theorem TrailingMonic.trailingCoeff {p : R[X]} (hp : p.TrailingMonic) : trailingCoeff p = 1 :=
   hp
 #align polynomial.trailing_monic.trailing_coeff Polynomial.TrailingMonic.trailingCoeff
+-/
 
+#print Polynomial.trailingDegree_zero /-
 @[simp]
 theorem trailingDegree_zero : trailingDegree (0 : R[X]) = ⊤ :=
   rfl
 #align polynomial.trailing_degree_zero Polynomial.trailingDegree_zero
+-/
 
+/- warning: polynomial.trailing_coeff_zero -> Polynomial.trailingCoeff_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) (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 _inst_1)))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_coeff_zero Polynomial.trailingCoeff_zeroₓ'. -/
 @[simp]
 theorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 :=
   rfl
 #align polynomial.trailing_coeff_zero Polynomial.trailingCoeff_zero
 
+#print Polynomial.natTrailingDegree_zero /-
 @[simp]
 theorem natTrailingDegree_zero : natTrailingDegree (0 : R[X]) = 0 :=
   rfl
 #align polynomial.nat_trailing_degree_zero Polynomial.natTrailingDegree_zero
+-/
 
+#print Polynomial.trailingDegree_eq_top /-
 theorem trailingDegree_eq_top : trailingDegree p = ⊤ ↔ p = 0 :=
   ⟨fun h => support_eq_empty.1 (Finset.min_eq_top.1 h), fun h => by simp [h]⟩
 #align polynomial.trailing_degree_eq_top Polynomial.trailingDegree_eq_top
+-/
 
+#print Polynomial.trailingDegree_eq_natTrailingDegree /-
 theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) :
     trailingDegree p = (natTrailingDegree p : ℕ∞) :=
   by
@@ -110,12 +147,16 @@ theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) :
   have hn : trailingDegree p = n := Classical.not_not.1 hn
   rw [nat_trailing_degree, hn] <;> rfl
 #align polynomial.trailing_degree_eq_nat_trailing_degree Polynomial.trailingDegree_eq_natTrailingDegree
+-/
 
+#print Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq /-
 theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
     p.trailingDegree = n ↔ p.natTrailingDegree = n := by
   rw [trailing_degree_eq_nat_trailing_degree hp, WithTop.coe_eq_coe]
 #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq
+-/
 
+#print Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos /-
 theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : 0 < n) :
     p.trailingDegree = n ↔ p.natTrailingDegree = n :=
   by
@@ -132,14 +173,23 @@ theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (
     rw [H] at hn
     exact lt_irrefl _ hn
 #align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
+-/
 
+#print Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some /-
 theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
     (h : trailingDegree p = n) : natTrailingDegree p = n :=
   have hp0 : p ≠ 0 := fun hp0 => by rw [hp0] at h <;> exact Option.noConfusion h
   Option.some_inj.1 <|
     show (natTrailingDegree p : ℕ∞) = n by rwa [← trailing_degree_eq_nat_trailing_degree hp0]
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
+-/
 
+/- warning: polynomial.nat_trailing_degree_le_trailing_degree -> Polynomial.natTrailingDegree_le_trailingDegree is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (Polynomial.trailingDegree.{u1} R _inst_1 p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (Polynomial.trailingDegree.{u1} R _inst_1 p)
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegreeₓ'. -/
 @[simp]
 theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p :=
   by
@@ -150,15 +200,29 @@ theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trail
   exact le_rfl
 #align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegree
 
+#print Polynomial.natTrailingDegree_eq_of_trailingDegree_eq /-
 theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
     (h : trailingDegree p = trailingDegree q) : natTrailingDegree p = natTrailingDegree q := by
   unfold nat_trailing_degree <;> rw [h]
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
+-/
 
+/- warning: polynomial.le_trailing_degree_of_ne_zero -> Polynomial.le_trailingDegree_of_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n))
+but is expected to have type
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n))
+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zeroₓ'. -/
 theorem le_trailingDegree_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
   show @LE.le ℕ∞ _ p.support.min n from min_le (mem_support_iff.2 h)
 #align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zero
 
+/- warning: polynomial.nat_trailing_degree_le_of_ne_zero -> Polynomial.natTrailingDegree_le_of_ne_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1)))))))) -> (LE.le.{0} Nat Nat.hasLe (Polynomial.natTrailingDegree.{u1} R _inst_1 p) n)
+but is expected to have type
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 p) n)
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zeroₓ'. -/
 theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n :=
   by
   rw [← WithTop.coe_le_coe, ← trailing_degree_eq_nat_trailing_degree]
@@ -168,6 +232,12 @@ theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegre
     exact h rfl
 #align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero
 
+/- warning: polynomial.trailing_degree_le_trailing_degree -> Polynomial.trailingDegree_le_trailingDegree is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1)))))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 q) (Polynomial.trailingDegree.{u1} R _inst_1 p))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 q) (Polynomial.trailingDegree.{u1} R _inst_1 p))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegreeₓ'. -/
 theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
     trailingDegree q ≤ trailingDegree p :=
   by
@@ -178,11 +248,19 @@ theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠
     exact le_trailing_degree_of_ne_zero h
 #align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegree
 
+#print Polynomial.trailingDegree_ne_of_natTrailingDegree_ne /-
 theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
     p.natTrailingDegree ≠ n → trailingDegree p ≠ n :=
   mt fun h => by rw [nat_trailing_degree, h, Option.getD_coe]
 #align polynomial.trailing_degree_ne_of_nat_trailing_degree_ne Polynomial.trailingDegree_ne_of_natTrailingDegree_ne
+-/
 
+/- warning: polynomial.nat_trailing_degree_le_of_trailing_degree_le -> Polynomial.natTrailingDegree_le_of_trailingDegree_le is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat} {hp : Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (LE.le.{0} Nat Nat.hasLe n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat} {hp : Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 p)) -> (LE.le.{0} Nat instLENat n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_leₓ'. -/
 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p :=
   by
@@ -190,6 +268,12 @@ theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
   exact with_top.coe_le_coe.mp H
 #align polynomial.nat_trailing_degree_le_of_trailing_degree_le Polynomial.natTrailingDegree_le_of_trailingDegree_le
 
+/- warning: polynomial.nat_trailing_degree_le_nat_trailing_degree -> Polynomial.natTrailingDegree_le_natTrailingDegree is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} {hq : Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))}, (LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (LE.le.{0} Nat instLENat (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegreeₓ'. -/
 theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     (hpq : p.trailingDegree ≤ q.trailingDegree) : p.natTrailingDegree ≤ q.natTrailingDegree :=
   by
@@ -200,66 +284,142 @@ theorem natTrailingDegree_le_natTrailingDegree {hq : q ≠ 0}
     WithTop.coe_le_coe] at hpq
 #align polynomial.nat_trailing_degree_le_nat_trailing_degree Polynomial.natTrailingDegree_le_natTrailingDegree
 
+/- warning: polynomial.trailing_degree_monomial -> Polynomial.trailingDegree_monomial is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomialₓ'. -/
 @[simp]
 theorem trailingDegree_monomial (ha : a ≠ 0) : trailingDegree (monomial n a) = n := by
   rw [trailing_degree, support_monomial n ha, min_singleton]
 #align polynomial.trailing_degree_monomial Polynomial.trailingDegree_monomial
 
+/- warning: polynomial.nat_trailing_degree_monomial -> Polynomial.natTrailingDegree_monomial is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomialₓ'. -/
 theorem natTrailingDegree_monomial (ha : a ≠ 0) : natTrailingDegree (monomial n a) = n := by
   rw [nat_trailing_degree, trailing_degree_monomial ha] <;> rfl
 #align polynomial.nat_trailing_degree_monomial Polynomial.natTrailingDegree_monomial
 
+/- warning: polynomial.nat_trailing_degree_monomial_le -> Polynomial.natTrailingDegree_monomial_le is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_leₓ'. -/
 theorem natTrailingDegree_monomial_le : natTrailingDegree (monomial n a) ≤ n :=
   if ha : a = 0 then by simp [ha] else (natTrailingDegree_monomial ha).le
 #align polynomial.nat_trailing_degree_monomial_le Polynomial.natTrailingDegree_monomial_le
 
+/- warning: polynomial.le_trailing_degree_monomial -> Polynomial.le_trailingDegree_monomial is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomialₓ'. -/
 theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
   if ha : a = 0 then by simp [ha] else (trailingDegree_monomial ha).ge
 #align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomial
 
+/- warning: polynomial.trailing_degree_C -> Polynomial.trailingDegree_C is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C Polynomial.trailingDegree_Cₓ'. -/
 @[simp]
-theorem trailingDegree_c (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
+theorem trailingDegree_C (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
   trailingDegree_monomial ha
-#align polynomial.trailing_degree_C Polynomial.trailingDegree_c
-
-theorem le_trailingDegree_c : (0 : ℕ∞) ≤ trailingDegree (C a) :=
+#align polynomial.trailing_degree_C Polynomial.trailingDegree_C
+
+/- warning: polynomial.le_trailing_degree_C -> Polynomial.le_trailingDegree_C is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_Cₓ'. -/
+theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
-#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_c
-
+#align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_C
+
+/- warning: polynomial.trailing_degree_one_le -> Polynomial.trailingDegree_one_le is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 0 (OfNat.mk.{0} ENat 0 (Zero.zero.{0} ENat ENat.hasZero))) (Polynomial.trailingDegree.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 1 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 1 (One.one.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.hasOne.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Polynomial.trailingDegree.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 1 (One.toOfNat1.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.one.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_leₓ'. -/
 theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
   rw [← C_1] <;> exact le_trailing_degree_C
 #align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_le
 
+/- warning: polynomial.nat_trailing_degree_C -> Polynomial.natTrailingDegree_C is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_Cₓ'. -/
 @[simp]
-theorem natTrailingDegree_c (a : R) : natTrailingDegree (C a) = 0 :=
+theorem natTrailingDegree_C (a : R) : natTrailingDegree (C a) = 0 :=
   nonpos_iff_eq_zero.1 natTrailingDegree_monomial_le
-#align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_c
+#align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_C
 
+#print Polynomial.natTrailingDegree_one /-
 @[simp]
 theorem natTrailingDegree_one : natTrailingDegree (1 : R[X]) = 0 :=
-  natTrailingDegree_c 1
+  natTrailingDegree_C 1
 #align polynomial.nat_trailing_degree_one Polynomial.natTrailingDegree_one
+-/
 
+#print Polynomial.natTrailingDegree_nat_cast /-
 @[simp]
 theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0 := by
   simp only [← C_eq_nat_cast, nat_trailing_degree_C]
 #align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
+-/
 
+/- warning: polynomial.trailing_degree_C_mul_X_pow -> Polynomial.trailingDegree_C_mul_X_pow is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_powₓ'. -/
 @[simp]
-theorem trailingDegree_c_mul_x_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
+theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
   rw [C_mul_X_pow_eq_monomial, trailing_degree_monomial ha]
-#align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_c_mul_x_pow
-
-theorem le_trailingDegree_c_mul_x_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
+#align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_C_mul_X_pow
+
+/- warning: polynomial.le_trailing_degree_C_mul_X_pow -> Polynomial.le_trailingDegree_C_mul_X_pow is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_powₓ'. -/
+theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by
   rw [C_mul_X_pow_eq_monomial]
   exact le_trailing_degree_monomial
-#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_c_mul_x_pow
-
+#align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
+
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_ltₓ'. -/
 theorem coeff_eq_zero_of_trailingDegree_lt (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
   Classical.not_not.1 (mt le_trailingDegree_of_ne_zero (not_le_of_gt h))
 #align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_lt
 
+/- warning: polynomial.coeff_eq_zero_of_lt_nat_trailing_degree -> Polynomial.coeff_eq_zero_of_lt_natTrailingDegree is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat}, (LT.lt.{0} Nat Nat.hasLt n (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (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 _inst_1))))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : Nat}, (LT.lt.{0} Nat instLTNat n (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_eq_zero_of_lt_nat_trailing_degree Polynomial.coeff_eq_zero_of_lt_natTrailingDegreeₓ'. -/
 theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
     p.coeff n = 0 := by
   apply coeff_eq_zero_of_trailing_degree_lt
@@ -269,6 +429,12 @@ theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.na
   · rwa [trailing_degree_eq_nat_trailing_degree hp, WithTop.coe_lt_coe]
 #align polynomial.coeff_eq_zero_of_lt_nat_trailing_degree Polynomial.coeff_eq_zero_of_lt_natTrailingDegree
 
+/- warning: polynomial.coeff_nat_trailing_degree_pred_eq_zero -> Polynomial.coeff_natTrailingDegree_pred_eq_zero is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {hp : LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 0 (Zero.toOfNat0.{0} ENat instENatZero)) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) (Polynomial.natTrailingDegree.{u1} R _inst_1 p))}, Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zeroₓ'. -/
 @[simp]
 theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < natTrailingDegree p} :
     p.coeff (p.natTrailingDegree - 1) = 0 :=
@@ -276,18 +442,38 @@ theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < nat
     Nat.sub_lt ((WithTop.zero_lt_coe (natTrailingDegree p)).mp hp) Nat.one_pos
 #align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zero
 
-theorem le_trailingDegree_x_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by
+/- warning: polynomial.le_trailing_degree_X_pow -> Polynomial.le_trailingDegree_X_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat ENat (HasLiftT.mk.{1, 1} Nat ENat (CoeTCₓ.coe.{1, 1} Nat ENat ENat.hasCoeT)) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Nat.cast.{0} ENat (CanonicallyOrderedCommSemiring.toNatCast.{0} ENat instENatCanonicallyOrderedCommSemiring) n) (Polynomial.trailingDegree.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))
+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_X_powₓ'. -/
+theorem le_trailingDegree_X_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by
   simpa only [C_1, one_mul] using le_trailing_degree_C_mul_X_pow n (1 : R)
-#align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_x_pow
-
-theorem le_trailingDegree_x : (1 : ℕ∞) ≤ trailingDegree (X : R[X]) :=
+#align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_X_pow
+
+/- warning: polynomial.le_trailing_degree_X -> Polynomial.le_trailingDegree_X is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (OfNat.ofNat.{0} ENat 1 (OfNat.mk.{0} ENat 1 (One.one.{0} ENat (AddMonoidWithOne.toOne.{0} ENat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} ENat ENat.addCommMonoidWithOne))))) (Polynomial.trailingDegree.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (OfNat.ofNat.{0} ENat 1 (One.toOfNat1.{0} ENat (CanonicallyOrderedCommSemiring.toOne.{0} ENat instENatCanonicallyOrderedCommSemiring))) (Polynomial.trailingDegree.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1))
+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_Xₓ'. -/
+theorem le_trailingDegree_X : (1 : ℕ∞) ≤ trailingDegree (X : R[X]) :=
   le_trailingDegree_monomial
-#align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_x
+#align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_X
 
-theorem natTrailingDegree_x_le : (X : R[X]).natTrailingDegree ≤ 1 :=
+#print Polynomial.natTrailingDegree_X_le /-
+theorem natTrailingDegree_X_le : (X : R[X]).natTrailingDegree ≤ 1 :=
   natTrailingDegree_monomial_le
-#align polynomial.nat_trailing_degree_X_le Polynomial.natTrailingDegree_x_le
+#align polynomial.nat_trailing_degree_X_le Polynomial.natTrailingDegree_X_le
+-/
 
+/- warning: polynomial.trailing_coeff_eq_zero -> Polynomial.trailingCoeff_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 p) (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 _inst_1)))))))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 p) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_coeff_eq_zero Polynomial.trailingCoeff_eq_zeroₓ'. -/
 @[simp]
 theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
   ⟨fun h =>
@@ -297,18 +483,29 @@ theorem trailingCoeff_eq_zero : trailingCoeff p = 0 ↔ p = 0 :=
     fun h => h.symm ▸ leadingCoeff_zero⟩
 #align polynomial.trailing_coeff_eq_zero Polynomial.trailingCoeff_eq_zero
 
+/- warning: polynomial.trailing_coeff_nonzero_iff_nonzero -> Polynomial.trailingCoeff_nonzero_iff_nonzero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Ne.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 p) (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 _inst_1)))))))) (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Ne.{succ u1} R (Polynomial.trailingCoeff.{u1} R _inst_1 p) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.trailing_coeff_nonzero_iff_nonzero Polynomial.trailingCoeff_nonzero_iff_nonzeroₓ'. -/
 theorem trailingCoeff_nonzero_iff_nonzero : trailingCoeff p ≠ 0 ↔ p ≠ 0 :=
   not_congr trailingCoeff_eq_zero
 #align polynomial.trailing_coeff_nonzero_iff_nonzero Polynomial.trailingCoeff_nonzero_iff_nonzero
 
+#print Polynomial.natTrailingDegree_mem_support_of_nonzero /-
 theorem natTrailingDegree_mem_support_of_nonzero : p ≠ 0 → natTrailingDegree p ∈ p.support :=
   mem_support_iff.mpr ∘ trailingCoeff_nonzero_iff_nonzero.mpr
 #align polynomial.nat_trailing_degree_mem_support_of_nonzero Polynomial.natTrailingDegree_mem_support_of_nonzero
+-/
 
+#print Polynomial.natTrailingDegree_le_of_mem_supp /-
 theorem natTrailingDegree_le_of_mem_supp (a : ℕ) : a ∈ p.support → natTrailingDegree p ≤ a :=
   natTrailingDegree_le_of_ne_zero ∘ mem_support_iff.mp
 #align polynomial.nat_trailing_degree_le_of_mem_supp Polynomial.natTrailingDegree_le_of_mem_supp
+-/
 
+#print Polynomial.natTrailingDegree_eq_support_min' /-
 theorem natTrailingDegree_eq_support_min' (h : p ≠ 0) :
     natTrailingDegree p = p.support.min' (nonempty_support_iff.mpr h) :=
   by
@@ -319,21 +516,31 @@ theorem natTrailingDegree_eq_support_min' (h : p ≠ 0) :
   · apply Finset.min'_le
     exact mem_support_iff.mpr (trailing_coeff_nonzero_iff_nonzero.mpr h)
 #align polynomial.nat_trailing_degree_eq_support_min' Polynomial.natTrailingDegree_eq_support_min'
+-/
 
+/- warning: polynomial.le_nat_trailing_degree -> Polynomial.le_natTrailingDegree is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) -> (forall (m : Nat), (LT.lt.{0} Nat Nat.hasLt m n) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p m) (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 _inst_1))))))))) -> (LE.le.{0} Nat Nat.hasLe n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
+but is expected to have type
+  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (m : Nat), (LT.lt.{0} Nat instLTNat m n) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p m) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))) -> (LE.le.{0} Nat instLENat n (Polynomial.natTrailingDegree.{u1} R _inst_1 p))
+Case conversion may be inaccurate. Consider using '#align polynomial.le_nat_trailing_degree Polynomial.le_natTrailingDegreeₓ'. -/
 theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) : n ≤ p.natTrailingDegree :=
   by
   rw [nat_trailing_degree_eq_support_min' hp]
   exact Finset.le_min' _ _ _ fun m hm => not_lt.1 fun hmn => mem_support_iff.1 hm <| hn _ hmn
 #align polynomial.le_nat_trailing_degree Polynomial.le_natTrailingDegree
 
+#print Polynomial.natTrailingDegree_le_natDegree /-
 theorem natTrailingDegree_le_natDegree (p : R[X]) : p.natTrailingDegree ≤ p.natDegree :=
   by
   by_cases hp : p = 0
   · rw [hp, nat_degree_zero, nat_trailing_degree_zero]
   · exact le_nat_degree_of_ne_zero (mt trailing_coeff_eq_zero.mp hp)
 #align polynomial.nat_trailing_degree_le_nat_degree Polynomial.natTrailingDegree_le_natDegree
+-/
 
-theorem natTrailingDegree_mul_x_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
+#print Polynomial.natTrailingDegree_mul_X_pow /-
+theorem natTrailingDegree_mul_X_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
     (p * X ^ n).natTrailingDegree = p.natTrailingDegree + n :=
   by
   apply le_antisymm
@@ -346,8 +553,15 @@ theorem natTrailingDegree_mul_x_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
       exact by_contra fun h => hy (if_neg h)
     rw [mem_support_iff, coeff_mul_X_pow', if_pos key] at hy
     exact (le_tsub_iff_right key).mp (nat_trailing_degree_le_of_ne_zero hy)
-#align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_x_pow
+#align polynomial.nat_trailing_degree_mul_X_pow Polynomial.natTrailingDegree_mul_X_pow
+-/
 
+/- warning: polynomial.le_trailing_degree_mul -> Polynomial.le_trailingDegree_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (HAdd.hAdd.{0, 0, 0} ENat ENat ENat (instHAdd.{0} ENat (Distrib.toHasAdd.{0} ENat (NonUnitalNonAssocSemiring.toDistrib.{0} ENat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENat (Semiring.toNonAssocSemiring.{0} ENat (OrderedSemiring.toSemiring.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, LE.le.{0} ENat (Preorder.toLE.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (HAdd.hAdd.{0, 0, 0} ENat ENat ENat (instHAdd.{0} ENat (Distrib.toAdd.{0} ENat (NonUnitalNonAssocSemiring.toDistrib.{0} ENat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENat (Semiring.toNonAssocSemiring.{0} ENat (OrderedSemiring.toSemiring.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring)))))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) (Polynomial.trailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q))
+Case conversion may be inaccurate. Consider using '#align polynomial.le_trailing_degree_mul Polynomial.le_trailingDegree_mulₓ'. -/
 theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).trailingDegree :=
   by
   refine' Finset.le_min fun n hn => _
@@ -360,6 +574,7 @@ theorem le_trailingDegree_mul : p.trailingDegree + q.trailingDegree ≤ (p * q).
   rwa [← WithTop.coe_add, WithTop.coe_eq_coe, ← nat.mem_antidiagonal]
 #align polynomial.le_trailing_degree_mul Polynomial.le_trailingDegree_mul
 
+#print Polynomial.le_natTrailingDegree_mul /-
 theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
     p.natTrailingDegree + q.natTrailingDegree ≤ (p * q).natTrailingDegree :=
   by
@@ -369,7 +584,14 @@ theorem le_natTrailingDegree_mul (h : p * q ≠ 0) :
     trailing_degree_eq_nat_trailing_degree hq, ← trailing_degree_eq_nat_trailing_degree h]
   exact le_trailing_degree_mul
 #align polynomial.le_nat_trailing_degree_mul Polynomial.le_natTrailingDegree_mul
+-/
 
+/- warning: polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree -> Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (Polynomial.trailingCoeff.{u1} R _inst_1 p) (Polynomial.trailingCoeff.{u1} R _inst_1 q))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegreeₓ'. -/
 theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
     (p * q).coeff (p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff :=
   by
@@ -388,6 +610,12 @@ theorem coeff_mul_natTrailingDegree_add_natTrailingDegree :
   exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
 #align polynomial.coeff_mul_nat_trailing_degree_add_nat_trailing_degree Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree
 
+/- warning: polynomial.trailing_degree_mul' -> Polynomial.trailingDegree_mul' 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.trailing_degree_mul' Polynomial.trailingDegree_mul'ₓ'. -/
 theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).trailingDegree = p.trailingDegree + q.trailingDegree :=
   by
@@ -400,6 +628,12 @@ theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
   rwa [coeff_mul_nat_trailing_degree_add_nat_trailing_degree]
 #align polynomial.trailing_degree_mul' Polynomial.trailingDegree_mul'
 
+/- warning: polynomial.nat_trailing_degree_mul' -> Polynomial.natTrailingDegree_mul' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (Polynomial.trailingCoeff.{u1} R _inst_1 p) (Polynomial.trailingCoeff.{u1} R _inst_1 q)) (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 _inst_1)))))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (Ne.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Polynomial.trailingCoeff.{u1} R _inst_1 p) (Polynomial.trailingCoeff.{u1} R _inst_1 q)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q)))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_mul' Polynomial.natTrailingDegree_mul'ₓ'. -/
 theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   by
@@ -410,6 +644,12 @@ theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
     trailing_degree_eq_nat_trailing_degree hq]
 #align polynomial.nat_trailing_degree_mul' Polynomial.natTrailingDegree_mul'
 
+/- warning: polynomial.nat_trailing_degree_mul -> Polynomial.natTrailingDegree_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} [_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))))], (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1} [_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))], (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) q (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (Eq.{1} Nat (Polynomial.natTrailingDegree.{u1} R _inst_1 (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (Polynomial.natTrailingDegree.{u1} R _inst_1 p) (Polynomial.natTrailingDegree.{u1} R _inst_1 q)))
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_trailing_degree_mul Polynomial.natTrailingDegree_mulₓ'. -/
 theorem natTrailingDegree_mul [NoZeroDivisors R] (hp : p ≠ 0) (hq : q ≠ 0) :
     (p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree :=
   natTrailingDegree_mul'
@@ -422,20 +662,26 @@ section NonzeroSemiring
 
 variable [Semiring R] [Nontrivial R] {p q : R[X]}
 
+#print Polynomial.trailingDegree_one /-
 @[simp]
 theorem trailingDegree_one : trailingDegree (1 : R[X]) = (0 : ℕ∞) :=
-  trailingDegree_c one_ne_zero
+  trailingDegree_C one_ne_zero
 #align polynomial.trailing_degree_one Polynomial.trailingDegree_one
+-/
 
+#print Polynomial.trailingDegree_X /-
 @[simp]
-theorem trailingDegree_x : trailingDegree (X : R[X]) = 1 :=
+theorem trailingDegree_X : trailingDegree (X : R[X]) = 1 :=
   trailingDegree_monomial one_ne_zero
-#align polynomial.trailing_degree_X Polynomial.trailingDegree_x
+#align polynomial.trailing_degree_X Polynomial.trailingDegree_X
+-/
 
+#print Polynomial.natTrailingDegree_X /-
 @[simp]
-theorem natTrailingDegree_x : (X : R[X]).natTrailingDegree = 1 :=
+theorem natTrailingDegree_X : (X : R[X]).natTrailingDegree = 1 :=
   natTrailingDegree_monomial one_ne_zero
-#align polynomial.nat_trailing_degree_X Polynomial.natTrailingDegree_x
+#align polynomial.nat_trailing_degree_X Polynomial.natTrailingDegree_X
+-/
 
 end NonzeroSemiring
 
@@ -443,20 +689,26 @@ section Ring
 
 variable [Ring R]
 
+#print Polynomial.trailingDegree_neg /-
 @[simp]
 theorem trailingDegree_neg (p : R[X]) : trailingDegree (-p) = trailingDegree p := by
   unfold trailing_degree <;> rw [support_neg]
 #align polynomial.trailing_degree_neg Polynomial.trailingDegree_neg
+-/
 
+#print Polynomial.natTrailingDegree_neg /-
 @[simp]
 theorem natTrailingDegree_neg (p : R[X]) : natTrailingDegree (-p) = natTrailingDegree p := by
   simp [nat_trailing_degree]
 #align polynomial.nat_trailing_degree_neg Polynomial.natTrailingDegree_neg
+-/
 
+#print Polynomial.natTrailingDegree_int_cast /-
 @[simp]
 theorem natTrailingDegree_int_cast (n : ℤ) : natTrailingDegree (n : R[X]) = 0 := by
   simp only [← C_eq_int_cast, nat_trailing_degree_C]
 #align polynomial.nat_trailing_degree_int_cast Polynomial.natTrailingDegree_int_cast
+-/
 
 end Ring
 
@@ -464,18 +716,27 @@ section Semiring
 
 variable [Semiring R]
 
+#print Polynomial.nextCoeffUp /-
 /-- The second-lowest coefficient, or 0 for constants -/
 def nextCoeffUp (p : R[X]) : R :=
   if p.natTrailingDegree = 0 then 0 else p.coeff (p.natTrailingDegree + 1)
 #align polynomial.next_coeff_up Polynomial.nextCoeffUp
+-/
 
+/- warning: polynomial.next_coeff_up_C_eq_zero -> Polynomial.nextCoeffUp_C_eq_zero 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.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zeroₓ'. -/
 @[simp]
-theorem nextCoeffUp_c_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=
+theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=
   by
   rw [next_coeff_up]
   simp
-#align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_c_eq_zero
+#align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
 
+#print Polynomial.nextCoeffUp_of_pos_natTrailingDegree /-
 theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingDegree) :
     nextCoeffUp p = p.coeff (p.natTrailingDegree + 1) :=
   by
@@ -483,6 +744,7 @@ theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingD
   contrapose! hp
   simpa
 #align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_pos_natTrailingDegree
+-/
 
 end Semiring
 
@@ -490,11 +752,23 @@ section Semiring
 
 variable [Semiring R] {p q : R[X]} {ι : Type _}
 
+/- warning: polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt -> Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (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 _inst_1))))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) (Polynomial.trailingDegree.{u1} R _inst_1 q)) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 q (Polynomial.natTrailingDegree.{u1} R _inst_1 p)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
+Case conversion may be inaccurate. Consider using '#align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_ltₓ'. -/
 theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
     (h : trailingDegree p < trailingDegree q) : coeff q (natTrailingDegree p) = 0 :=
   coeff_eq_zero_of_trailingDegree_lt <| natTrailingDegree_le_trailingDegree.trans_lt h
 #align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
 
+/- warning: polynomial.ne_zero_of_trailing_degree_lt -> Polynomial.ne_zero_of_trailingDegree_lt is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedAddCommMonoid.toPartialOrder.{0} ENat (OrderedSemiring.toOrderedAddCommMonoid.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat ENat.canonicallyOrderedCommSemiring)))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {n : ENat}, (LT.lt.{0} ENat (Preorder.toLT.{0} ENat (PartialOrder.toPreorder.{0} ENat (OrderedSemiring.toPartialOrder.{0} ENat (OrderedCommSemiring.toOrderedSemiring.{0} ENat (CanonicallyOrderedCommSemiring.toOrderedCommSemiring.{0} ENat instENatCanonicallyOrderedCommSemiring))))) (Polynomial.trailingDegree.{u1} R _inst_1 p) n) -> (Ne.{succ u1} (Polynomial.{u1} R _inst_1) p (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))
+Case conversion may be inaccurate. Consider using '#align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_ltₓ'. -/
 theorem ne_zero_of_trailingDegree_lt {n : ℕ∞} (h : trailingDegree p < n) : p ≠ 0 := fun h₀ =>
   h.not_le (by simp [h₀])
 #align polynomial.ne_zero_of_trailing_degree_lt Polynomial.ne_zero_of_trailingDegree_lt

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

@@ -218,11 +218,11 @@ theorem le_trailingDegree_monomial : ↑n ≤ trailingDegree (monomial n a) :=
 #align polynomial.le_trailing_degree_monomial Polynomial.le_trailingDegree_monomial
 
 @[simp]
-theorem trailingDegree_c (ha : a ≠ 0) : trailingDegree (c a) = (0 : ℕ∞) :=
+theorem trailingDegree_c (ha : a ≠ 0) : trailingDegree (C a) = (0 : ℕ∞) :=
   trailingDegree_monomial ha
 #align polynomial.trailing_degree_C Polynomial.trailingDegree_c
 
-theorem le_trailingDegree_c : (0 : ℕ∞) ≤ trailingDegree (c a) :=
+theorem le_trailingDegree_c : (0 : ℕ∞) ≤ trailingDegree (C a) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_C Polynomial.le_trailingDegree_c
 
@@ -231,7 +231,7 @@ theorem trailingDegree_one_le : (0 : ℕ∞) ≤ trailingDegree (1 : R[X]) := by
 #align polynomial.trailing_degree_one_le Polynomial.trailingDegree_one_le
 
 @[simp]
-theorem natTrailingDegree_c (a : R) : natTrailingDegree (c a) = 0 :=
+theorem natTrailingDegree_c (a : R) : natTrailingDegree (C a) = 0 :=
   nonpos_iff_eq_zero.1 natTrailingDegree_monomial_le
 #align polynomial.nat_trailing_degree_C Polynomial.natTrailingDegree_c
 
@@ -246,11 +246,11 @@ theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0
 #align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
 
 @[simp]
-theorem trailingDegree_c_mul_x_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (c a * x ^ n) = n := by
+theorem trailingDegree_c_mul_x_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
   rw [C_mul_X_pow_eq_monomial, trailing_degree_monomial ha]
 #align polynomial.trailing_degree_C_mul_X_pow Polynomial.trailingDegree_c_mul_x_pow
 
-theorem le_trailingDegree_c_mul_x_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (c a * x ^ n) :=
+theorem le_trailingDegree_c_mul_x_pow (n : ℕ) (a : R) : (n : ℕ∞) ≤ trailingDegree (C a * X ^ n) :=
   by
   rw [C_mul_X_pow_eq_monomial]
   exact le_trailing_degree_monomial
@@ -276,15 +276,15 @@ theorem coeff_natTrailingDegree_pred_eq_zero {p : R[X]} {hp : (0 : ℕ∞) < nat
     Nat.sub_lt ((WithTop.zero_lt_coe (natTrailingDegree p)).mp hp) Nat.one_pos
 #align polynomial.coeff_nat_trailing_degree_pred_eq_zero Polynomial.coeff_natTrailingDegree_pred_eq_zero
 
-theorem le_trailingDegree_x_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (x ^ n : R[X]) := by
+theorem le_trailingDegree_x_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by
   simpa only [C_1, one_mul] using le_trailing_degree_C_mul_X_pow n (1 : R)
 #align polynomial.le_trailing_degree_X_pow Polynomial.le_trailingDegree_x_pow
 
-theorem le_trailingDegree_x : (1 : ℕ∞) ≤ trailingDegree (x : R[X]) :=
+theorem le_trailingDegree_x : (1 : ℕ∞) ≤ trailingDegree (X : R[X]) :=
   le_trailingDegree_monomial
 #align polynomial.le_trailing_degree_X Polynomial.le_trailingDegree_x
 
-theorem natTrailingDegree_x_le : (x : R[X]).natTrailingDegree ≤ 1 :=
+theorem natTrailingDegree_x_le : (X : R[X]).natTrailingDegree ≤ 1 :=
   natTrailingDegree_monomial_le
 #align polynomial.nat_trailing_degree_X_le Polynomial.natTrailingDegree_x_le
 
@@ -334,7 +334,7 @@ theorem natTrailingDegree_le_natDegree (p : R[X]) : p.natTrailingDegree ≤ p.na
 #align polynomial.nat_trailing_degree_le_nat_degree Polynomial.natTrailingDegree_le_natDegree
 
 theorem natTrailingDegree_mul_x_pow {p : R[X]} (hp : p ≠ 0) (n : ℕ) :
-    (p * x ^ n).natTrailingDegree = p.natTrailingDegree + n :=
+    (p * X ^ n).natTrailingDegree = p.natTrailingDegree + n :=
   by
   apply le_antisymm
   · refine' nat_trailing_degree_le_of_ne_zero fun h => mt trailing_coeff_eq_zero.mp hp _
@@ -428,12 +428,12 @@ theorem trailingDegree_one : trailingDegree (1 : R[X]) = (0 : ℕ∞) :=
 #align polynomial.trailing_degree_one Polynomial.trailingDegree_one
 
 @[simp]
-theorem trailingDegree_x : trailingDegree (x : R[X]) = 1 :=
+theorem trailingDegree_x : trailingDegree (X : R[X]) = 1 :=
   trailingDegree_monomial one_ne_zero
 #align polynomial.trailing_degree_X Polynomial.trailingDegree_x
 
 @[simp]
-theorem natTrailingDegree_x : (x : R[X]).natTrailingDegree = 1 :=
+theorem natTrailingDegree_x : (X : R[X]).natTrailingDegree = 1 :=
   natTrailingDegree_monomial one_ne_zero
 #align polynomial.nat_trailing_degree_X Polynomial.natTrailingDegree_x
 
@@ -470,7 +470,7 @@ def nextCoeffUp (p : R[X]) : R :=
 #align polynomial.next_coeff_up Polynomial.nextCoeffUp
 
 @[simp]
-theorem nextCoeffUp_c_eq_zero (c : R) : nextCoeffUp (c c) = 0 :=
+theorem nextCoeffUp_c_eq_zero (c : R) : nextCoeffUp (C c) = 0 :=
   by
   rw [next_coeff_up]
   simp

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

@@ -562,15 +562,28 @@ lemma natTrailingDegree_eq_zero_of_constantCoeff_ne_zero (h : constantCoeff p 
     p.natTrailingDegree = 0 :=
   le_antisymm (natTrailingDegree_le_of_ne_zero h) zero_le'
 
-lemma Monic.eq_X_pow_of_natTrailingDegree_eq_natDegree
-    (h₁ : p.Monic) (h₂ : p.natTrailingDegree = p.natDegree) :
-    p = X ^ p.natDegree := by
-  ext n
-  rw [coeff_X_pow]
-  obtain hn | rfl | hn := lt_trichotomy n p.natDegree
-  · rw [if_neg hn.ne, coeff_eq_zero_of_lt_natTrailingDegree (h₂ ▸ hn)]
-  · simpa only [if_pos rfl] using h₁.leadingCoeff
-  · rw [if_neg hn.ne', coeff_eq_zero_of_natDegree_lt hn]
+namespace Monic
+
+lemma eq_X_pow_iff_natDegree_le_natTrailingDegree (h₁ : p.Monic) :
+    p = X ^ p.natDegree ↔ p.natDegree ≤ p.natTrailingDegree := by
+  refine ⟨fun h => ?_, fun h => ?_⟩
+  · nontriviality R
+    rw [h, natTrailingDegree_X_pow, ← h]
+  · ext n
+    rw [coeff_X_pow]
+    obtain hn | rfl | hn := lt_trichotomy n p.natDegree
+    · rw [if_neg hn.ne, coeff_eq_zero_of_lt_natTrailingDegree (hn.trans_le h)]
+    · simpa only [if_pos rfl] using h₁.leadingCoeff
+    · rw [if_neg hn.ne', coeff_eq_zero_of_natDegree_lt hn]
+
+lemma eq_X_pow_iff_natTrailingDegree_eq_natDegree (h₁ : p.Monic) :
+    p = X ^ p.natDegree ↔ p.natTrailingDegree = p.natDegree :=
+  h₁.eq_X_pow_iff_natDegree_le_natTrailingDegree.trans (natTrailingDegree_le_natDegree p).ge_iff_eq
+
+@[deprecated] -- 2024-04-26
+alias ⟨_, eq_X_pow_of_natTrailingDegree_eq_natDegree⟩ := eq_X_pow_iff_natTrailingDegree_eq_natDegree
+
+end Monic
 
 end Semiring
 

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

@@ -278,9 +278,9 @@ theorem natTrailingDegree_one : natTrailingDegree (1 : R[X]) = 0 :=
 #align polynomial.nat_trailing_degree_one Polynomial.natTrailingDegree_one
 
 @[simp]
-theorem natTrailingDegree_nat_cast (n : ℕ) : natTrailingDegree (n : R[X]) = 0 := by
-  simp only [← C_eq_nat_cast, natTrailingDegree_C]
-#align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_nat_cast
+theorem natTrailingDegree_natCast (n : ℕ) : natTrailingDegree (n : R[X]) = 0 := by
+  simp only [← C_eq_natCast, natTrailingDegree_C]
+#align polynomial.nat_trailing_degree_nat_cast Polynomial.natTrailingDegree_natCast
 
 @[simp]
 theorem trailingDegree_C_mul_X_pow (n : ℕ) (ha : a ≠ 0) : trailingDegree (C a * X ^ n) = n := by
@@ -512,9 +512,9 @@ theorem natTrailingDegree_neg (p : R[X]) : natTrailingDegree (-p) = natTrailingD
 #align polynomial.nat_trailing_degree_neg Polynomial.natTrailingDegree_neg
 
 @[simp]
-theorem natTrailingDegree_int_cast (n : ℤ) : natTrailingDegree (n : R[X]) = 0 := by
-  simp only [← C_eq_int_cast, natTrailingDegree_C]
-#align polynomial.nat_trailing_degree_int_cast Polynomial.natTrailingDegree_int_cast
+theorem natTrailingDegree_intCast (n : ℤ) : natTrailingDegree (n : R[X]) = 0 := by
+  simp only [← C_eq_intCast, natTrailingDegree_C]
+#align polynomial.nat_trailing_degree_int_cast Polynomial.natTrailingDegree_intCast
 
 end Ring
 

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

@@ -67,11 +67,9 @@ def TrailingMonic (p : R[X]) :=
   trailingCoeff p = (1 : R)
 #align polynomial.trailing_monic Polynomial.TrailingMonic
 
--- Adaptation note: 2024-03-15: this was called `def`.
--- Should lean be changed to allow that as a name again?
-theorem TrailingMonic.definition : TrailingMonic p ↔ trailingCoeff p = 1 :=
+theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
   Iff.rfl
-#align polynomial.trailing_monic.def Polynomial.TrailingMonic.definition
+#align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
 
 instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) :=
   inferInstanceAs <| Decidable (trailingCoeff p = (1 : R))

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

@@ -3,8 +3,8 @@ Copyright (c) 2020 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
+import Mathlib.Algebra.Polynomial.Degree.Definitions
 import Mathlib.Data.ENat.Basic
-import Mathlib.Data.Polynomial.Degree.Definitions
 
 #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
 

Characterise when trailingDegree p = 0 and natTrailingDegree p = 0. Also fix a few names.

@@ -153,26 +153,60 @@ theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
   rw [h]
 #align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
 
-theorem le_trailingDegree_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
+theorem trailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n :=
   show @LE.le ℕ∞ _ p.support.min n from min_le (mem_support_iff.2 h)
-#align polynomial.le_trailing_degree_of_ne_zero Polynomial.le_trailingDegree_of_ne_zero
+#align polynomial.le_trailing_degree_of_ne_zero Polynomial.trailingDegree_le_of_ne_zero
 
 theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n := by
   have : WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) := rfl
   rw [← WithTop.coe_le_coe, this, ← trailingDegree_eq_natTrailingDegree]
-  · exact le_trailingDegree_of_ne_zero h
+  · exact trailingDegree_le_of_ne_zero h
   · intro h
     subst h
     exact h rfl
 #align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero
 
-theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
+@[simp] lemma coeff_natTrailingDegree_eq_zero : coeff p p.natTrailingDegree = 0 ↔ p = 0 := by
+  constructor
+  · rintro h
+    by_contra hp
+    obtain ⟨n, hpn, hn⟩ := by simpa using min_mem_image_coe $ support_nonempty.2 hp
+    obtain rfl := (trailingDegree_eq_iff_natTrailingDegree_eq hp).1 hn.symm
+    exact hpn h
+  · rintro rfl
+    simp
+
+lemma coeff_natTrailingDegree_ne_zero : coeff p p.natTrailingDegree ≠ 0 ↔ p ≠ 0 :=
+  coeff_natTrailingDegree_eq_zero.not
+
+@[simp] lemma natTrailingDegree_eq_zero : natTrailingDegree p = 0 ↔ p = 0 ∨ coeff p 0 ≠ 0 := by
+  constructor
+  · rw [or_iff_not_imp_left]
+    rintro h hp
+    rwa [← h, coeff_natTrailingDegree_ne_zero]
+  · rintro (rfl | h)
+    · simp
+    · exact nonpos_iff_eq_zero.1 $ natTrailingDegree_le_of_ne_zero h
+
+lemma trailingDegree_eq_zero : trailingDegree p = 0 ↔ coeff p 0 ≠ 0 := by
+  obtain rfl | hp := eq_or_ne p 0
+  · simp [WithTop.top_ne_zero (α := ℕ)]
+  · exact (trailingDegree_eq_iff_natTrailingDegree_eq hp).trans $
+      natTrailingDegree_eq_zero.trans $ or_iff_right hp
+
+lemma natTrailingDegree_ne_zero : natTrailingDegree p ≠ 0 ↔ p ≠ 0 ∧ coeff p 0 = 0 :=
+  natTrailingDegree_eq_zero.not.trans $ by rw [not_or, not_ne_iff]
+
+lemma trailingDegree_ne_zero : trailingDegree p ≠ 0 ↔ coeff p 0 = 0 :=
+  trailingDegree_eq_zero.not_left
+
+@[simp] theorem trailingDegree_le_trailingDegree (h : coeff q (natTrailingDegree p) ≠ 0) :
     trailingDegree q ≤ trailingDegree p := by
   by_cases hp : p = 0
   · rw [hp]
     exact le_top
   · rw [trailingDegree_eq_natTrailingDegree hp]
-    exact le_trailingDegree_of_ne_zero h
+    exact trailingDegree_le_of_ne_zero h
 #align polynomial.trailing_degree_le_trailing_degree Polynomial.trailingDegree_le_trailingDegree
 
 theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
@@ -263,13 +297,13 @@ theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) :
 set_option linter.uppercaseLean3 false in
 #align polynomial.le_trailing_degree_C_mul_X_pow Polynomial.le_trailingDegree_C_mul_X_pow
 
-theorem coeff_eq_zero_of_trailingDegree_lt (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
-  Classical.not_not.1 (mt le_trailingDegree_of_ne_zero (not_le_of_gt h))
-#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_trailingDegree_lt
+theorem coeff_eq_zero_of_lt_trailingDegree (h : (n : ℕ∞) < trailingDegree p) : coeff p n = 0 :=
+  Classical.not_not.1 (mt trailingDegree_le_of_ne_zero (not_le_of_gt h))
+#align polynomial.coeff_eq_zero_of_trailing_degree_lt Polynomial.coeff_eq_zero_of_lt_trailingDegree
 
 theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
     p.coeff n = 0 := by
-  apply coeff_eq_zero_of_trailingDegree_lt
+  apply coeff_eq_zero_of_lt_trailingDegree
   by_cases hp : p = 0
   · rw [hp, trailingDegree_zero]
     exact WithTop.coe_lt_top n
@@ -405,7 +439,7 @@ theorem trailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
   refine' le_antisymm _ le_trailingDegree_mul
   rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq, ←
     ENat.coe_add]
-  apply le_trailingDegree_of_ne_zero
+  apply trailingDegree_le_of_ne_zero
   rwa [coeff_mul_natTrailingDegree_add_natTrailingDegree]
 #align polynomial.trailing_degree_mul' Polynomial.trailingDegree_mul'
 
@@ -495,6 +529,8 @@ def nextCoeffUp (p : R[X]) : R :=
   if p.natTrailingDegree = 0 then 0 else p.coeff (p.natTrailingDegree + 1)
 #align polynomial.next_coeff_up Polynomial.nextCoeffUp
 
+@[simp] lemma nextCoeffUp_zero : nextCoeffUp (0 : R[X]) = 0 := by simp [nextCoeffUp]
+
 @[simp]
 theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by
   rw [nextCoeffUp]
@@ -502,12 +538,12 @@ theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by
 set_option linter.uppercaseLean3 false in
 #align polynomial.next_coeff_up_C_eq_zero Polynomial.nextCoeffUp_C_eq_zero
 
-theorem nextCoeffUp_of_pos_natTrailingDegree (p : R[X]) (hp : 0 < p.natTrailingDegree) :
+theorem nextCoeffUp_of_constantCoeff_eq_zero (p : R[X]) (hp : coeff p 0 = 0) :
     nextCoeffUp p = p.coeff (p.natTrailingDegree + 1) := by
-  rw [nextCoeffUp, if_neg]
-  contrapose! hp
-  simpa
-#align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_pos_natTrailingDegree
+  obtain rfl | hp₀ := eq_or_ne p 0
+  · simp
+  · rw [nextCoeffUp, if_neg (natTrailingDegree_ne_zero.2 ⟨hp₀, hp⟩)]
+#align polynomial.next_coeff_up_of_pos_nat_trailing_degree Polynomial.nextCoeffUp_of_constantCoeff_eq_zero
 
 end Semiring
 
@@ -517,7 +553,7 @@ variable [Semiring R] {p q : R[X]}
 
 theorem coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
     (h : trailingDegree p < trailingDegree q) : coeff q (natTrailingDegree p) = 0 :=
-  coeff_eq_zero_of_trailingDegree_lt <| natTrailingDegree_le_trailingDegree.trans_lt h
+  coeff_eq_zero_of_lt_trailingDegree <| natTrailingDegree_le_trailingDegree.trans_lt h
 #align polynomial.coeff_nat_trailing_degree_eq_zero_of_trailing_degree_lt Polynomial.coeff_natTrailingDegree_eq_zero_of_trailingDegree_lt
 
 theorem ne_zero_of_trailingDegree_lt {n : ℕ∞} (h : trailingDegree p < n) : p ≠ 0 := fun h₀ =>

This will become an error in 2024-03-16 nightly, possibly not permanently.

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -67,9 +67,11 @@ def TrailingMonic (p : R[X]) :=
   trailingCoeff p = (1 : R)
 #align polynomial.trailing_monic Polynomial.TrailingMonic
 
-theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 :=
+-- Adaptation note: 2024-03-15: this was called `def`.
+-- Should lean be changed to allow that as a name again?
+theorem TrailingMonic.definition : TrailingMonic p ↔ trailingCoeff p = 1 :=
   Iff.rfl
-#align polynomial.trailing_monic.def Polynomial.TrailingMonic.def
+#align polynomial.trailing_monic.def Polynomial.TrailingMonic.definition
 
 instance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) :=
   inferInstanceAs <| Decidable (trailingCoeff p = (1 : R))