data.polynomial.inductions | Mathlib porting status

Changes since the initial port

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

Changes in mathlib3

57e09a12

(last sync)

feat(algebra/monoid_algebra): add division by a generator (#15905)

This generalizes polynomial.div_X.

Zulip thread

Diff

@@ -3,6 +3,7 @@ Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 -/
+import algebra.monoid_algebra.division
 import data.nat.interval
 import data.polynomial.degree.definitions
 import data.polynomial.induction
@@ -31,21 +32,16 @@ variables [semiring R] {p q : R[X]}
 /-- `div_X p` returns a polynomial `q` such that `q * X + C (p.coeff 0) = p`.
   It can be used in a semiring where the usual division algorithm is not possible -/
 def div_X (p : R[X]) : R[X] :=
-∑ n in Ico 0 p.nat_degree, monomial n (p.coeff (n + 1))
+⟨add_monoid_algebra.div_of p.to_finsupp 1⟩
 
 @[simp] lemma coeff_div_X : (div_X p).coeff n = p.coeff (n+1) :=
-begin
-  simp only [div_X, coeff_monomial, true_and, finset_sum_coeff, not_lt,
-    mem_Ico, zero_le, finset.sum_ite_eq', ite_eq_left_iff],
-  intro h,
-  rw coeff_eq_zero_of_nat_degree_lt (nat.lt_succ_of_le h)
-end
+by { rw [add_comm], cases p, refl }
 
 lemma div_X_mul_X_add (p : R[X]) : div_X p * X + C (p.coeff 0) = p :=
 ext $ by rintro ⟨_|_⟩; simp [coeff_C, nat.succ_ne_zero, coeff_mul_X]
 
 @[simp] lemma div_X_C (a : R) : div_X (C a) = 0 :=
-ext $ λ n, by simp [div_X, coeff_C]; simp [coeff]
+ext $ λ n, by simp [coeff_div_X, coeff_C, finsupp.single_eq_of_ne _]
 
 lemma div_X_eq_zero_iff : div_X p = 0 ↔ p = C (p.coeff 0) :=
 ⟨λ h, by simpa [eq_comm, h] using div_X_mul_X_add p,

70fd9563

(first ported)

Changes in mathlib3port

mathlib3

mathlib3port

57e09a12

bump to nightly-2024-04-07-08

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

b03b8656

Diff

@@ -5,8 +5,8 @@ Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wage
 -/
 import Algebra.MonoidAlgebra.Division
 import Data.Nat.Interval
-import Data.Polynomial.Degree.Definitions
-import Data.Polynomial.Induction
+import Algebra.Polynomial.Degree.Definitions
+import Algebra.Polynomial.Induction
 
 #align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"

57e09a12

bump to nightly-2024-03-17-08

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

22f01ce4

Diff

@@ -147,11 +147,11 @@ See `nat_degree_ne_zero_induction_on` for a similar statement involving no expli
 theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree p)
     (hC : ∀ {a}, a ≠ 0 → P (C a * X)) (hX : ∀ {p}, 0 < degree p → P p → P (p * X))
     (hadd : ∀ {p} {a}, 0 < degree p → P p → P (p + C a)) : P p :=
-  recOnHorner p (fun h => by rw [degree_zero] at h  <;> exact absurd h (by decide))
+  recOnHorner p (fun h => by rw [degree_zero] at h <;> exact absurd h (by decide))
     (fun p a _ _ ih h0 =>
       have : 0 < degree p :=
         lt_of_not_ge fun h =>
-          not_lt_of_ge degree_C_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0 
+          not_lt_of_ge degree_C_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0
       hadd this (ih this))
     (fun p _ ih h0' =>
       if h0 : 0 < degree p then hX h0 (ih h0)

57e09a12

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,10 +3,10 @@ Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 -/
-import Mathbin.Algebra.MonoidAlgebra.Division
-import Mathbin.Data.Nat.Interval
-import Mathbin.Data.Polynomial.Degree.Definitions
-import Mathbin.Data.Polynomial.Induction
+import Algebra.MonoidAlgebra.Division
+import Data.Nat.Interval
+import Data.Polynomial.Degree.Definitions
+import Data.Polynomial.Induction
 
 #align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"

57e09a12

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
-
-! This file was ported from Lean 3 source module data.polynomial.inductions
-! leanprover-community/mathlib commit 57e09a1296bfb4330ddf6624f1028ba186117d82
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.MonoidAlgebra.Division
 import Mathbin.Data.Nat.Interval
 import Mathbin.Data.Polynomial.Degree.Definitions
 import Mathbin.Data.Polynomial.Induction
 
+#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"
+
 /-!
 # Induction on polynomials

57e09a12

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -53,18 +53,24 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by rw [add_comm]; cas
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 -/
 
+#print Polynomial.divX_mul_X_add /-
 theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
 #align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
+-/
 
+#print Polynomial.divX_C /-
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
   ext fun n => by simp [coeff_div_X, coeff_C, Finsupp.single_eq_of_ne _]
 #align polynomial.div_X_C Polynomial.divX_C
+-/
 
+#print Polynomial.divX_eq_zero_iff /-
 theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
 #align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
+-/
 
 #print Polynomial.divX_add /-
 theorem divX_add : divX (p + q) = divX p + divX q :=
@@ -105,6 +111,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
 -/
 
+#print Polynomial.recOnHorner /-
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
 noncomputable def recOnHorner {M : R[X] → Sort _} :
@@ -127,7 +134,9 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
                 show M (div_X p * X) by rw [hpX0, MulZeroClass.zero_mul] <;> exact M0
               else MX (div_X p) hpX0 (rec_on_horner _ M0 MC MX))
 #align polynomial.rec_on_horner Polynomial.recOnHorner
+-/
 
+#print Polynomial.degree_pos_induction_on /-
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for
 * `a * X`, with `a ∈ R`,
@@ -154,7 +163,9 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
           exact hC fun h : coeff p 0 = 0 => by simpa [h, Nat.not_lt_zero] using h0')
     h0
 #align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_on
+-/
 
+#print Polynomial.natDegree_ne_zero_induction_on /-
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for
 * `p + a`, with `a ∈ R`, `p ∈ R[X]`,
@@ -190,6 +201,7 @@ theorem natDegree_ne_zero_induction_on {M : R[X] → Prop} {f : R[X]} (f0 : f.na
       rw [C_mul_X_pow_eq_monomial]
       exact h_monomial a0 n.succ_ne_zero
 #align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_on
+-/
 
 end Semiring

57e09a12

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -100,10 +100,8 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
                         rw [zero_le_degree_iff, Ne.def, div_X_eq_zero_iff] <;>
                           exact fun h0 => h (h0.symm ▸ degree_C_le))
                       le_rfl
-              
           rw [degree_add_eq_left_of_degree_lt this] <;> exact degree_lt_degree_mul_X hXp0
       _ = p.degree := congr_arg _ (div_X_mul_X_add _)
-      
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
 -/

57e09a12

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -143,11 +143,11 @@ See `nat_degree_ne_zero_induction_on` for a similar statement involving no expli
 theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree p)
     (hC : ∀ {a}, a ≠ 0 → P (C a * X)) (hX : ∀ {p}, 0 < degree p → P p → P (p * X))
     (hadd : ∀ {p} {a}, 0 < degree p → P p → P (p + C a)) : P p :=
-  recOnHorner p (fun h => by rw [degree_zero] at h <;> exact absurd h (by decide))
+  recOnHorner p (fun h => by rw [degree_zero] at h  <;> exact absurd h (by decide))
     (fun p a _ _ ih h0 =>
       have : 0 < degree p :=
         lt_of_not_ge fun h =>
-          not_lt_of_ge degree_C_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0
+          not_lt_of_ge degree_C_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0 
       hadd this (ih this))
     (fun p _ ih h0' =>
       if h0 : 0 < degree p then hX h0 (ih h0)

57e09a12

bump to nightly-2023-05-28-18

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

8d353152

Diff

@@ -25,7 +25,7 @@ This file contains lemmas dealing with different flavours of induction on polyno
 
 noncomputable section
 
-open Classical BigOperators Polynomial
+open scoped Classical BigOperators Polynomial
 
 open Finset
 
@@ -72,6 +72,7 @@ theorem divX_add : divX (p + q) = divX p + divX q :=
 #align polynomial.div_X_add Polynomial.divX_add
 -/
 
+#print Polynomial.degree_divX_lt /-
 theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := nontrivial.of_polynomial_ne hp0 <;>
     calc
@@ -104,6 +105,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
       _ = p.degree := congr_arg _ (div_X_mul_X_add _)
       
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
+-/
 
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]

57e09a12

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -53,33 +53,15 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by rw [add_comm]; cas
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 -/
 
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-Case conversion may be inaccurate. Consider using '#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_addₓ'. -/
 theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
 #align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
 
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 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
   ext fun n => by simp [coeff_div_X, coeff_C, Finsupp.single_eq_of_ne _]
 #align polynomial.div_X_C Polynomial.divX_C
 
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 theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
 #align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
@@ -90,12 +72,6 @@ theorem divX_add : divX (p + q) = divX p + divX q :=
 #align polynomial.div_X_add Polynomial.divX_add
 -/
 
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 theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := nontrivial.of_polynomial_ne hp0 <;>
     calc
@@ -129,12 +105,6 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
       
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
 
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 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
 noncomputable def recOnHorner {M : R[X] → Sort _} :
@@ -158,9 +128,6 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
               else MX (div_X p) hpX0 (rec_on_horner _ M0 MC MX))
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
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 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for
 * `a * X`, with `a ∈ R`,
@@ -188,12 +155,6 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
     h0
 #align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_on
 
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-Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for
 * `p + a`, with `a ∈ R`, `p ∈ R[X]`,

57e09a12

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -49,11 +49,7 @@ def divX (p : R[X]) : R[X] :=
 
 #print Polynomial.coeff_divX /-
 @[simp]
-theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
-  by
-  rw [add_comm]
-  cases p
-  rfl
+theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by rw [add_comm]; cases p; rfl
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 -/

57e09a12

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -163,10 +163,7 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
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(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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+<too large>
 Case conversion may be inaccurate. Consider using '#align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_onₓ'. -/
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for

57e09a12

bump to nightly-2023-05-24-06

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

fbc7b476

Diff

@@ -199,7 +199,7 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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)))))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (M (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))) -> (M f)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for

57e09a12

bump to nightly-2023-05-19-16

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

a31df521

Diff

@@ -61,7 +61,7 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))))) p
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) p
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) p
 Case conversion may be inaccurate. Consider using '#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_addₓ'. -/
 theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
@@ -71,7 +71,7 @@ theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{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] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{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.div_X_C Polynomial.divX_Cₓ'. -/
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
@@ -82,7 +82,7 @@ theorem divX_C (a : R) : divX (C a) = 0 :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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))))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (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] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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)))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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)))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))
 Case conversion may be inaccurate. Consider using '#align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iffₓ'. -/
 theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
@@ -137,7 +137,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (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 (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (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} 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)))))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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))))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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)))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (forall (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)))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 Case conversion may be inaccurate. Consider using '#align polynomial.rec_on_horner Polynomial.recOnHornerₓ'. -/
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
@@ -166,7 +166,7 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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)))))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 Case conversion may be inaccurate. Consider using '#align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_onₓ'. -/
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for
@@ -199,7 +199,7 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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)))))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (M (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))) -> (M f)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for

57e09a12

bump to nightly-2023-05-18-03

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

b46e9d53

Diff

@@ -199,7 +199,7 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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)))))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (M (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))) -> (M f)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for

57e09a12

bump to nightly-2023-05-16-16

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

9cb92e8c

Diff

@@ -94,7 +94,12 @@ theorem divX_add : divX (p + q) = divX p + divX q :=
 #align polynomial.div_X_add Polynomial.divX_add
 -/
 
-#print Polynomial.degree_divX_lt /-
+/- warning: polynomial.degree_div_X_lt -> Polynomial.degree_divX_lt is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_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))))) -> (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (Polynomial.degree.{u1} R _inst_1 (Polynomial.divX.{u1} R _inst_1 p)) (Polynomial.degree.{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}, (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)))) -> (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (Polynomial.degree.{u1} R _inst_1 (Polynomial.divX.{u1} R _inst_1 p)) (Polynomial.degree.{u1} R _inst_1 p))
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_div_X_lt Polynomial.degree_divX_ltₓ'. -/
 theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := nontrivial.of_polynomial_ne hp0 <;>
     calc
@@ -127,7 +132,6 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
       _ = p.degree := congr_arg _ (div_X_mul_X_add _)
       
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
--/
 
 /- warning: polynomial.rec_on_horner -> Polynomial.recOnHorner is a dubious translation:
 lean 3 declaration is
@@ -160,7 +164,7 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
 
 /- warning: polynomial.degree_pos_induction_on -> Polynomial.degree_pos_induction_on is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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)))))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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)))))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toHasLt.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 Case conversion may be inaccurate. Consider using '#align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_onₓ'. -/

57e09a12

bump to nightly-2023-03-23-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/57e09a1296bfb4330ddf6624f1028ba186117d82

9354dcbd

Diff

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 
 ! This file was ported from Lean 3 source module data.polynomial.inductions
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
+! leanprover-community/mathlib commit 57e09a1296bfb4330ddf6624f1028ba186117d82
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Algebra.MonoidAlgebra.Division
 import Mathbin.Data.Nat.Interval
 import Mathbin.Data.Polynomial.Degree.Definitions
 import Mathbin.Data.Polynomial.Induction
@@ -42,7 +43,7 @@ variable [Semiring R] {p q : R[X]}
 /-- `div_X p` returns a polynomial `q` such that `q * X + C (p.coeff 0) = p`.
   It can be used in a semiring where the usual division algorithm is not possible -/
 def divX (p : R[X]) : R[X] :=
-  ∑ n in Ico 0 p.natDegree, monomial n (p.coeff (n + 1))
+  ⟨AddMonoidAlgebra.divOf p.toFinsupp 1⟩
 #align polynomial.div_X Polynomial.divX
 -/
 
@@ -50,10 +51,9 @@ def divX (p : R[X]) : R[X] :=
 @[simp]
 theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
   by
-  simp only [div_X, coeff_monomial, true_and_iff, finset_sum_coeff, not_lt, mem_Ico, zero_le,
-    Finset.sum_ite_eq', ite_eq_left_iff]
-  intro h
-  rw [coeff_eq_zero_of_nat_degree_lt (Nat.lt_succ_of_le h)]
+  rw [add_comm]
+  cases p
+  rfl
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 -/
 
@@ -75,7 +75,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align polynomial.div_X_C Polynomial.divX_Cₓ'. -/
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
-  ext fun n => by simp [div_X, coeff_C] <;> simp [coeff]
+  ext fun n => by simp [coeff_div_X, coeff_C, Finsupp.single_eq_of_ne _]
 #align polynomial.div_X_C Polynomial.divX_C
 
 /- warning: polynomial.div_X_eq_zero_iff -> Polynomial.divX_eq_zero_iff is a dubious translation:

69c6a5a1

bump to nightly-2023-03-14-02

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

d4b38a42

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 
 ! This file was ported from Lean 3 source module data.polynomial.inductions
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Data.Polynomial.Induction
 /-!
 # Induction on polynomials
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file contains lemmas dealing with different flavours of induction on polynomials.
 -/

70fd9563

bump to nightly-2023-03-11-22

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

a8ee822f

Diff

@@ -58,7 +58,7 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))))) p
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) p
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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)) (Polynomial.divX.{u1} R _inst_1 p) (Polynomial.X.{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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))) p
 Case conversion may be inaccurate. Consider using '#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_addₓ'. -/
 theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
@@ -68,7 +68,7 @@ theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{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] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : R), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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.{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.div_X_C Polynomial.divX_Cₓ'. -/
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
@@ -79,7 +79,7 @@ theorem divX_C (a : R) : divX (C a) = 0 :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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))))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (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] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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)))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.divX.{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)))) (Eq.{succ u1} (Polynomial.{u1} R _inst_1) p (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) (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))
 Case conversion may be inaccurate. Consider using '#align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iffₓ'. -/
 theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
@@ -130,7 +130,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (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 (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (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} 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)))))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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))))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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)))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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)))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 Case conversion may be inaccurate. Consider using '#align polynomial.rec_on_horner Polynomial.recOnHornerₓ'. -/
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
@@ -159,7 +159,7 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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)))))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
 Case conversion may be inaccurate. Consider using '#align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_onₓ'. -/
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for
@@ -192,7 +192,7 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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)))))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (M (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))) -> (M f)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
 Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for

70fd9563

bump to nightly-2023-03-11-16

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

cd44fb92

Diff

@@ -99,7 +99,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
         if h : degree p ≤ 0 then
           by
           have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
-          rw [eq_C_of_degree_le_zero h, div_X_C, degree_zero, zero_mul, zero_add]
+          rw [eq_C_of_degree_le_zero h, div_X_C, degree_zero, MulZeroClass.zero_mul, zero_add]
           exact lt_of_le_of_ne bot_le (Ne.symm (mt degree_eq_bot.1 <| by simp [h']))
         else
           by
@@ -150,7 +150,8 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
                 MX _ (fun h : div_X p = 0 => by simpa [h, hcp0] using hp) (rec_on_horner _ M0 MC MX)
           else
             MC _ _ (coeff_mul_X_zero _) hcp0
-              (if hpX0 : div_X p = 0 then show M (div_X p * X) by rw [hpX0, zero_mul] <;> exact M0
+              (if hpX0 : div_X p = 0 then
+                show M (div_X p * X) by rw [hpX0, MulZeroClass.zero_mul] <;> exact M0
               else MX (div_X p) hpX0 (rec_on_horner _ M0 MC MX))
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
@@ -223,7 +224,7 @@ theorem natDegree_ne_zero_induction_on {M : R[X] → Prop} {f : R[X]} (f0 : f.na
     · exact Or.inr (h_add hp hq)
   · intro n a hi
     by_cases a0 : a = 0
-    · exact Or.inl (by rw [a0, C_0, zero_mul, nat_degree_zero])
+    · exact Or.inl (by rw [a0, C_0, MulZeroClass.zero_mul, nat_degree_zero])
     · refine' Or.inr _
       rw [C_mul_X_pow_eq_monomial]
       exact h_monomial a0 n.succ_ne_zero

70fd9563

bump to nightly-2023-03-09-18

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

3063fab7

Diff

@@ -35,12 +35,15 @@ section Semiring
 
 variable [Semiring R] {p q : R[X]}
 
+#print Polynomial.divX /-
 /-- `div_X p` returns a polynomial `q` such that `q * X + C (p.coeff 0) = p`.
   It can be used in a semiring where the usual division algorithm is not possible -/
 def divX (p : R[X]) : R[X] :=
   ∑ n in Ico 0 p.natDegree, monomial n (p.coeff (n + 1))
 #align polynomial.div_X Polynomial.divX
+-/
 
+#print Polynomial.coeff_divX /-
 @[simp]
 theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
   by
@@ -49,24 +52,46 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
   intro h
   rw [coeff_eq_zero_of_nat_degree_lt (Nat.lt_succ_of_le h)]
 #align polynomial.coeff_div_X Polynomial.coeff_divX
+-/
 
-theorem divX_mul_x_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
+/- warning: polynomial.div_X_mul_X_add -> Polynomial.divX_mul_X_add is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_addₓ'. -/
+theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
-#align polynomial.div_X_mul_X_add Polynomial.divX_mul_x_add
+#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
 
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+Case conversion may be inaccurate. Consider using '#align polynomial.div_X_C Polynomial.divX_Cₓ'. -/
 @[simp]
-theorem divX_c (a : R) : divX (C a) = 0 :=
+theorem divX_C (a : R) : divX (C a) = 0 :=
   ext fun n => by simp [div_X, coeff_C] <;> simp [coeff]
-#align polynomial.div_X_C Polynomial.divX_c
+#align polynomial.div_X_C Polynomial.divX_C
 
+/- warning: polynomial.div_X_eq_zero_iff -> Polynomial.divX_eq_zero_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iffₓ'. -/
 theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
 #align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
 
+#print Polynomial.divX_add /-
 theorem divX_add : divX (p + q) = divX p + divX q :=
   ext <| by simp
 #align polynomial.div_X_add Polynomial.divX_add
+-/
 
+#print Polynomial.degree_divX_lt /-
 theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := nontrivial.of_polynomial_ne hp0 <;>
     calc
@@ -99,12 +124,13 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
       _ = p.degree := congr_arg _ (div_X_mul_X_add _)
       
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
+-/
 
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 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {M : (Polynomial.{u2} R _inst_1) -> Sort.{u1}} (p : Polynomial.{u2} R _inst_1), (M (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1))))) -> (forall (p : Polynomial.{u2} R _inst_1) (a : R), (Eq.{succ u2} R (Polynomial.coeff.{u2} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (M p) -> (M (HAdd.hAdd.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHAdd.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.add'.{u2} R _inst_1)) p (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (fun (_x : RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) => R -> (Polynomial.{u2} R _inst_1)) (RingHom.hasCoeToFun.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (Polynomial.C.{u2} R _inst_1) a)))) -> (forall (p : Polynomial.{u2} R _inst_1), (Ne.{succ u2} (Polynomial.{u2} R _inst_1) p (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1))))) -> (M p) -> (M (HMul.hMul.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHMul.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.mul'.{u2} R _inst_1)) p (Polynomial.X.{u2} R _inst_1)))) -> (M p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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)))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 Case conversion may be inaccurate. Consider using '#align polynomial.rec_on_horner Polynomial.recOnHornerₓ'. -/
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
@@ -128,6 +154,12 @@ noncomputable def recOnHorner {M : R[X] → Sort _} :
               else MX (div_X p) hpX0 (rec_on_horner _ M0 MC MX))
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
+/- warning: polynomial.degree_pos_induction_on -> Polynomial.degree_pos_induction_on is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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)))))))) -> (P (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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring))))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (OfNat.mk.{0} (WithBot.{0} Nat) 0 (Zero.zero.{0} (WithBot.{0} Nat) (WithBot.hasZero.{0} Nat Nat.hasZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {P : (Polynomial.{u1} R _inst_1) -> Prop} (p : Polynomial.{u1} R _inst_1), (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (forall {a : 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))))) -> (P (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) 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(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) (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (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 (Polynomial.X.{u1} R _inst_1)))) -> (forall {p : Polynomial.{u1} R _inst_1} {a : R}, (LT.lt.{0} (WithBot.{0} Nat) (Preorder.toLT.{0} (WithBot.{0} Nat) (WithBot.preorder.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring)))) (OfNat.ofNat.{0} (WithBot.{0} Nat) 0 (Zero.toOfNat0.{0} (WithBot.{0} Nat) (WithBot.zero.{0} Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)))) (Polynomial.degree.{u1} R _inst_1 p)) -> (P p) -> (P (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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)))) -> (P p)
+Case conversion may be inaccurate. Consider using '#align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_onₓ'. -/
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`
 if it holds for
 * `a * X`, with `a ∈ R`,
@@ -155,6 +187,12 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
     h0
 #align polynomial.degree_pos_induction_on Polynomial.degree_pos_induction_on
 
+/- warning: polynomial.nat_degree_ne_zero_induction_on -> Polynomial.natDegree_ne_zero_induction_on is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Prop} {f : Polynomial.{u1} R _inst_1}, (Ne.{1} Nat (Polynomial.natDegree.{u1} R _inst_1 f) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall {a : R} {p : Polynomial.{u1} R _inst_1}, (M p) -> (M (HAdd.hAdd.{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) (instHAdd.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.add'.{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) p))) -> (forall {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, (M p) -> (M q) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p q))) -> (forall {n : Nat} {a : 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))))) -> (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (M (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))) -> (M f)
+Case conversion may be inaccurate. Consider using '#align polynomial.nat_degree_ne_zero_induction_on Polynomial.natDegree_ne_zero_induction_onₓ'. -/
 /-- A property holds for all polynomials of non-zero `nat_degree` with coefficients in a
 semiring `R` if it holds for
 * `p + a`, with `a ∈ R`, `p ∈ R[X]`,

70fd9563

bump to nightly-2023-03-08-12

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

f8a90967

Diff

@@ -145,7 +145,7 @@ theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree
     (fun p a _ _ ih h0 =>
       have : 0 < degree p :=
         lt_of_not_ge fun h =>
-          not_lt_of_ge degree_c_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0
+          not_lt_of_ge degree_C_le <| by rwa [eq_C_of_degree_le_zero h, ← C_add] at h0
       hadd this (ih this))
     (fun p _ ih h0' =>
       if h0 : 0 < degree p then hX h0 (ih h0)

70fd9563

bump to nightly-2023-03-08-10

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

422cc012

Diff

@@ -50,16 +50,16 @@ theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) :=
   rw [coeff_eq_zero_of_nat_degree_lt (Nat.lt_succ_of_le h)]
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 
-theorem divX_mul_x_add (p : R[X]) : divX p * x + c (p.coeff 0) = p :=
+theorem divX_mul_x_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
   ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
 #align polynomial.div_X_mul_X_add Polynomial.divX_mul_x_add
 
 @[simp]
-theorem divX_c (a : R) : divX (c a) = 0 :=
+theorem divX_c (a : R) : divX (C a) = 0 :=
   ext fun n => by simp [div_X, coeff_C] <;> simp [coeff]
 #align polynomial.div_X_C Polynomial.divX_c
 
-theorem divX_eq_zero_iff : divX p = 0 ↔ p = c (p.coeff 0) :=
+theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
   ⟨fun h => by simpa [eq_comm, h] using div_X_mul_X_add p, fun h => by rw [h, div_X_C]⟩
 #align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
 
@@ -102,16 +102,16 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
 
 /- warning: polynomial.rec_on_horner -> Polynomial.recOnHorner is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (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.hasZero.{u1} R _inst_1))))) -> (forall (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (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} 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)))))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.hasAdd.{u1} R _inst_1)) p (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 (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.hasZero.{u1} R _inst_1))))) -> (M p) -> (M (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.hasMul.{u1} R _inst_1)) p (Polynomial.x.{u1} R _inst_1)))) -> (M p)
+  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : (Polynomial.{u1} R _inst_1) -> Sort.{u2}} (p : Polynomial.{u1} R _inst_1), (M (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 (p : Polynomial.{u1} R _inst_1) (a : R), (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (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} 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)))))))) -> (M p) -> (M (HAdd.hAdd.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHAdd.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1)) p (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 (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))))) -> (M p) -> (M (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 (Polynomial.X.{u1} R _inst_1)))) -> (M p)
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {M : (Polynomial.{u2} R _inst_1) -> Sort.{u1}} (p : Polynomial.{u2} R _inst_1), (M (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.hasZero.{u2} R _inst_1))))) -> (forall (p : Polynomial.{u2} R _inst_1) (a : R), (Eq.{succ u2} R (Polynomial.coeff.{u2} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (M p) -> (M (HAdd.hAdd.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHAdd.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.hasAdd.{u2} R _inst_1)) p (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (fun (_x : RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) => R -> (Polynomial.{u2} R _inst_1)) (RingHom.hasCoeToFun.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (Polynomial.c.{u2} R _inst_1) a)))) -> (forall (p : Polynomial.{u2} R _inst_1), (Ne.{succ u2} (Polynomial.{u2} R _inst_1) p (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.hasZero.{u2} R _inst_1))))) -> (M p) -> (M (HMul.hMul.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHMul.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.hasMul.{u2} R _inst_1)) p (Polynomial.x.{u2} R _inst_1)))) -> (M p)
+  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {M : (Polynomial.{u2} R _inst_1) -> Sort.{u1}} (p : Polynomial.{u2} R _inst_1), (M (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1))))) -> (forall (p : Polynomial.{u2} R _inst_1) (a : R), (Eq.{succ u2} R (Polynomial.coeff.{u2} R _inst_1 p (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) -> (M p) -> (M (HAdd.hAdd.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHAdd.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.add'.{u2} R _inst_1)) p (coeFn.{succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (fun (_x : RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) => R -> (Polynomial.{u2} R _inst_1)) (RingHom.hasCoeToFun.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (Polynomial.C.{u2} R _inst_1) a)))) -> (forall (p : Polynomial.{u2} R _inst_1), (Ne.{succ u2} (Polynomial.{u2} R _inst_1) p (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (OfNat.mk.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.zero.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1))))) -> (M p) -> (M (HMul.hMul.{u2, u2, u2} (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHMul.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.mul'.{u2} R _inst_1)) p (Polynomial.X.{u2} R _inst_1)))) -> (M p)
 Case conversion may be inaccurate. Consider using '#align polynomial.rec_on_horner Polynomial.recOnHornerₓ'. -/
 /-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
 noncomputable def recOnHorner {M : R[X] → Sort _} :
     ∀ p : R[X],
       M 0 →
-        (∀ p a, coeff p 0 = 0 → a ≠ 0 → M p → M (p + c a)) → (∀ p, p ≠ 0 → M p → M (p * x)) → M p
+        (∀ p a, coeff p 0 = 0 → a ≠ 0 → M p → M (p + C a)) → (∀ p, p ≠ 0 → M p → M (p * X)) → M p
   | p => fun M0 MC MX =>
     if hp : p = 0 then Eq.recOn hp.symm M0
     else by
@@ -139,8 +139,8 @@ See `nat_degree_ne_zero_induction_on` for a similar statement involving no expli
  -/
 @[elab_as_elim]
 theorem degree_pos_induction_on {P : R[X] → Prop} (p : R[X]) (h0 : 0 < degree p)
-    (hC : ∀ {a}, a ≠ 0 → P (c a * x)) (hX : ∀ {p}, 0 < degree p → P p → P (p * x))
-    (hadd : ∀ {p} {a}, 0 < degree p → P p → P (p + c a)) : P p :=
+    (hC : ∀ {a}, a ≠ 0 → P (C a * X)) (hX : ∀ {p}, 0 < degree p → P p → P (p * X))
+    (hadd : ∀ {p} {a}, 0 < degree p → P p → P (p + C a)) : P p :=
   recOnHorner p (fun h => by rw [degree_zero] at h <;> exact absurd h (by decide))
     (fun p a _ _ ih h0 =>
       have : 0 < degree p :=
@@ -167,7 +167,7 @@ See `degree_pos_induction_on` for a similar statement involving more explicit mu
  -/
 @[elab_as_elim]
 theorem natDegree_ne_zero_induction_on {M : R[X] → Prop} {f : R[X]} (f0 : f.natDegree ≠ 0)
-    (h_C_add : ∀ {a p}, M p → M (c a + p)) (h_add : ∀ {p q}, M p → M q → M (p + q))
+    (h_C_add : ∀ {a p}, M p → M (C a + p)) (h_add : ∀ {p q}, M p → M q → M (p + q))
     (h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (monomial n a)) : M f :=
   by
   suffices f.natDegree = 0 ∨ M f from Or.dcases_on this (fun h => (f0 h).elim) id

70fd9563

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

57e09a12

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

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

56e439ec

Diff

@@ -4,10 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 -/
 import Mathlib.Algebra.MonoidAlgebra.Division
+import Mathlib.Algebra.Polynomial.Degree.Definitions
+import Mathlib.Algebra.Polynomial.Induction
+import Mathlib.Algebra.Polynomial.EraseLead
 import Mathlib.Data.Nat.Interval
-import Mathlib.Data.Polynomial.Degree.Definitions
-import Mathlib.Data.Polynomial.Induction
-import Mathlib.Data.Polynomial.EraseLead
 
 #align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"

57e09a12

chore: avoid Ne.def (adaptation for nightly-2024-03-27) (#11801)

1b40c7bc

Diff

@@ -136,7 +136,7 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
             _ ≤ degree (divX p * X) := by
               rw [← zero_add (degree X), degree_mul' this]
               exact add_le_add
-                (by rw [zero_le_degree_iff, Ne.def, divX_eq_zero_iff]
+                (by rw [zero_le_degree_iff, Ne, divX_eq_zero_iff]
                     exact fun h0 => h (h0.symm ▸ degree_C_le))
                     le_rfl
         rw [degree_add_eq_left_of_degree_lt this]; exact degree_lt_degree_mul_X hXp0

57e09a12

chore: move to v4.6.0-rc1, merging adaptations from bump/v4.6.0 (#10176)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

490d2d48

Diff

@@ -163,7 +163,7 @@ noncomputable def recOnHorner {M : R[X] → Sort*} (p : R[X]) (M0 : M 0)
         MC _ _ (coeff_mul_X_zero _) hcp0
           (if hpX0 : divX p = 0 then show M (divX p * X) by rw [hpX0, zero_mul]; exact M0
           else MX (divX p) hpX0 (recOnHorner _ M0 MC MX))
-termination_by _ => p.degree
+termination_by p.degree
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`

57e09a12

chore: tidy various files (#8409)

f5f44c59

Diff

@@ -94,9 +94,9 @@ theorem divX_X_pow : divX (X ^ n : R[X]) = if (n = 0) then 0 else X ^ (n - 1) :=
 /-- `divX` as an additive homomorphism. -/
 noncomputable
 def divX_hom : R[X] →+ R[X] :=
-{ toFun := divX
-  map_zero' := divX_zero
-  map_add' := fun _ _ => divX_add }
+  { toFun := divX
+    map_zero' := divX_zero
+    map_add' := fun _ _ => divX_add }
 
 @[simp] theorem divX_hom_toFun : divX_hom p = divX p := rfl
 
@@ -111,7 +111,7 @@ theorem natDegree_divX_eq_natDegree_tsub_one : p.divX.natDegree = p.natDegree -
     · exact natDegree_C_mul_X_pow (n - 1) c c0
 
 theorem natDegree_divX_le : p.divX.natDegree ≤ p.natDegree :=
-natDegree_divX_eq_natDegree_tsub_one.trans_le (Nat.pred_le _)
+  natDegree_divX_eq_natDegree_tsub_one.trans_le (Nat.pred_le _)
 
 theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by
   simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]

57e09a12

refactor(Data/Polynomial): remove open Classical (#7706)

This doesn't change any polynomial operations, but:

Makes some Decidable values computable (otherwise, they're pointless!)
Add a few missing arguments to lemmas here and there to make them more general

This is exhaustive, within the directories it touches.

Once again, the use of letI := Classical.decEq R instead of classical here is because of the weird style of proofs in these files, where if is preferred to by_cases.

4a7ff0e3

Diff

@@ -20,7 +20,7 @@ This file contains lemmas dealing with different flavours of induction on polyno
 
 noncomputable section
 
-open Classical BigOperators Polynomial
+open BigOperators Polynomial
 
 open Finset
 
@@ -149,6 +149,7 @@ set_option linter.uppercaseLean3 false in
 noncomputable def recOnHorner {M : R[X] → Sort*} (p : R[X]) (M0 : M 0)
     (MC : ∀ p a, coeff p 0 = 0 → a ≠ 0 → M p → M (p + C a))
     (MX : ∀ p, p ≠ 0 → M p → M (p * X)) : M p :=
+  letI := Classical.decEq R
   if hp : p = 0 then hp ▸ M0
   else by
     have wf : degree (divX p) < degree p := degree_divX_lt hp

57e09a12

feat: expand API for Polynomial.divX and some related results (#6428)

Various lemmas about Polynomial.natDegree

Co-authored-by: Pratyush Mishra

48d20bd1

Diff

@@ -7,6 +7,7 @@ import Mathlib.Algebra.MonoidAlgebra.Division
 import Mathlib.Data.Nat.Interval
 import Mathlib.Data.Polynomial.Degree.Definitions
 import Mathlib.Data.Polynomial.Induction
+import Mathlib.Data.Polynomial.EraseLead
 
 #align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"
 
@@ -51,6 +52,10 @@ theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
 set_option linter.uppercaseLean3 false in
 #align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
 
+@[simp]
+theorem X_mul_divX_add (p : R[X]) : X * divX p + C (p.coeff 0) = p :=
+  ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
+
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
   ext fun n => by simp [coeff_divX, coeff_C, Finsupp.single_eq_of_ne _]
@@ -67,6 +72,50 @@ theorem divX_add : divX (p + q) = divX p + divX q :=
 set_option linter.uppercaseLean3 false in
 #align polynomial.div_X_add Polynomial.divX_add
 
+@[simp]
+theorem divX_zero : divX (0 : R[X]) = 0 := leadingCoeff_eq_zero.mp rfl
+
+@[simp]
+theorem divX_one : divX (1 : R[X]) = 0 := by
+  ext
+  simpa only [coeff_divX, coeff_zero] using coeff_one
+
+@[simp]
+theorem divX_C_mul : divX (C a * p) = C a * divX p := by
+  ext
+  simp
+
+theorem divX_X_pow : divX (X ^ n : R[X]) = if (n = 0) then 0 else X ^ (n - 1) := by
+  cases n
+  · simp
+  · ext n
+    simp [coeff_X_pow]
+
+/-- `divX` as an additive homomorphism. -/
+noncomputable
+def divX_hom : R[X] →+ R[X] :=
+{ toFun := divX
+  map_zero' := divX_zero
+  map_add' := fun _ _ => divX_add }
+
+@[simp] theorem divX_hom_toFun : divX_hom p = divX p := rfl
+
+theorem natDegree_divX_eq_natDegree_tsub_one : p.divX.natDegree = p.natDegree - 1 := by
+  apply map_natDegree_eq_sub (φ := divX_hom)
+  · intro f
+    simpa [divX_hom, divX_eq_zero_iff] using eq_C_of_natDegree_eq_zero
+  · intros n c c0
+    rw [← C_mul_X_pow_eq_monomial, divX_hom_toFun, divX_C_mul, divX_X_pow]
+    split_ifs with n0
+    · simp [n0]
+    · exact natDegree_C_mul_X_pow (n - 1) c c0
+
+theorem natDegree_divX_le : p.divX.natDegree ≤ p.natDegree :=
+natDegree_divX_eq_natDegree_tsub_one.trans_le (Nat.pred_le _)
+
+theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by
+  simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]
+
 theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := Nontrivial.of_polynomial_ne hp0
   calc

57e09a12

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

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

This has nice performance benefits.

32de3f67

Diff

@@ -95,9 +95,9 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
 set_option linter.uppercaseLean3 false in
 #align polynomial.degree_div_X_lt Polynomial.degree_divX_lt
 
-/-- An induction principle for polynomials, valued in Sort _ instead of Prop. -/
+/-- An induction principle for polynomials, valued in Sort* instead of Prop. -/
 @[elab_as_elim]
-noncomputable def recOnHorner {M : R[X] → Sort _} (p : R[X]) (M0 : M 0)
+noncomputable def recOnHorner {M : R[X] → Sort*} (p : R[X]) (M0 : M 0)
     (MC : ∀ p a, coeff p 0 = 0 → a ≠ 0 → M p → M (p + C a))
     (MX : ∀ p, p ≠ 0 → M p → M (p * X)) : M p :=
   if hp : p = 0 then hp ▸ M0

57e09a12

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

depends on: #5966

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

89aa3a3b

Diff

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
-
-! This file was ported from Lean 3 source module data.polynomial.inductions
-! leanprover-community/mathlib commit 57e09a1296bfb4330ddf6624f1028ba186117d82
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.MonoidAlgebra.Division
 import Mathlib.Data.Nat.Interval
 import Mathlib.Data.Polynomial.Degree.Definitions
 import Mathlib.Data.Polynomial.Induction
 
+#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"
+
 /-!
 # Induction on polynomials

57e09a12

chore: remove legacy termination_by' (#5426)

This adds a couple of WellFoundedRelation instances, like for example WellFoundedRelation (WithBot Nat). Longer-term, we should probably add a WellFoundedOrder class for types with a well-founded less-than relation and a [WellFoundOrder α] : WellFoundedRelation α instance (or maybe just [LT α] [IsWellFounded fun a b : α => a < b] : WellFoundedRelation α).

ceb42c50

Diff

@@ -116,7 +116,7 @@ noncomputable def recOnHorner {M : R[X] → Sort _} (p : R[X]) (M0 : M 0)
         MC _ _ (coeff_mul_X_zero _) hcp0
           (if hpX0 : divX p = 0 then show M (divX p * X) by rw [hpX0, zero_mul]; exact M0
           else MX (divX p) hpX0 (recOnHorner _ M0 MC MX))
-termination_by' invImage PSigma.fst ⟨_, degree_lt_wf⟩
+termination_by _ => p.degree
 #align polynomial.rec_on_horner Polynomial.recOnHorner
 
 /-- A property holds for all polynomials of positive `degree` with coefficients in a semiring `R`

57e09a12

chore: bye-bye, solo bys! (#3825)

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

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

14167e48

Diff

@@ -74,13 +74,11 @@ theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
   haveI := Nontrivial.of_polynomial_ne hp0
   calc
     degree (divX p) < (divX p * X + C (p.coeff 0)).degree :=
-      if h : degree p ≤ 0 then
-        by
+      if h : degree p ≤ 0 then by
         have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
         rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add]
         exact lt_of_le_of_ne bot_le (Ne.symm (mt degree_eq_bot.1 <| by simpa using h'))
-      else
-        by
+      else by
         have hXp0 : divX p ≠ 0 := by
           simpa [divX_eq_zero_iff, -not_le, degree_le_zero_iff] using h
         have : leadingCoeff (divX p) * leadingCoeff X ≠ 0 := by simpa

57e09a12

chore: forward port leanprover-community/mathlib#15905 at Data.Polynomial.Inductions (#3098)

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

705192ea

Diff

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
 
 ! This file was ported from Lean 3 source module data.polynomial.inductions
-! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! leanprover-community/mathlib commit 57e09a1296bfb4330ddf6624f1028ba186117d82
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathlib.Algebra.MonoidAlgebra.Division
 import Mathlib.Data.Nat.Interval
 import Mathlib.Data.Polynomial.Degree.Definitions
 import Mathlib.Data.Polynomial.Induction
@@ -38,16 +39,13 @@ variable [Semiring R] {p q : R[X]}
 /-- `divX p` returns a polynomial `q` such that `q * X + C (p.coeff 0) = p`.
   It can be used in a semiring where the usual division algorithm is not possible -/
 def divX (p : R[X]) : R[X] :=
-  ∑ n in Ico 0 p.natDegree, monomial n (p.coeff (n + 1))
+  ⟨AddMonoidAlgebra.divOf p.toFinsupp 1⟩
 set_option linter.uppercaseLean3 false in
 #align polynomial.div_X Polynomial.divX
 
 @[simp]
 theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by
-  simp only [divX, coeff_monomial, true_and_iff, finset_sum_coeff, not_lt, mem_Ico, zero_le,
-    Finset.sum_ite_eq', ite_eq_left_iff]
-  intro h
-  rw [coeff_eq_zero_of_natDegree_lt (Nat.lt_succ_of_le h)]
+  rw [add_comm]; cases p; rfl
 set_option linter.uppercaseLean3 false in
 #align polynomial.coeff_div_X Polynomial.coeff_divX
 
@@ -58,7 +56,7 @@ set_option linter.uppercaseLean3 false in
 
 @[simp]
 theorem divX_C (a : R) : divX (C a) = 0 :=
-  ext fun n => by simp [divX, coeff_C]
+  ext fun n => by simp [coeff_divX, coeff_C, Finsupp.single_eq_of_ne _]
 set_option linter.uppercaseLean3 false in
 #align polynomial.div_X_C Polynomial.divX_C

70fd9563

feat: port Data.Polynomial.Inductions (#2747)

Co-authored-by: Johan Commelin <johan@commelin.net>

60395bde

`data.polynomial.inductions` ⟷ `Mathlib.Data.Polynomial.Inductions`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 396

data.polynomial.inductions ⟷ Mathlib.Data.Polynomial.Inductions

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Dependencies 8 + 396

`data.polynomial.inductions` ⟷ `Mathlib.Data.Polynomial.Inductions`