/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker

! This file was ported from Lean 3 source module data.polynomial.basic
! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.MonoidAlgebra.Basic
import Mathbin.Data.Finset.Sort

/-!
# Theory of univariate polynomials

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

This file defines `polynomial R`, the type of univariate polynomials over the semiring `R`, builds
a semiring structure on it, and gives basic definitions that are expanded in other files in this
directory.

## Main definitions

* `monomial n a` is the polynomial `a X^n`. Note that `monomial n` is defined as an `R`-linear map.
* `C a` is the constant polynomial `a`. Note that `C` is defined as a ring homomorphism.
* `X` is the polynomial `X`, i.e., `monomial 1 1`.
* `p.sum f` is `∑ n in p.support, f n (p.coeff n)`, i.e., one sums the values of functions applied
  to coefficients of the polynomial `p`.
* `p.erase n` is the polynomial `p` in which one removes the `c X^n` term.

There are often two natural variants of lemmas involving sums, depending on whether one acts on the
polynomials, or on the function. The naming convention is that one adds `index` when acting on
the polynomials. For instance,
* `sum_add_index` states that `(p + q).sum f = p.sum f + q.sum f`;
* `sum_add` states that `p.sum (λ n x, f n x + g n x) = p.sum f + p.sum g`.
* Notation to refer to `polynomial R`, as `R[X]` or `R[t]`.

## Implementation

Polynomials are defined using `add_monoid_algebra R ℕ`, where `R` is a semiring.
The variable `X` commutes with every polynomial `p`: lemma `X_mul` proves the identity
`X * p = p * X`.  The relationship to `add_monoid_algebra R ℕ` is through a structure
to make polynomials irreducible from the point of view of the kernel. Most operations
are irreducible since Lean can not compute anyway with `add_monoid_algebra`. There are two
exceptions that we make semireducible:
* The zero polynomial, so that its coefficients are definitionally equal to `0`.
* The scalar action, to permit typeclass search to unfold it to resolve potential instance
  diamonds.

The raw implementation of the equivalence between `R[X]` and `add_monoid_algebra R ℕ` is
done through `of_finsupp` and `to_finsupp` (or, equivalently, `rcases p` when `p` is a polynomial
gives an element `q` of `add_monoid_algebra R ℕ`, and conversely `⟨q⟩` gives back `p`). The
equivalence is also registered as a ring equiv in `polynomial.to_finsupp_iso`. These should
in general not be used once the basic API for polynomials is constructed.
-/

noncomputable section

#print Polynomial /-
/-- `polynomial R` is the type of univariate polynomials over `R`.

Polynomials should be seen as (semi-)rings with the additional constructor `X`.
The embedding from `R` is called `C`. -/
structure Polynomial (R : Type _) [Semiring R] where ofFinsupp ::
  toFinsupp : AddMonoidAlgebra R ℕ
#align polynomial Polynomial
-/

-- mathport name: polynomial
scoped[Polynomial] notation:9000 R "[X]" => Polynomial R

open AddMonoidAlgebra Finsupp Function

open BigOperators Polynomial

namespace Polynomial

universe u

variable {R : Type u} {a b : R} {m n : ℕ}

section Semiring

variable [Semiring R] {p q : R[X]}

#print Polynomial.forall_iff_forall_finsupp /-
theorem forall_iff_forall_finsupp (P : R[X] → Prop) :
    (∀ p, P p) ↔ ∀ q : AddMonoidAlgebra R ℕ, P ⟨q⟩ :=
  ⟨fun h q => h ⟨q⟩, fun h ⟨p⟩ => h p⟩
#align polynomial.forall_iff_forall_finsupp Polynomial.forall_iff_forall_finsupp
-/

#print Polynomial.exists_iff_exists_finsupp /-
theorem exists_iff_exists_finsupp (P : R[X] → Prop) :
    (∃ p, P p) ↔ ∃ q : AddMonoidAlgebra R ℕ, P ⟨q⟩ :=
  ⟨fun ⟨⟨p⟩, hp⟩ => ⟨p, hp⟩, fun ⟨q, hq⟩ => ⟨⟨q⟩, hq⟩⟩
#align polynomial.exists_iff_exists_finsupp Polynomial.exists_iff_exists_finsupp
-/

#print Polynomial.eta /-
@[simp]
theorem eta (f : R[X]) : Polynomial.ofFinsupp f.toFinsupp = f := by cases f <;> rfl
#align polynomial.eta Polynomial.eta
-/

/-! ### Conversions to and from `add_monoid_algebra`

Since `R[X]` is not defeq to `add_monoid_algebra R ℕ`, but instead is a structure wrapping
it, we have to copy across all the arithmetic operators manually, along with the lemmas about how
they unfold around `polynomial.of_finsupp` and `polynomial.to_finsupp`.
-/

section AddMonoidAlgebra

private irreducible_def add : R[X] → R[X] → R[X]
  | ⟨a⟩, ⟨b⟩ => ⟨a + b⟩
#align polynomial.add polynomial.add

private irreducible_def neg {R : Type u} [Ring R] : R[X] → R[X]
  | ⟨a⟩ => ⟨-a⟩
#align polynomial.neg polynomial.neg

private irreducible_def mul : R[X] → R[X] → R[X]
  | ⟨a⟩, ⟨b⟩ => ⟨a * b⟩
#align polynomial.mul polynomial.mul

instance : Zero R[X] :=
  ⟨⟨0⟩⟩

instance : One R[X] :=
  ⟨⟨1⟩⟩

instance : Add R[X] :=
  ⟨add⟩

instance {R : Type u} [Ring R] : Neg R[X] :=
  ⟨neg⟩

instance {R : Type u} [Ring R] : Sub R[X] :=
  ⟨fun a b => a + -b⟩

instance : Mul R[X] :=
  ⟨mul⟩

instance {S : Type _} [SMulZeroClass S R] : SMulZeroClass S R[X]
    where
  smul r p := ⟨r • p.toFinsupp⟩
  smul_zero a := congr_arg ofFinsupp (smul_zero a)

#print Polynomial.pow /-
-- to avoid a bug in the `ring` tactic
instance (priority := 1) pow : Pow R[X] ℕ where pow p n := npowRec n p
#align polynomial.has_pow Polynomial.pow
-/

/- warning: polynomial.of_finsupp_zero -> Polynomial.ofFinsupp_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (OfNat.mk.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.zero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MulZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd))))))) (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], Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.toOfNat0.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MonoidWithZero.toZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Semiring.toMonoidWithZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.semiring.{u1, 0} R Nat _inst_1 Nat.addMonoid)))))) (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.of_finsupp_zero Polynomial.ofFinsupp_zeroₓ'. -/
@[simp]
theorem ofFinsupp_zero : (⟨0⟩ : R[X]) = 0 :=
  rfl
#align polynomial.of_finsupp_zero Polynomial.ofFinsupp_zero

#print Polynomial.ofFinsupp_one /-
@[simp]
theorem ofFinsupp_one : (⟨1⟩ : R[X]) = 1 :=
  rfl
#align polynomial.of_finsupp_one Polynomial.ofFinsupp_one
-/

/- warning: polynomial.of_finsupp_add -> Polynomial.ofFinsupp_add is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_add Polynomial.ofFinsupp_addₓ'. -/
@[simp]
theorem ofFinsupp_add {a b} : (⟨a + b⟩ : R[X]) = ⟨a⟩ + ⟨b⟩ :=
  show _ = add _ _ by rw [add]
#align polynomial.of_finsupp_add Polynomial.ofFinsupp_add

/- warning: polynomial.of_finsupp_neg -> Polynomial.ofFinsupp_neg is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_2 : Ring.{u1} R] {a : AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)}, Eq.{succ u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.ofFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) (Neg.neg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (SubNegMonoid.toHasNeg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroup.toSubNegMonoid.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroupWithOne.toAddGroup.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddCommGroupWithOne.toAddGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Ring.toAddCommGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.ring.{u1, 0} R Nat _inst_2 Nat.addMonoid)))))) a)) (Neg.neg.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.neg'.{u1} R _inst_2) (Polynomial.ofFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) a))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_neg Polynomial.ofFinsupp_negₓ'. -/
@[simp]
theorem ofFinsupp_neg {R : Type u} [Ring R] {a} : (⟨-a⟩ : R[X]) = -⟨a⟩ :=
  show _ = neg _ by rw [neg]
#align polynomial.of_finsupp_neg Polynomial.ofFinsupp_neg

/- warning: polynomial.of_finsupp_sub -> Polynomial.ofFinsupp_sub is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_sub Polynomial.ofFinsupp_subₓ'. -/
@[simp]
theorem ofFinsupp_sub {R : Type u} [Ring R] {a b} : (⟨a - b⟩ : R[X]) = ⟨a⟩ - ⟨b⟩ :=
  by
  rw [sub_eq_add_neg, of_finsupp_add, of_finsupp_neg]
  rfl
#align polynomial.of_finsupp_sub Polynomial.ofFinsupp_sub

#print Polynomial.ofFinsupp_mul /-
@[simp]
theorem ofFinsupp_mul (a b) : (⟨a * b⟩ : R[X]) = ⟨a⟩ * ⟨b⟩ :=
  show _ = mul _ _ by rw [mul]
#align polynomial.of_finsupp_mul Polynomial.ofFinsupp_mul
-/

/- warning: polynomial.of_finsupp_smul -> Polynomial.ofFinsupp_smul is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : SMulZeroClass.{u2, u1} S R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))] (a : S) (b : AddMonoidAlgebra.{u1, 0} R Nat _inst_1), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (SMul.smul.{u2, u1} S (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (SMulZeroClass.toHasSmul.{u2, u1} S (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoid.toAddZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddCommMonoid.toAddMonoid.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.addCommMonoid.{0, u1} R Nat _inst_1)))) (AddMonoidAlgebra.smulZeroClass.{u1, 0, u2} R Nat S _inst_1 _inst_2)) a b)) (SMul.smul.{u2, u1} S (Polynomial.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u2, u1} S (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1) (Polynomial.smulZeroClass.{u1, u2} R _inst_1 S _inst_2)) a (Polynomial.ofFinsupp.{u1} R _inst_1 b))
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_smul Polynomial.ofFinsupp_smulₓ'. -/
@[simp]
theorem ofFinsupp_smul {S : Type _} [SMulZeroClass S R] (a : S) (b) :
    (⟨a • b⟩ : R[X]) = (a • ⟨b⟩ : R[X]) :=
  rfl
#align polynomial.of_finsupp_smul Polynomial.ofFinsupp_smul

#print Polynomial.ofFinsupp_pow /-
@[simp]
theorem ofFinsupp_pow (a) (n : ℕ) : (⟨a ^ n⟩ : R[X]) = ⟨a⟩ ^ n :=
  by
  change _ = npowRec n _
  induction n
  · simp [npowRec]
  · simp [npowRec, n_ih, pow_succ]
#align polynomial.of_finsupp_pow Polynomial.ofFinsupp_pow
-/

/- warning: polynomial.to_finsupp_zero -> Polynomial.toFinsupp_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (OfNat.mk.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.zero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))))) (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (OfNat.mk.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.zero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MulZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1)))) (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.toOfNat0.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MonoidWithZero.toZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Semiring.toMonoidWithZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.semiring.{u1, 0} R Nat _inst_1 Nat.addMonoid)))))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_zero Polynomial.toFinsupp_zeroₓ'. -/
@[simp]
theorem toFinsupp_zero : (0 : R[X]).toFinsupp = 0 :=
  rfl
#align polynomial.to_finsupp_zero Polynomial.toFinsupp_zero

#print Polynomial.toFinsupp_one /-
@[simp]
theorem toFinsupp_one : (1 : R[X]).toFinsupp = 1 :=
  rfl
#align polynomial.to_finsupp_one Polynomial.toFinsupp_one
-/

/- warning: polynomial.to_finsupp_add -> Polynomial.toFinsupp_add is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : Polynomial.{u1} R _inst_1) (b : Polynomial.{u1} R _inst_1), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{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)) a b)) (HAdd.hAdd.{u1, u1, u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (instHAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Distrib.toHasAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd)))) (Polynomial.toFinsupp.{u1} R _inst_1 a) (Polynomial.toFinsupp.{u1} R _inst_1 b))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (a : Polynomial.{u1} R _inst_1) (b : Polynomial.{u1} R _inst_1), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{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)) a b)) (HAdd.hAdd.{u1, u1, u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (instHAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Distrib.toAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 instAddNat)))) (Polynomial.toFinsupp.{u1} R _inst_1 a) (Polynomial.toFinsupp.{u1} R _inst_1 b))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_add Polynomial.toFinsupp_addₓ'. -/
@[simp]
theorem toFinsupp_add (a b : R[X]) : (a + b).toFinsupp = a.toFinsupp + b.toFinsupp :=
  by
  cases a
  cases b
  rw [← of_finsupp_add]
#align polynomial.to_finsupp_add Polynomial.toFinsupp_add

/- warning: polynomial.to_finsupp_neg -> Polynomial.toFinsupp_neg is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_2 : Ring.{u1} R] (a : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) (Neg.neg.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.neg'.{u1} R _inst_2) a)) (Neg.neg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (SubNegMonoid.toHasNeg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroup.toSubNegMonoid.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroupWithOne.toAddGroup.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddCommGroupWithOne.toAddGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Ring.toAddCommGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.ring.{u1, 0} R Nat _inst_2 Nat.addMonoid)))))) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) a))
but is expected to have type
  forall {R : Type.{u1}} [_inst_2 : Ring.{u1} R] (a : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) (Neg.neg.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.neg'.{u1} R _inst_2) a)) (Neg.neg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Ring.toNeg.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.ring.{u1, 0} R Nat _inst_2 Nat.addMonoid)) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) a))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_neg Polynomial.toFinsupp_negₓ'. -/
@[simp]
theorem toFinsupp_neg {R : Type u} [Ring R] (a : R[X]) : (-a).toFinsupp = -a.toFinsupp :=
  by
  cases a
  rw [← of_finsupp_neg]
#align polynomial.to_finsupp_neg Polynomial.toFinsupp_neg

/- warning: polynomial.to_finsupp_sub -> Polynomial.toFinsupp_sub is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_2 : Ring.{u1} R] (a : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (b : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.sub.{u1} R _inst_2)) a b)) (HSub.hSub.{u1, u1, u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (instHSub.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (SubNegMonoid.toHasSub.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroup.toSubNegMonoid.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddGroupWithOne.toAddGroup.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddCommGroupWithOne.toAddGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Ring.toAddCommGroupWithOne.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.ring.{u1, 0} R Nat _inst_2 Nat.addMonoid))))))) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) a) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) b))
but is expected to have type
  forall {R : Type.{u1}} [_inst_2 : Ring.{u1} R] (a : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (b : Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) (HSub.hSub.{u1, u1, u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (instHSub.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Polynomial.sub.{u1} R _inst_2)) a b)) (HSub.hSub.{u1, u1, u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (instHSub.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (Ring.toSub.{u1} (AddMonoidAlgebra.{u1, 0} R Nat (Ring.toSemiring.{u1} R _inst_2)) (AddMonoidAlgebra.ring.{u1, 0} R Nat _inst_2 Nat.addMonoid))) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) a) (Polynomial.toFinsupp.{u1} R (Ring.toSemiring.{u1} R _inst_2) b))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_sub Polynomial.toFinsupp_subₓ'. -/
@[simp]
theorem toFinsupp_sub {R : Type u} [Ring R] (a b : R[X]) :
    (a - b).toFinsupp = a.toFinsupp - b.toFinsupp :=
  by
  rw [sub_eq_add_neg, ← to_finsupp_neg, ← to_finsupp_add]
  rfl
#align polynomial.to_finsupp_sub Polynomial.toFinsupp_sub

#print Polynomial.toFinsupp_mul /-
@[simp]
theorem toFinsupp_mul (a b : R[X]) : (a * b).toFinsupp = a.toFinsupp * b.toFinsupp :=
  by
  cases a
  cases b
  rw [← of_finsupp_mul]
#align polynomial.to_finsupp_mul Polynomial.toFinsupp_mul
-/

/- warning: polynomial.to_finsupp_smul -> Polynomial.toFinsupp_smul is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : SMulZeroClass.{u2, u1} S R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))] (a : S) (b : Polynomial.{u1} R _inst_1), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (SMul.smul.{u2, u1} S (Polynomial.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u2, u1} S (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1) (Polynomial.smulZeroClass.{u1, u2} R _inst_1 S _inst_2)) a b)) (SMul.smul.{u2, u1} S (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (SMulZeroClass.toHasSmul.{u2, u1} S (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoid.toAddZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddCommMonoid.toAddMonoid.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.addCommMonoid.{0, u1} R Nat _inst_1)))) (AddMonoidAlgebra.smulZeroClass.{u1, 0, u2} R Nat S _inst_1 _inst_2)) a (Polynomial.toFinsupp.{u1} R _inst_1 b))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : SMulZeroClass.{u1, u2} S R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))] (a : S) (b : Polynomial.{u2} R _inst_1), Eq.{succ u2} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u2} R _inst_1 (HSMul.hSMul.{u1, u2, u2} S (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHSMul.{u1, u2} S (Polynomial.{u2} R _inst_1) (SMulZeroClass.toSMul.{u1, u2} S (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1) (Polynomial.smulZeroClass.{u2, u1} R _inst_1 S _inst_2))) a b)) (HSMul.hSMul.{u1, u2, u2} S (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (instHSMul.{u1, u2} S (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (SMulZeroClass.toSMul.{u1, u2} S (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (MonoidWithZero.toZero.{u2} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (Semiring.toMonoidWithZero.{u2} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (AddMonoidAlgebra.semiring.{u2, 0} R Nat _inst_1 Nat.addMonoid))) (AddMonoidAlgebra.smulZeroClass.{u2, 0, u1} R Nat S _inst_1 _inst_2))) a (Polynomial.toFinsupp.{u2} R _inst_1 b))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_smul Polynomial.toFinsupp_smulₓ'. -/
@[simp]
theorem toFinsupp_smul {S : Type _} [SMulZeroClass S R] (a : S) (b : R[X]) :
    (a • b).toFinsupp = a • b.toFinsupp :=
  rfl
#align polynomial.to_finsupp_smul Polynomial.toFinsupp_smul

#print Polynomial.toFinsupp_pow /-
@[simp]
theorem toFinsupp_pow (a : R[X]) (n : ℕ) : (a ^ n).toFinsupp = a.toFinsupp ^ n :=
  by
  cases a
  rw [← of_finsupp_pow]
#align polynomial.to_finsupp_pow Polynomial.toFinsupp_pow
-/

/- warning: is_smul_regular.polynomial -> IsSMulRegular.polynomial is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : Monoid.{u2} S] [_inst_3 : DistribMulAction.{u2, u1} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))] {a : S}, (IsSMulRegular.{u2, u1} S R (SMulZeroClass.toHasSmul.{u2, u1} S R (AddZeroClass.toHasZero.{u1} R (AddMonoid.toAddZeroClass.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))) (DistribSMul.toSmulZeroClass.{u2, u1} S R (AddMonoid.toAddZeroClass.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (DistribMulAction.toDistribSMul.{u2, u1} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) _inst_3))) a) -> (IsSMulRegular.{u2, u1} S (Polynomial.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u2, u1} S (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1) (Polynomial.smulZeroClass.{u1, u2} R _inst_1 S (DistribSMul.toSmulZeroClass.{u2, u1} S R (AddMonoid.toAddZeroClass.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (DistribMulAction.toDistribSMul.{u2, u1} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) _inst_3)))) a)
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : Monoid.{u1} S] [_inst_3 : DistribMulAction.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))] {a : S}, (IsSMulRegular.{u1, u2} S R (SMulZeroClass.toSMul.{u1, u2} S R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (DistribSMul.toSMulZeroClass.{u1, u2} S R (AddMonoid.toAddZeroClass.{u2} R (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (DistribMulAction.toDistribSMul.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) _inst_3))) a) -> (IsSMulRegular.{u1, u2} S (Polynomial.{u2} R _inst_1) (SMulZeroClass.toSMul.{u1, u2} S (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1) (Polynomial.smulZeroClass.{u2, u1} R _inst_1 S (DistribSMul.toSMulZeroClass.{u1, u2} S R (AddMonoid.toAddZeroClass.{u2} R (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (DistribMulAction.toDistribSMul.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) _inst_3)))) a)
Case conversion may be inaccurate. Consider using '#align is_smul_regular.polynomial IsSMulRegular.polynomialₓ'. -/
theorem IsSMulRegular.polynomial {S : Type _} [Monoid S] [DistribMulAction S R] {a : S}
    (ha : IsSMulRegular R a) : IsSMulRegular R[X] a
  | ⟨x⟩, ⟨y⟩, h => congr_arg _ <| ha.Finsupp (Polynomial.ofFinsupp.inj h)
#align is_smul_regular.polynomial IsSMulRegular.polynomial

#print Polynomial.toFinsupp_injective /-
theorem toFinsupp_injective : Function.Injective (toFinsupp : R[X] → AddMonoidAlgebra _ _) :=
  fun ⟨x⟩ ⟨y⟩ => congr_arg _
#align polynomial.to_finsupp_injective Polynomial.toFinsupp_injective
-/

#print Polynomial.toFinsupp_inj /-
@[simp]
theorem toFinsupp_inj {a b : R[X]} : a.toFinsupp = b.toFinsupp ↔ a = b :=
  toFinsupp_injective.eq_iff
#align polynomial.to_finsupp_inj Polynomial.toFinsupp_inj
-/

/- warning: polynomial.to_finsupp_eq_zero -> Polynomial.toFinsupp_eq_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {a : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 a) (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (OfNat.mk.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.zero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MulZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd))))))) (Eq.{succ u1} (Polynomial.{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 : Polynomial.{u1} R _inst_1}, Iff (Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 a) (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.toOfNat0.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MonoidWithZero.toZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Semiring.toMonoidWithZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.semiring.{u1, 0} R Nat _inst_1 Nat.addMonoid)))))) (Eq.{succ u1} (Polynomial.{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.to_finsupp_eq_zero Polynomial.toFinsupp_eq_zeroₓ'. -/
@[simp]
theorem toFinsupp_eq_zero {a : R[X]} : a.toFinsupp = 0 ↔ a = 0 := by
  rw [← to_finsupp_zero, to_finsupp_inj]
#align polynomial.to_finsupp_eq_zero Polynomial.toFinsupp_eq_zero

#print Polynomial.toFinsupp_eq_one /-
@[simp]
theorem toFinsupp_eq_one {a : R[X]} : a.toFinsupp = 1 ↔ a = 1 := by
  rw [← to_finsupp_one, to_finsupp_inj]
#align polynomial.to_finsupp_eq_one Polynomial.toFinsupp_eq_one
-/

#print Polynomial.ofFinsupp_inj /-
/-- A more convenient spelling of `polynomial.of_finsupp.inj_eq` in terms of `iff`. -/
theorem ofFinsupp_inj {a b} : (⟨a⟩ : R[X]) = ⟨b⟩ ↔ a = b :=
  iff_of_eq ofFinsupp.inj_eq
#align polynomial.of_finsupp_inj Polynomial.ofFinsupp_inj
-/

/- warning: polynomial.of_finsupp_eq_zero -> Polynomial.ofFinsupp_eq_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {a : AddMonoidAlgebra.{u1, 0} R Nat _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{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))))) (Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) a (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (OfNat.mk.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.zero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MulZeroClass.toHasZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd)))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {a : AddMonoidAlgebra.{u1, 0} R Nat _inst_1}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{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)))) (Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) a (OfNat.ofNat.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) 0 (Zero.toOfNat0.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (MonoidWithZero.toZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Semiring.toMonoidWithZero.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.semiring.{u1, 0} R Nat _inst_1 Nat.addMonoid))))))
Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_eq_zero Polynomial.ofFinsupp_eq_zeroₓ'. -/
@[simp]
theorem ofFinsupp_eq_zero {a} : (⟨a⟩ : R[X]) = 0 ↔ a = 0 := by
  rw [← of_finsupp_zero, of_finsupp_inj]
#align polynomial.of_finsupp_eq_zero Polynomial.ofFinsupp_eq_zero

#print Polynomial.ofFinsupp_eq_one /-
@[simp]
theorem ofFinsupp_eq_one {a} : (⟨a⟩ : R[X]) = 1 ↔ a = 1 := by rw [← of_finsupp_one, of_finsupp_inj]
#align polynomial.of_finsupp_eq_one Polynomial.ofFinsupp_eq_one
-/

instance : Inhabited R[X] :=
  ⟨0⟩

instance : NatCast R[X] :=
  ⟨fun n => Polynomial.ofFinsupp n⟩

instance : Semiring R[X] :=
  Function.Injective.semiring toFinsupp toFinsupp_injective toFinsupp_zero toFinsupp_one
    toFinsupp_add toFinsupp_mul (fun _ _ => toFinsupp_smul _ _) toFinsupp_pow fun _ => rfl

instance {S} [Monoid S] [DistribMulAction S R] : DistribMulAction S R[X] :=
  Function.Injective.distribMulAction ⟨toFinsupp, toFinsupp_zero, toFinsupp_add⟩ toFinsupp_injective
    toFinsupp_smul

instance {S} [Monoid S] [DistribMulAction S R] [FaithfulSMul S R] : FaithfulSMul S R[X]
    where eq_of_smul_eq_smul s₁ s₂ h :=
    eq_of_smul_eq_smul fun a : ℕ →₀ R => congr_arg toFinsupp (h ⟨a⟩)

instance {S} [Semiring S] [Module S R] : Module S R[X] :=
  Function.Injective.module _ ⟨toFinsupp, toFinsupp_zero, toFinsupp_add⟩ toFinsupp_injective
    toFinsupp_smul

instance {S₁ S₂} [Monoid S₁] [Monoid S₂] [DistribMulAction S₁ R] [DistribMulAction S₂ R]
    [SMulCommClass S₁ S₂ R] : SMulCommClass S₁ S₂ R[X] :=
  ⟨by
    rintro _ _ ⟨⟩
    simp_rw [← of_finsupp_smul, smul_comm]⟩

instance {S₁ S₂} [SMul S₁ S₂] [Monoid S₁] [Monoid S₂] [DistribMulAction S₁ R]
    [DistribMulAction S₂ R] [IsScalarTower S₁ S₂ R] : IsScalarTower S₁ S₂ R[X] :=
  ⟨by
    rintro _ _ ⟨⟩
    simp_rw [← of_finsupp_smul, smul_assoc]⟩

/- warning: polynomial.is_scalar_tower_right -> Polynomial.isScalarTower_right is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {K : Type.{u2}} [_inst_2 : Semiring.{u2} K] [_inst_3 : DistribSMul.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2)))))] [_inst_4 : IsScalarTower.{u1, u2, u2} α K K (SMulZeroClass.toHasSmul.{u1, u2} α K (AddZeroClass.toHasZero.{u2} K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2)))))) (DistribSMul.toSmulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3)) (Mul.toSMul.{u2} K (Distrib.toHasMul.{u2} K (NonUnitalNonAssocSemiring.toDistrib.{u2} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) (SMulZeroClass.toHasSmul.{u1, u2} α K (AddZeroClass.toHasZero.{u2} K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2)))))) (DistribSMul.toSmulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3))], IsScalarTower.{u1, u2, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.{u2} K _inst_2) (SMulZeroClass.toHasSmul.{u1, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (Polynomial.smulZeroClass.{u2, u1} K _inst_2 α (DistribSMul.toSmulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3))) (Mul.toSMul.{u2} (Polynomial.{u2} K _inst_2) (Polynomial.mul'.{u2} K _inst_2)) (SMulZeroClass.toHasSmul.{u1, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (Polynomial.smulZeroClass.{u2, u1} K _inst_2 α (DistribSMul.toSmulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3)))
but is expected to have type
  forall {α : Type.{u1}} {K : Type.{u2}} [_inst_2 : Semiring.{u2} K] [_inst_3 : DistribSMul.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2)))))] [_inst_4 : IsScalarTower.{u1, u2, u2} α K K (SMulZeroClass.toSMul.{u1, u2} α K (MonoidWithZero.toZero.{u2} K (Semiring.toMonoidWithZero.{u2} K _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3)) (SMulZeroClass.toSMul.{u2, u2} K K (MonoidWithZero.toZero.{u2} K (Semiring.toMonoidWithZero.{u2} K _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u2} K K (MonoidWithZero.toZero.{u2} K (Semiring.toMonoidWithZero.{u2} K _inst_2)) (MonoidWithZero.toZero.{u2} K (Semiring.toMonoidWithZero.{u2} K _inst_2)) (MulZeroClass.toSMulWithZero.{u2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} K (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2)))))) (SMulZeroClass.toSMul.{u1, u2} α K (MonoidWithZero.toZero.{u2} K (Semiring.toMonoidWithZero.{u2} K _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3))], IsScalarTower.{u1, u2, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.{u2} K _inst_2) (SMulZeroClass.toSMul.{u1, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (Polynomial.smulZeroClass.{u2, u1} K _inst_2 α (DistribSMul.toSMulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3))) (SMulZeroClass.toSMul.{u2, u2} (Polynomial.{u2} K _inst_2) (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (SMulWithZero.toSMulZeroClass.{u2, u2} (Polynomial.{u2} K _inst_2) (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (MulZeroClass.toSMulWithZero.{u2} (Polynomial.{u2} K _inst_2) (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} (Polynomial.{u2} K _inst_2) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} K _inst_2) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} K _inst_2) (Polynomial.semiring.{u2} K _inst_2))))))) (SMulZeroClass.toSMul.{u1, u2} α (Polynomial.{u2} K _inst_2) (Polynomial.zero.{u2} K _inst_2) (Polynomial.smulZeroClass.{u2, u1} K _inst_2 α (DistribSMul.toSMulZeroClass.{u1, u2} α K (AddMonoid.toAddZeroClass.{u2} K (AddMonoidWithOne.toAddMonoid.{u2} K (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} K (NonAssocSemiring.toAddCommMonoidWithOne.{u2} K (Semiring.toNonAssocSemiring.{u2} K _inst_2))))) _inst_3)))
Case conversion may be inaccurate. Consider using '#align polynomial.is_scalar_tower_right Polynomial.isScalarTower_rightₓ'. -/
instance isScalarTower_right {α K : Type _} [Semiring K] [DistribSMul α K] [IsScalarTower α K K] :
    IsScalarTower α K[X] K[X] :=
  ⟨by
    rintro _ ⟨⟩ ⟨⟩ <;>
      simp_rw [smul_eq_mul, ← of_finsupp_smul, ← of_finsupp_mul, ← of_finsupp_smul, smul_mul_assoc]⟩
#align polynomial.is_scalar_tower_right Polynomial.isScalarTower_right

instance {S} [Monoid S] [DistribMulAction S R] [DistribMulAction Sᵐᵒᵖ R] [IsCentralScalar S R] :
    IsCentralScalar S R[X] :=
  ⟨by
    rintro _ ⟨⟩
    simp_rw [← of_finsupp_smul, op_smul_eq_smul]⟩

instance [Subsingleton R] : Unique R[X] :=
  { Polynomial.inhabited with
    uniq := by
      rintro ⟨x⟩
      refine' congr_arg of_finsupp _
      simp }

variable (R)

/- warning: polynomial.to_finsupp_iso -> Polynomial.toFinsuppIso is a dubious translation:
lean 3 declaration is
  forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R], RingEquiv.{u1, u1} (Polynomial.{u1} R _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.mul'.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1) (AddMonoidAlgebra.hasMul.{u1, 0} R Nat _inst_1 Nat.hasAdd) (Distrib.toHasAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 Nat.hasAdd)))
but is expected to have type
  forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R], RingEquiv.{u1, u1} (Polynomial.{u1} R _inst_1) (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.mul'.{u1} R _inst_1) (AddMonoidAlgebra.hasMul.{u1, 0} R Nat _inst_1 instAddNat) (Polynomial.add'.{u1} R _inst_1) (Distrib.toAdd.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (AddMonoidAlgebra.nonUnitalNonAssocSemiring.{u1, 0} R Nat _inst_1 instAddNat)))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_iso Polynomial.toFinsuppIsoₓ'. -/
/-- Ring isomorphism between `R[X]` and `add_monoid_algebra R ℕ`. This is just an
implementation detail, but it can be useful to transfer results from `finsupp` to polynomials. -/
@[simps apply symm_apply]
def toFinsuppIso : R[X] ≃+* AddMonoidAlgebra R ℕ
    where
  toFun := toFinsupp
  invFun := ofFinsupp
  left_inv := fun ⟨p⟩ => rfl
  right_inv p := rfl
  map_mul' := toFinsupp_mul
  map_add' := toFinsupp_add
#align polynomial.to_finsupp_iso Polynomial.toFinsuppIso

end AddMonoidAlgebra

variable {R}

/- warning: polynomial.of_finsupp_sum -> Polynomial.ofFinsupp_sum is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (AddMonoidAlgebra.{u1, 0} R Nat _inst_1)), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (Finset.sum.{u1, u2} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) ι (AddMonoidAlgebra.addCommMonoid.{0, u1} R Nat _inst_1) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} (Polynomial.{u1} R _inst_1) ι (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)))) s (fun (i : ι) => Polynomial.ofFinsupp.{u1} R _inst_1 (f i)))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (AddMonoidAlgebra.{u2, 0} R Nat _inst_1)), Eq.{succ u2} (Polynomial.{u2} R _inst_1) (Polynomial.ofFinsupp.{u2} R _inst_1 (Finset.sum.{u2, u1} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) ι (AddMonoidAlgebra.addCommMonoid.{u2, 0} R Nat _inst_1) s (fun (i : ι) => f i))) (Finset.sum.{u2, u1} (Polynomial.{u2} R _inst_1) ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) s (fun (i : ι) => Polynomial.ofFinsupp.{u2} R _inst_1 (f i)))
Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_sum Polynomial.ofFinsupp_sumₓ'. -/
theorem ofFinsupp_sum {ι : Type _} (s : Finset ι) (f : ι → AddMonoidAlgebra R ℕ) :
    (⟨∑ i in s, f i⟩ : R[X]) = ∑ i in s, ⟨f i⟩ :=
  map_sum (toFinsuppIso R).symm f s
#align polynomial.of_finsupp_sum Polynomial.ofFinsupp_sum

/- warning: polynomial.to_finsupp_sum -> Polynomial.toFinsupp_sum is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (Polynomial.{u1} R _inst_1)), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (Finset.sum.{u1, u2} (Polynomial.{u1} R _inst_1) ι (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)))) s (fun (i : ι) => f i))) (Finset.sum.{u1, u2} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) ι (AddMonoidAlgebra.addCommMonoid.{0, u1} R Nat _inst_1) s (fun (i : ι) => Polynomial.toFinsupp.{u1} R _inst_1 (f i)))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (Polynomial.{u2} R _inst_1)), Eq.{succ u2} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u2} R _inst_1 (Finset.sum.{u2, u1} (Polynomial.{u2} R _inst_1) ι (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) s (fun (i : ι) => f i))) (Finset.sum.{u2, u1} (AddMonoidAlgebra.{u2, 0} R Nat _inst_1) ι (AddMonoidAlgebra.addCommMonoid.{u2, 0} R Nat _inst_1) s (fun (i : ι) => Polynomial.toFinsupp.{u2} R _inst_1 (f i)))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_sum Polynomial.toFinsupp_sumₓ'. -/
theorem toFinsupp_sum {ι : Type _} (s : Finset ι) (f : ι → R[X]) :
    (∑ i in s, f i : R[X]).toFinsupp = ∑ i in s, (f i).toFinsupp :=
  map_sum (toFinsuppIso R) f s
#align polynomial.to_finsupp_sum Polynomial.toFinsupp_sum

#print Polynomial.support /-
/-- The set of all `n` such that `X^n` has a non-zero coefficient.
-/
@[simp]
def support : R[X] → Finset ℕ
  | ⟨p⟩ => p.support
#align polynomial.support Polynomial.support
-/

#print Polynomial.support_ofFinsupp /-
@[simp]
theorem support_ofFinsupp (p) : support (⟨p⟩ : R[X]) = p.support := by rw [support]
#align polynomial.support_of_finsupp Polynomial.support_ofFinsupp
-/

#print Polynomial.support_zero /-
@[simp]
theorem support_zero : (0 : R[X]).support = ∅ :=
  rfl
#align polynomial.support_zero Polynomial.support_zero
-/

#print Polynomial.support_eq_empty /-
@[simp]
theorem support_eq_empty : p.support = ∅ ↔ p = 0 :=
  by
  rcases p with ⟨⟩
  simp [support]
#align polynomial.support_eq_empty Polynomial.support_eq_empty
-/

#print Polynomial.card_support_eq_zero /-
theorem card_support_eq_zero : p.support.card = 0 ↔ p = 0 := by simp
#align polynomial.card_support_eq_zero Polynomial.card_support_eq_zero
-/

#print Polynomial.monomial /-
/-- `monomial s a` is the monomial `a * X^s` -/
def monomial (n : ℕ) : R →ₗ[R] R[X]
    where
  toFun t := ⟨Finsupp.single n t⟩
  map_add' := by simp
  map_smul' := by simp [← of_finsupp_smul]
#align polynomial.monomial Polynomial.monomial
-/

/- warning: polynomial.to_finsupp_monomial -> Polynomial.toFinsupp_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_monomial Polynomial.toFinsupp_monomialₓ'. -/
@[simp]
theorem toFinsupp_monomial (n : ℕ) (r : R) : (monomial n r).toFinsupp = Finsupp.single n r := by
  simp [monomial]
#align polynomial.to_finsupp_monomial Polynomial.toFinsupp_monomial

/- warning: polynomial.of_finsupp_single -> Polynomial.ofFinsupp_single is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_single Polynomial.ofFinsupp_singleₓ'. -/
@[simp]
theorem ofFinsupp_single (n : ℕ) (r : R) : (⟨Finsupp.single n r⟩ : R[X]) = monomial n r := by
  simp [monomial]
#align polynomial.of_finsupp_single Polynomial.ofFinsupp_single

/- warning: polynomial.monomial_zero_right -> Polynomial.monomial_zero_right is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_zero_right Polynomial.monomial_zero_rightₓ'. -/
@[simp]
theorem monomial_zero_right (n : ℕ) : monomial n (0 : R) = 0 :=
  (monomial n).map_zero
#align polynomial.monomial_zero_right Polynomial.monomial_zero_right

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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_zero_one Polynomial.monomial_zero_oneₓ'. -/
-- This is not a `simp` lemma as `monomial_zero_left` is more general.
theorem monomial_zero_one : monomial 0 (1 : R) = 1 :=
  rfl
#align polynomial.monomial_zero_one Polynomial.monomial_zero_one

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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_add Polynomial.monomial_addₓ'. -/
-- TODO: can't we just delete this one?
theorem monomial_add (n : ℕ) (r s : R) : monomial n (r + s) = monomial n r + monomial n s :=
  (monomial n).map_add _ _
#align polynomial.monomial_add Polynomial.monomial_add

/- warning: polynomial.monomial_mul_monomial -> Polynomial.monomial_mul_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_mul_monomial Polynomial.monomial_mul_monomialₓ'. -/
theorem monomial_mul_monomial (n m : ℕ) (r s : R) :
    monomial n r * monomial m s = monomial (n + m) (r * s) :=
  toFinsupp_injective <| by
    simp only [to_finsupp_monomial, to_finsupp_mul, AddMonoidAlgebra.single_mul_single]
#align polynomial.monomial_mul_monomial Polynomial.monomial_mul_monomial

/- warning: polynomial.monomial_pow -> Polynomial.monomial_pow is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_pow Polynomial.monomial_powₓ'. -/
@[simp]
theorem monomial_pow (n : ℕ) (r : R) (k : ℕ) : monomial n r ^ k = monomial (n * k) (r ^ k) :=
  by
  induction' k with k ih
  · simp [pow_zero, monomial_zero_one]
  · simp [pow_succ, ih, monomial_mul_monomial, Nat.succ_eq_add_one, mul_add, add_comm]
#align polynomial.monomial_pow Polynomial.monomial_pow

/- warning: polynomial.smul_monomial -> Polynomial.smul_monomial is a dubious translation:
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(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) (SMul.smul.{u2, u1} S R (SMulZeroClass.toHasSmul.{u2, u1} S R (AddZeroClass.toHasZero.{u1} R (AddMonoid.toAddZeroClass.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))) (DistribSMul.toSmulZeroClass.{u2, u1} S R (AddMonoid.toAddZeroClass.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (DistribMulAction.toDistribSMul.{u2, u1} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) _inst_3))) a b))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : Monoid.{u1} S] [_inst_3 : DistribMulAction.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))] (a : S) (n : Nat) (b : R), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) b) (HSMul.hSMul.{u1, u2, u2} S ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) b) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) b) (instHSMul.{u1, u2} S ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) b) (SMulZeroClass.toSMul.{u1, u2} S ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) b) (Polynomial.zero.{u2} R _inst_1) (Polynomial.smulZeroClass.{u2, u1} R _inst_1 S (DistribSMul.toSMulZeroClass.{u1, u2} S R (AddMonoid.toAddZeroClass.{u2} R (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (DistribMulAction.toDistribSMul.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) _inst_3))))) a (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u2, u2, u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) 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(Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Polynomial.monomial.{u2} R _inst_1 n) b)) (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u2, u2, u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R _inst_1) (Polynomial.module.{u2, u2} R _inst_1 R _inst_1 (Semiring.toModule.{u2} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u2, u2} R R R (Polynomial.{u2} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R _inst_1) (Polynomial.module.{u2, u2} R _inst_1 R _inst_1 (Semiring.toModule.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Polynomial.monomial.{u2} R _inst_1 n) (HSMul.hSMul.{u1, u2, u2} S R R (instHSMul.{u1, u2} S R (SMulZeroClass.toSMul.{u1, u2} S R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (DistribSMul.toSMulZeroClass.{u1, u2} S R (AddMonoid.toAddZeroClass.{u2} R (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (DistribMulAction.toDistribSMul.{u1, u2} S R _inst_2 (AddMonoidWithOne.toAddMonoid.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) _inst_3)))) a b))
Case conversion may be inaccurate. Consider using '#align polynomial.smul_monomial Polynomial.smul_monomialₓ'. -/
theorem smul_monomial {S} [Monoid S] [DistribMulAction S R] (a : S) (n : ℕ) (b : R) :
    a • monomial n b = monomial n (a • b) :=
  toFinsupp_injective <| by simp
#align polynomial.smul_monomial Polynomial.smul_monomial

/- warning: polynomial.monomial_injective -> Polynomial.monomial_injective is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), Function.Injective.{succ u1, succ u1} R (Polynomial.{u1} R _inst_1) (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), Function.Injective.{succ u1, succ u1} R (Polynomial.{u1} R _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n))
Case conversion may be inaccurate. Consider using '#align polynomial.monomial_injective Polynomial.monomial_injectiveₓ'. -/
theorem monomial_injective (n : ℕ) : Function.Injective (monomial n : R → R[X]) :=
  (toFinsuppIso R).symm.Injective.comp (single_injective n)
#align polynomial.monomial_injective Polynomial.monomial_injective

/- warning: polynomial.monomial_eq_zero_iff -> Polynomial.monomial_eq_zero_iff is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_eq_zero_iff Polynomial.monomial_eq_zero_iffₓ'. -/
@[simp]
theorem monomial_eq_zero_iff (t : R) (n : ℕ) : monomial n t = 0 ↔ t = 0 :=
  LinearMap.map_eq_zero_iff _ (Polynomial.monomial_injective n)
#align polynomial.monomial_eq_zero_iff Polynomial.monomial_eq_zero_iff

/- warning: polynomial.support_add -> Polynomial.support_add is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.support_add Polynomial.support_addₓ'. -/
theorem support_add : (p + q).support ⊆ p.support ∪ q.support :=
  by
  rcases p with ⟨⟩; rcases q with ⟨⟩
  simp only [← of_finsupp_add, support]
  exact support_add
#align polynomial.support_add Polynomial.support_add

#print Polynomial.C /-
/-- `C a` is the constant polynomial `a`.
`C` is provided as a ring homomorphism.
-/
def C : R →+* R[X] :=
  { monomial 0 with
    map_one' := by simp [monomial_zero_one]
    map_mul' := by simp [monomial_mul_monomial]
    map_zero' := by simp }
#align polynomial.C Polynomial.C
-/

/- warning: polynomial.monomial_zero_left -> Polynomial.monomial_zero_left is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_zero_left Polynomial.monomial_zero_leftₓ'. -/
@[simp]
theorem monomial_zero_left (a : R) : monomial 0 a = C a :=
  rfl
#align polynomial.monomial_zero_left Polynomial.monomial_zero_left

/- warning: polynomial.to_finsupp_C -> Polynomial.toFinsupp_C is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_C Polynomial.toFinsupp_Cₓ'. -/
@[simp]
theorem toFinsupp_C (a : R) : (C a).toFinsupp = single 0 a :=
  rfl
#align polynomial.to_finsupp_C Polynomial.toFinsupp_C

/- warning: polynomial.C_0 -> Polynomial.C_0 is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_0 Polynomial.C_0ₓ'. -/
theorem C_0 : C (0 : R) = 0 := by simp
#align polynomial.C_0 Polynomial.C_0

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Case conversion may be inaccurate. Consider using '#align polynomial.C_1 Polynomial.C_1ₓ'. -/
theorem C_1 : C (1 : R) = 1 :=
  rfl
#align polynomial.C_1 Polynomial.C_1

/- warning: polynomial.C_mul -> Polynomial.C_mul is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_mul Polynomial.C_mulₓ'. -/
theorem C_mul : C (a * b) = C a * C b :=
  C.map_mul a b
#align polynomial.C_mul Polynomial.C_mul

/- warning: polynomial.C_add -> Polynomial.C_add is a dubious translation:
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(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) b))
Case conversion may be inaccurate. Consider using '#align polynomial.C_add Polynomial.C_addₓ'. -/
theorem C_add : C (a + b) = C a + C b :=
  C.map_add a b
#align polynomial.C_add Polynomial.C_add

/- warning: polynomial.smul_C -> Polynomial.smul_C is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.smul_C Polynomial.smul_Cₓ'. -/
@[simp]
theorem smul_C {S} [Monoid S] [DistribMulAction S R] (s : S) (r : R) : s • C r = C (s • r) :=
  smul_monomial _ _ r
#align polynomial.smul_C Polynomial.smul_C

/- warning: polynomial.C_bit0 -> Polynomial.C_bit0 is a dubious translation:
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but is expected to have type
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(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) (bit0.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) a)) (bit0.{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) 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(Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
Case conversion may be inaccurate. Consider using '#align polynomial.C_bit0 Polynomial.C_bit0ₓ'. -/
@[simp]
theorem C_bit0 : C (bit0 a) = bit0 (C a) :=
  C_add
#align polynomial.C_bit0 Polynomial.C_bit0

/- warning: polynomial.C_bit1 -> Polynomial.C_bit1 is a dubious translation:
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but is expected to have type
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(RingHom.{u1, u1} R (Polynomial.{u1} 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) (bit1.{u1} R (Semiring.toOne.{u1} R _inst_1) (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) a)) (bit1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.one.{u1} R _inst_1) (Polynomial.add'.{u1} R _inst_1) 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(RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a))
Case conversion may be inaccurate. Consider using '#align polynomial.C_bit1 Polynomial.C_bit1ₓ'. -/
@[simp]
theorem C_bit1 : C (bit1 a) = bit1 (C a) := by simp [bit1, C_bit0]
#align polynomial.C_bit1 Polynomial.C_bit1

/- warning: polynomial.C_pow -> Polynomial.C_pow is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_pow Polynomial.C_powₓ'. -/
theorem C_pow : C (a ^ n) = C a ^ n :=
  C.map_pow a n
#align polynomial.C_pow Polynomial.C_pow

/- warning: polynomial.C_eq_nat_cast -> Polynomial.C_eq_nat_cast is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.C_eq_nat_cast Polynomial.C_eq_nat_castₓ'. -/
@[simp]
theorem C_eq_nat_cast (n : ℕ) : C (n : R) = (n : R[X]) :=
  map_natCast C n
#align polynomial.C_eq_nat_cast Polynomial.C_eq_nat_cast

/- warning: polynomial.C_mul_monomial -> Polynomial.C_mul_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_mul_monomial Polynomial.C_mul_monomialₓ'. -/
@[simp]
theorem C_mul_monomial : C a * monomial n b = monomial n (a * b) := by
  simp only [← monomial_zero_left, monomial_mul_monomial, zero_add]
#align polynomial.C_mul_monomial Polynomial.C_mul_monomial

/- warning: polynomial.monomial_mul_C -> Polynomial.monomial_mul_C is a dubious translation:
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(Polynomial.monomial.{u1} R _inst_1 n) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) a b))
Case conversion may be inaccurate. Consider using '#align polynomial.monomial_mul_C Polynomial.monomial_mul_Cₓ'. -/
@[simp]
theorem monomial_mul_C : monomial n a * C b = monomial n (a * b) := by
  simp only [← monomial_zero_left, monomial_mul_monomial, add_zero]
#align polynomial.monomial_mul_C Polynomial.monomial_mul_C

#print Polynomial.X /-
/-- `X` is the polynomial variable (aka indeterminate). -/
def X : R[X] :=
  monomial 1 1
#align polynomial.X Polynomial.X
-/

/- warning: polynomial.monomial_one_one_eq_X -> Polynomial.monomial_one_one_eq_X is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_one_one_eq_X Polynomial.monomial_one_one_eq_Xₓ'. -/
theorem monomial_one_one_eq_X : monomial 1 (1 : R) = X :=
  rfl
#align polynomial.monomial_one_one_eq_X Polynomial.monomial_one_one_eq_X

/- warning: polynomial.monomial_one_right_eq_X_pow -> Polynomial.monomial_one_right_eq_X_pow is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_one_right_eq_X_pow Polynomial.monomial_one_right_eq_X_powₓ'. -/
theorem monomial_one_right_eq_X_pow (n : ℕ) : monomial n (1 : R) = X ^ n :=
  by
  induction' n with n ih
  · simp [monomial_zero_one]
  · rw [pow_succ, ← ih, ← monomial_one_one_eq_X, monomial_mul_monomial, add_comm, one_mul]
#align polynomial.monomial_one_right_eq_X_pow Polynomial.monomial_one_right_eq_X_pow

/- warning: polynomial.to_finsupp_X -> Polynomial.toFinsupp_X is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1)) (Finsupp.single.{0, u1} Nat R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1)) (Finsupp.single.{0, u1} Nat R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1))))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_X Polynomial.toFinsupp_Xₓ'. -/
@[simp]
theorem toFinsupp_X : X.toFinsupp = Finsupp.single 1 (1 : R) :=
  rfl
#align polynomial.to_finsupp_X Polynomial.toFinsupp_X

#print Polynomial.X_mul /-
/-- `X` commutes with everything, even when the coefficients are noncommutative. -/
theorem X_mul : X * p = p * X := by
  rcases p with ⟨⟩
  simp only [X, ← of_finsupp_single, ← of_finsupp_mul, LinearMap.coe_mk]
  ext
  simp [AddMonoidAlgebra.mul_apply, sum_single_index, add_comm]
#align polynomial.X_mul Polynomial.X_mul
-/

#print Polynomial.X_pow_mul /-
theorem X_pow_mul {n : ℕ} : X ^ n * p = p * X ^ n :=
  by
  induction' n with n ih
  · simp
  · conv_lhs => rw [pow_succ']
    rw [mul_assoc, X_mul, ← mul_assoc, ih, mul_assoc, ← pow_succ']
#align polynomial.X_pow_mul Polynomial.X_pow_mul
-/

/- warning: polynomial.X_mul_C -> Polynomial.X_mul_C is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.X_mul_C Polynomial.X_mul_Cₓ'. -/
/-- Prefer putting constants to the left of `X`.

This lemma is the loop-avoiding `simp` version of `polynomial.X_mul`. -/
@[simp]
theorem X_mul_C (r : R) : X * C r = C r * X :=
  X_mul
#align polynomial.X_mul_C Polynomial.X_mul_C

/- warning: polynomial.X_pow_mul_C -> Polynomial.X_pow_mul_C is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.X_pow_mul_C Polynomial.X_pow_mul_Cₓ'. -/
/-- Prefer putting constants to the left of `X ^ n`.

This lemma is the loop-avoiding `simp` version of `X_pow_mul`. -/
@[simp]
theorem X_pow_mul_C (r : R) (n : ℕ) : X ^ n * C r = C r * X ^ n :=
  X_pow_mul
#align polynomial.X_pow_mul_C Polynomial.X_pow_mul_C

#print Polynomial.X_pow_mul_assoc /-
theorem X_pow_mul_assoc {n : ℕ} : p * X ^ n * q = p * q * X ^ n := by
  rw [mul_assoc, X_pow_mul, ← mul_assoc]
#align polynomial.X_pow_mul_assoc Polynomial.X_pow_mul_assoc
-/

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Case conversion may be inaccurate. Consider using '#align polynomial.X_pow_mul_assoc_C Polynomial.X_pow_mul_assoc_Cₓ'. -/
/-- Prefer putting constants to the left of `X ^ n`.

This lemma is the loop-avoiding `simp` version of `X_pow_mul_assoc`. -/
@[simp]
theorem X_pow_mul_assoc_C {n : ℕ} (r : R) : p * X ^ n * C r = p * C r * X ^ n :=
  X_pow_mul_assoc
#align polynomial.X_pow_mul_assoc_C Polynomial.X_pow_mul_assoc_C

#print Polynomial.commute_X /-
theorem commute_X (p : R[X]) : Commute X p :=
  X_mul
#align polynomial.commute_X Polynomial.commute_X
-/

#print Polynomial.commute_X_pow /-
theorem commute_X_pow (p : R[X]) (n : ℕ) : Commute (X ^ n) p :=
  X_pow_mul
#align polynomial.commute_X_pow Polynomial.commute_X_pow
-/

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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_mul_X Polynomial.monomial_mul_Xₓ'. -/
@[simp]
theorem monomial_mul_X (n : ℕ) (r : R) : monomial n r * X = monomial (n + 1) r := by
  erw [monomial_mul_monomial, mul_one]
#align polynomial.monomial_mul_X Polynomial.monomial_mul_X

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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_mul_X_pow Polynomial.monomial_mul_X_powₓ'. -/
@[simp]
theorem monomial_mul_X_pow (n : ℕ) (r : R) (k : ℕ) : monomial n r * X ^ k = monomial (n + k) r :=
  by
  induction' k with k ih
  · simp
  · simp [ih, pow_succ', ← mul_assoc, add_assoc]
#align polynomial.monomial_mul_X_pow Polynomial.monomial_mul_X_pow

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Case conversion may be inaccurate. Consider using '#align polynomial.X_mul_monomial Polynomial.X_mul_monomialₓ'. -/
@[simp]
theorem X_mul_monomial (n : ℕ) (r : R) : X * monomial n r = monomial (n + 1) r := by
  rw [X_mul, monomial_mul_X]
#align polynomial.X_mul_monomial Polynomial.X_mul_monomial

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Case conversion may be inaccurate. Consider using '#align polynomial.X_pow_mul_monomial Polynomial.X_pow_mul_monomialₓ'. -/
@[simp]
theorem X_pow_mul_monomial (k n : ℕ) (r : R) : X ^ k * monomial n r = monomial (n + k) r := by
  rw [X_pow_mul, monomial_mul_X_pow]
#align polynomial.X_pow_mul_monomial Polynomial.X_pow_mul_monomial

#print Polynomial.coeff /-
/-- `coeff p n` (often denoted `p.coeff n`) is the coefficient of `X^n` in `p`. -/
@[simp]
def coeff : R[X] → ℕ → R
  | ⟨p⟩ => p
#align polynomial.coeff Polynomial.coeff
-/

#print Polynomial.coeff_injective /-
theorem coeff_injective : Injective (coeff : R[X] → ℕ → R) :=
  by
  rintro ⟨p⟩ ⟨q⟩
  simp only [coeff, FunLike.coe_fn_eq, imp_self]
#align polynomial.coeff_injective Polynomial.coeff_injective
-/

#print Polynomial.coeff_inj /-
@[simp]
theorem coeff_inj : p.coeff = q.coeff ↔ p = q :=
  coeff_injective.eq_iff
#align polynomial.coeff_inj Polynomial.coeff_inj
-/

#print Polynomial.toFinsupp_apply /-
theorem toFinsupp_apply (f : R[X]) (i) : f.toFinsupp i = f.coeff i := by cases f <;> rfl
#align polynomial.to_finsupp_apply Polynomial.toFinsupp_apply
-/

/- warning: polynomial.coeff_monomial -> Polynomial.coeff_monomial is a dubious translation:
lean 3 declaration is
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but is expected to have type
  forall {R : Type.{u1}} {a : R} {m : Nat} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a) m) (ite.{succ u1} R (Eq.{1} Nat n m) (instDecidableEqNat n m) a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_monomial Polynomial.coeff_monomialₓ'. -/
theorem coeff_monomial : coeff (monomial n a) m = if n = m then a else 0 :=
  by
  simp only [← of_finsupp_single, coeff, LinearMap.coe_mk]
  rw [Finsupp.single_apply]
#align polynomial.coeff_monomial Polynomial.coeff_monomial

/- warning: polynomial.coeff_zero -> Polynomial.coeff_zero 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] (n : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (OfNat.ofNat.{u1} (Polynomial.{u1} R _inst_1) 0 (Zero.toOfNat0.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1))) n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_zero Polynomial.coeff_zeroₓ'. -/
@[simp]
theorem coeff_zero (n : ℕ) : coeff (0 : R[X]) n = 0 :=
  rfl
#align polynomial.coeff_zero Polynomial.coeff_zero

#print Polynomial.coeff_one_zero /-
@[simp]
theorem coeff_one_zero : coeff (1 : R[X]) 0 = 1 :=
  by
  rw [← monomial_zero_one, coeff_monomial]
  simp
#align polynomial.coeff_one_zero Polynomial.coeff_one_zero
-/

#print Polynomial.coeff_X_one /-
@[simp]
theorem coeff_X_one : coeff (X : R[X]) 1 = 1 :=
  coeff_monomial
#align polynomial.coeff_X_one Polynomial.coeff_X_one
-/

/- warning: polynomial.coeff_X_zero -> Polynomial.coeff_X_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) (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)))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) (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))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_X_zero Polynomial.coeff_X_zeroₓ'. -/
@[simp]
theorem coeff_X_zero : coeff (X : R[X]) 0 = 0 :=
  coeff_monomial
#align polynomial.coeff_X_zero Polynomial.coeff_X_zero

/- warning: polynomial.coeff_monomial_succ -> Polynomial.coeff_monomial_succ is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) a) (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)))))))
but is expected to have type
  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) a) (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))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_monomial_succ Polynomial.coeff_monomial_succₓ'. -/
@[simp]
theorem coeff_monomial_succ : coeff (monomial (n + 1) a) 0 = 0 := by simp [coeff_monomial]
#align polynomial.coeff_monomial_succ Polynomial.coeff_monomial_succ

/- warning: polynomial.coeff_X -> Polynomial.coeff_X is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) n) (ite.{succ u1} R (Eq.{1} Nat (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) n) (Nat.decidableEq (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) n) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
but is expected to have type
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) n) (ite.{succ u1} R (Eq.{1} Nat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) n) (instDecidableEqNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) n) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1))) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_X Polynomial.coeff_Xₓ'. -/
theorem coeff_X : coeff (X : R[X]) n = if 1 = n then 1 else 0 :=
  coeff_monomial
#align polynomial.coeff_X Polynomial.coeff_X

/- warning: polynomial.coeff_X_of_ne_one -> Polynomial.coeff_X_of_ne_one is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) n) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.X.{u1} R _inst_1) n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_X_of_ne_one Polynomial.coeff_X_of_ne_oneₓ'. -/
theorem coeff_X_of_ne_one {n : ℕ} (hn : n ≠ 1) : coeff (X : R[X]) n = 0 := by
  rw [coeff_X, if_neg hn.symm]
#align polynomial.coeff_X_of_ne_one Polynomial.coeff_X_of_ne_one

/- warning: polynomial.mem_support_iff -> Polynomial.mem_support_iff is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Membership.Mem.{0, 0} Nat (Finset.{0} Nat) (Finset.hasMem.{0} Nat) n (Polynomial.support.{u1} R _inst_1 p)) (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
but is expected to have type
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Membership.mem.{0, 0} Nat (Finset.{0} Nat) (Finset.instMembershipFinset.{0} Nat) n (Polynomial.support.{u1} R _inst_1 p)) (Ne.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.mem_support_iff Polynomial.mem_support_iffₓ'. -/
@[simp]
theorem mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 :=
  by
  rcases p with ⟨⟩
  simp
#align polynomial.mem_support_iff Polynomial.mem_support_iff

/- warning: polynomial.not_mem_support_iff -> Polynomial.not_mem_support_iff is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Not (Membership.Mem.{0, 0} Nat (Finset.{0} Nat) (Finset.hasMem.{0} Nat) n (Polynomial.support.{u1} R _inst_1 p))) (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))
but is expected to have type
  forall {R : Type.{u1}} {n : Nat} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1}, Iff (Not (Membership.mem.{0, 0} Nat (Finset.{0} Nat) (Finset.instMembershipFinset.{0} Nat) n (Polynomial.support.{u1} R _inst_1 p))) (Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 p n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.not_mem_support_iff Polynomial.not_mem_support_iffₓ'. -/
theorem not_mem_support_iff : n ∉ p.support ↔ p.coeff n = 0 := by simp
#align polynomial.not_mem_support_iff Polynomial.not_mem_support_iff

/- warning: polynomial.coeff_C -> Polynomial.coeff_C is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{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) n) (ite.{succ u1} R (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (Nat.decidableEq n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) 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))))))))
but is expected to have type
  forall {R : Type.{u1}} {a : R} {n : Nat} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{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) n) (ite.{succ u1} R (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (instDecidableEqNat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_C Polynomial.coeff_Cₓ'. -/
theorem coeff_C : coeff (C a) n = ite (n = 0) a 0 :=
  by
  convert coeff_monomial using 2
  simp [eq_comm]
#align polynomial.coeff_C Polynomial.coeff_C

/- warning: polynomial.coeff_C_zero -> Polynomial.coeff_C_zero is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} {a : R} [_inst_1 : Semiring.{u1} R], Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) a
but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.coeff_C_zero Polynomial.coeff_C_zeroₓ'. -/
@[simp]
theorem coeff_C_zero : coeff (C a) 0 = a :=
  coeff_monomial
#align polynomial.coeff_C_zero Polynomial.coeff_C_zero

/- warning: polynomial.coeff_C_ne_zero -> Polynomial.coeff_C_ne_zero is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.coeff_C_ne_zero Polynomial.coeff_C_ne_zeroₓ'. -/
theorem coeff_C_ne_zero (h : n ≠ 0) : (C a).coeff n = 0 := by rw [coeff_C, if_neg h]
#align polynomial.coeff_C_ne_zero Polynomial.coeff_C_ne_zero

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Case conversion may be inaccurate. Consider using '#align polynomial.C_mul_X_pow_eq_monomial Polynomial.C_mul_X_pow_eq_monomialₓ'. -/
theorem C_mul_X_pow_eq_monomial : ∀ {n : ℕ}, C a * X ^ n = monomial n a
  | 0 => mul_one _
  | n + 1 => by
    rw [pow_succ', ← mul_assoc, C_mul_X_pow_eq_monomial, X, monomial_mul_monomial, mul_one]
#align polynomial.C_mul_X_pow_eq_monomial Polynomial.C_mul_X_pow_eq_monomial

/- warning: polynomial.to_finsupp_C_mul_X_pow -> Polynomial.toFinsupp_C_mul_X_pow is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_C_mul_X_pow Polynomial.toFinsupp_C_mul_X_powₓ'. -/
@[simp]
theorem toFinsupp_C_mul_X_pow (a : R) (n : ℕ) : (C a * X ^ n).toFinsupp = Finsupp.single n a := by
  rw [C_mul_X_pow_eq_monomial, to_finsupp_monomial]
#align polynomial.to_finsupp_C_mul_X_pow Polynomial.toFinsupp_C_mul_X_pow

/- warning: polynomial.C_mul_X_eq_monomial -> Polynomial.C_mul_X_eq_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_mul_X_eq_monomial Polynomial.C_mul_X_eq_monomialₓ'. -/
theorem C_mul_X_eq_monomial : C a * X = monomial 1 a := by rw [← C_mul_X_pow_eq_monomial, pow_one]
#align polynomial.C_mul_X_eq_monomial Polynomial.C_mul_X_eq_monomial

/- warning: polynomial.to_finsupp_C_mul_X -> Polynomial.toFinsupp_C_mul_X is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_C_mul_X Polynomial.toFinsupp_C_mul_Xₓ'. -/
@[simp]
theorem toFinsupp_C_mul_X (a : R) : (C a * X).toFinsupp = Finsupp.single 1 a := by
  rw [C_mul_X_eq_monomial, to_finsupp_monomial]
#align polynomial.to_finsupp_C_mul_X Polynomial.toFinsupp_C_mul_X

/- warning: polynomial.C_injective -> Polynomial.C_injective is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.C_injective Polynomial.C_injectiveₓ'. -/
theorem C_injective : Injective (C : R → R[X]) :=
  monomial_injective 0
#align polynomial.C_injective Polynomial.C_injective

/- warning: polynomial.C_inj -> Polynomial.C_inj is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.C_inj Polynomial.C_injₓ'. -/
@[simp]
theorem C_inj : C a = C b ↔ a = b :=
  C_injective.eq_iff
#align polynomial.C_inj Polynomial.C_inj

/- warning: polynomial.C_eq_zero -> Polynomial.C_eq_zero is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.C_eq_zero Polynomial.C_eq_zeroₓ'. -/
@[simp]
theorem C_eq_zero : C a = 0 ↔ a = 0 :=
  C_injective.eq_iff' (map_zero C)
#align polynomial.C_eq_zero Polynomial.C_eq_zero

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Case conversion may be inaccurate. Consider using '#align polynomial.C_ne_zero Polynomial.C_ne_zeroₓ'. -/
theorem C_ne_zero : C a ≠ 0 ↔ a ≠ 0 :=
  C_eq_zero.Not
#align polynomial.C_ne_zero Polynomial.C_ne_zero

#print Polynomial.subsingleton_iff_subsingleton /-
theorem subsingleton_iff_subsingleton : Subsingleton R[X] ↔ Subsingleton R :=
  ⟨@Injective.subsingleton _ _ _ C_injective, by
    intro
    infer_instance⟩
#align polynomial.subsingleton_iff_subsingleton Polynomial.subsingleton_iff_subsingleton
-/

#print Polynomial.Nontrivial.of_polynomial_ne /-
theorem Nontrivial.of_polynomial_ne (h : p ≠ q) : Nontrivial R :=
  (subsingleton_or_nontrivial R).resolve_left fun hI => h <| Subsingleton.elim _ _
#align polynomial.nontrivial.of_polynomial_ne Polynomial.Nontrivial.of_polynomial_ne
-/

#print Polynomial.forall_eq_iff_forall_eq /-
theorem forall_eq_iff_forall_eq : (∀ f g : R[X], f = g) ↔ ∀ a b : R, a = b := by
  simpa only [← subsingleton_iff] using subsingleton_iff_subsingleton
#align polynomial.forall_eq_iff_forall_eq Polynomial.forall_eq_iff_forall_eq
-/

#print Polynomial.ext_iff /-
theorem ext_iff {p q : R[X]} : p = q ↔ ∀ n, coeff p n = coeff q n :=
  by
  rcases p with ⟨⟩
  rcases q with ⟨⟩
  simp [coeff, Finsupp.ext_iff]
#align polynomial.ext_iff Polynomial.ext_iff
-/

#print Polynomial.ext /-
@[ext]
theorem ext {p q : R[X]} : (∀ n, coeff p n = coeff q n) → p = q :=
  ext_iff.2
#align polynomial.ext Polynomial.ext
-/

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Case conversion may be inaccurate. Consider using '#align polynomial.add_submonoid_closure_set_of_eq_monomial Polynomial.addSubmonoid_closure_setOf_eq_monomialₓ'. -/
/-- Monomials generate the additive monoid of polynomials. -/
theorem addSubmonoid_closure_setOf_eq_monomial :
    AddSubmonoid.closure { p : R[X] | ∃ n a, p = monomial n a } = ⊤ :=
  by
  apply top_unique
  rw [← AddSubmonoid.map_equiv_top (to_finsupp_iso R).symm.toAddEquiv, ←
    Finsupp.add_closure_setOf_eq_single, AddMonoidHom.map_mclosure]
  refine' AddSubmonoid.closure_mono (Set.image_subset_iff.2 _)
  rintro _ ⟨n, a, rfl⟩
  exact ⟨n, a, Polynomial.ofFinsupp_single _ _⟩
#align polynomial.add_submonoid_closure_set_of_eq_monomial Polynomial.addSubmonoid_closure_setOf_eq_monomial

/- warning: polynomial.add_hom_ext -> Polynomial.addHom_ext is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.add_hom_ext Polynomial.addHom_extₓ'. -/
theorem addHom_ext {M : Type _} [AddMonoid M] {f g : R[X] →+ M}
    (h : ∀ n a, f (monomial n a) = g (monomial n a)) : f = g :=
  AddMonoidHom.eq_of_eqOn_denseM addSubmonoid_closure_setOf_eq_monomial <|
    by
    rintro p ⟨n, a, rfl⟩
    exact h n a
#align polynomial.add_hom_ext Polynomial.addHom_ext

/- warning: polynomial.add_hom_ext' -> Polynomial.addHom_ext' is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.add_hom_ext' Polynomial.addHom_ext'ₓ'. -/
@[ext]
theorem addHom_ext' {M : Type _} [AddMonoid M] {f g : R[X] →+ M}
    (h : ∀ n, f.comp (monomial n).toAddMonoidHom = g.comp (monomial n).toAddMonoidHom) : f = g :=
  addHom_ext fun n => AddMonoidHom.congr_fun (h n)
#align polynomial.add_hom_ext' Polynomial.addHom_ext'

/- warning: polynomial.lhom_ext' -> Polynomial.lhom_ext' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.lhom_ext' Polynomial.lhom_ext'ₓ'. -/
@[ext]
theorem lhom_ext' {M : Type _} [AddCommMonoid M] [Module R M] {f g : R[X] →ₗ[R] M}
    (h : ∀ n, f.comp (monomial n) = g.comp (monomial n)) : f = g :=
  LinearMap.toAddMonoidHom_injective <| addHom_ext fun n => LinearMap.congr_fun (h n)
#align polynomial.lhom_ext' Polynomial.lhom_ext'

/- warning: polynomial.eq_zero_of_eq_zero -> Polynomial.eq_zero_of_eq_zero is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.eq_zero_of_eq_zero Polynomial.eq_zero_of_eq_zeroₓ'. -/
-- this has the same content as the subsingleton
theorem eq_zero_of_eq_zero (h : (0 : R) = (1 : R)) (p : R[X]) : p = 0 := by
  rw [← one_smul R p, ← h, zero_smul]
#align polynomial.eq_zero_of_eq_zero Polynomial.eq_zero_of_eq_zero

section Fewnomials

/- warning: polynomial.support_monomial -> Polynomial.support_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_monomial Polynomial.support_monomialₓ'. -/
theorem support_monomial (n) {a : R} (H : a ≠ 0) : (monomial n a).support = singleton n := by
  rw [← of_finsupp_single, support, Finsupp.support_single_ne_zero _ H]
#align polynomial.support_monomial Polynomial.support_monomial

/- warning: polynomial.support_monomial' -> Polynomial.support_monomial' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_monomial' Polynomial.support_monomial'ₓ'. -/
theorem support_monomial' (n) (a : R) : (monomial n a).support ⊆ singleton n :=
  by
  rw [← of_finsupp_single, support]
  exact Finsupp.support_single_subset
#align polynomial.support_monomial' Polynomial.support_monomial'

/- warning: polynomial.support_C_mul_X -> Polynomial.support_C_mul_X is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_C_mul_X Polynomial.support_C_mul_Xₓ'. -/
theorem support_C_mul_X {c : R} (h : c ≠ 0) : (C c * X).support = singleton 1 := by
  rw [C_mul_X_eq_monomial, support_monomial 1 h]
#align polynomial.support_C_mul_X Polynomial.support_C_mul_X

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Case conversion may be inaccurate. Consider using '#align polynomial.support_C_mul_X' Polynomial.support_C_mul_X'ₓ'. -/
theorem support_C_mul_X' (c : R) : (C c * X).support ⊆ singleton 1 := by
  simpa only [C_mul_X_eq_monomial] using support_monomial' 1 c
#align polynomial.support_C_mul_X' Polynomial.support_C_mul_X'

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Case conversion may be inaccurate. Consider using '#align polynomial.support_C_mul_X_pow Polynomial.support_C_mul_X_powₓ'. -/
theorem support_C_mul_X_pow (n : ℕ) {c : R} (h : c ≠ 0) : (C c * X ^ n).support = singleton n := by
  rw [C_mul_X_pow_eq_monomial, support_monomial n h]
#align polynomial.support_C_mul_X_pow Polynomial.support_C_mul_X_pow

/- warning: polynomial.support_C_mul_X_pow' -> Polynomial.support_C_mul_X_pow' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_C_mul_X_pow' Polynomial.support_C_mul_X_pow'ₓ'. -/
theorem support_C_mul_X_pow' (n : ℕ) (c : R) : (C c * X ^ n).support ⊆ singleton n := by
  simpa only [C_mul_X_pow_eq_monomial] using support_monomial' n c
#align polynomial.support_C_mul_X_pow' Polynomial.support_C_mul_X_pow'

open Finset

/- warning: polynomial.support_binomial' -> Polynomial.support_binomial' is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_binomial' Polynomial.support_binomial'ₓ'. -/
theorem support_binomial' (k m : ℕ) (x y : R) : (C x * X ^ k + C y * X ^ m).support ⊆ {k, m} :=
  support_add.trans
    (union_subset
      ((support_C_mul_X_pow' k x).trans (singleton_subset_iff.mpr (mem_insert_self k {m})))
      ((support_C_mul_X_pow' m y).trans
        (singleton_subset_iff.mpr (mem_insert_of_mem (mem_singleton_self m)))))
#align polynomial.support_binomial' Polynomial.support_binomial'

/- warning: polynomial.support_trinomial' -> Polynomial.support_trinomial' is a dubious translation:
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(Finset.instInsertFinset.{0} Nat (fun (a : Nat) (b : Nat) => instDecidableEqNat a b)) k (Insert.insert.{0, 0} Nat (Finset.{0} Nat) (Finset.instInsertFinset.{0} Nat (fun (a : Nat) (b : Nat) => instDecidableEqNat a b)) m (Singleton.singleton.{0, 0} Nat (Finset.{0} Nat) (Finset.instSingletonFinset.{0} Nat) n)))
Case conversion may be inaccurate. Consider using '#align polynomial.support_trinomial' Polynomial.support_trinomial'ₓ'. -/
theorem support_trinomial' (k m n : ℕ) (x y z : R) :
    (C x * X ^ k + C y * X ^ m + C z * X ^ n).support ⊆ {k, m, n} :=
  support_add.trans
    (union_subset
      (support_add.trans
        (union_subset
          ((support_C_mul_X_pow' k x).trans (singleton_subset_iff.mpr (mem_insert_self k {m, n})))
          ((support_C_mul_X_pow' m y).trans
            (singleton_subset_iff.mpr (mem_insert_of_mem (mem_insert_self m {n}))))))
      ((support_C_mul_X_pow' n z).trans
        (singleton_subset_iff.mpr (mem_insert_of_mem (mem_insert_of_mem (mem_singleton_self n))))))
#align polynomial.support_trinomial' Polynomial.support_trinomial'

end Fewnomials

/- warning: polynomial.X_pow_eq_monomial -> Polynomial.X_pow_eq_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.X_pow_eq_monomial Polynomial.X_pow_eq_monomialₓ'. -/
theorem X_pow_eq_monomial (n) : X ^ n = monomial n (1 : R) :=
  by
  induction' n with n hn
  · rw [pow_zero, monomial_zero_one]
  · rw [pow_succ', hn, X, monomial_mul_monomial, one_mul]
#align polynomial.X_pow_eq_monomial Polynomial.X_pow_eq_monomial

/- warning: polynomial.to_finsupp_X_pow -> Polynomial.toFinsupp_X_pow is a dubious translation:
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but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (n : Nat), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n)) (Finsupp.single.{0, u1} Nat R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) n (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1))))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_X_pow Polynomial.toFinsupp_X_powₓ'. -/
@[simp]
theorem toFinsupp_X_pow (n : ℕ) : (X ^ n).toFinsupp = Finsupp.single n (1 : R) := by
  rw [X_pow_eq_monomial, to_finsupp_monomial]
#align polynomial.to_finsupp_X_pow Polynomial.toFinsupp_X_pow

/- warning: polynomial.smul_X_eq_monomial -> Polynomial.smul_X_eq_monomial is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.smul_X_eq_monomial Polynomial.smul_X_eq_monomialₓ'. -/
theorem smul_X_eq_monomial {n} : a • X ^ n = monomial n (a : R) := by
  rw [X_pow_eq_monomial, smul_monomial, smul_eq_mul, mul_one]
#align polynomial.smul_X_eq_monomial Polynomial.smul_X_eq_monomial

/- warning: polynomial.support_X_pow -> Polynomial.support_X_pow is a dubious translation:
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.support_X_pow Polynomial.support_X_powₓ'. -/
theorem support_X_pow (H : ¬(1 : R) = 0) (n : ℕ) : (X ^ n : R[X]).support = singleton n :=
  by
  convert support_monomial n H
  exact X_pow_eq_monomial n
#align polynomial.support_X_pow Polynomial.support_X_pow

/- warning: polynomial.support_X_empty -> Polynomial.support_X_empty is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_X_empty Polynomial.support_X_emptyₓ'. -/
theorem support_X_empty (H : (1 : R) = 0) : (X : R[X]).support = ∅ := by
  rw [X, H, monomial_zero_right, support_zero]
#align polynomial.support_X_empty Polynomial.support_X_empty

/- warning: polynomial.support_X -> Polynomial.support_X is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.support_X Polynomial.support_Xₓ'. -/
theorem support_X (H : ¬(1 : R) = 0) : (X : R[X]).support = singleton 1 := by
  rw [← pow_one X, support_X_pow H 1]
#align polynomial.support_X Polynomial.support_X

/- warning: polynomial.monomial_left_inj -> Polynomial.monomial_left_inj is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_left_inj Polynomial.monomial_left_injₓ'. -/
theorem monomial_left_inj {a : R} (ha : a ≠ 0) {i j : ℕ} : monomial i a = monomial j a ↔ i = j := by
  simp_rw [← of_finsupp_single, Finsupp.single_left_inj ha]
#align polynomial.monomial_left_inj Polynomial.monomial_left_inj

/- warning: polynomial.binomial_eq_binomial -> Polynomial.binomial_eq_binomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.binomial_eq_binomial Polynomial.binomial_eq_binomialₓ'. -/
theorem binomial_eq_binomial {k l m n : ℕ} {u v : R} (hu : u ≠ 0) (hv : v ≠ 0) :
    C u * X ^ k + C v * X ^ l = C u * X ^ m + C v * X ^ n ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n :=
  by
  simp_rw [C_mul_X_pow_eq_monomial, ← to_finsupp_inj, to_finsupp_add, to_finsupp_monomial]
  exact Finsupp.single_add_single_eq_single_add_single hu hv
#align polynomial.binomial_eq_binomial Polynomial.binomial_eq_binomial

#print Polynomial.nat_cast_mul /-
theorem nat_cast_mul (n : ℕ) (p : R[X]) : (n : R[X]) * p = n • p :=
  (nsmul_eq_mul _ _).symm
#align polynomial.nat_cast_mul Polynomial.nat_cast_mul
-/

#print Polynomial.sum /-
/-- Summing the values of a function applied to the coefficients of a polynomial -/
def sum {S : Type _} [AddCommMonoid S] (p : R[X]) (f : ℕ → R → S) : S :=
  ∑ n in p.support, f n (p.coeff n)
#align polynomial.sum Polynomial.sum
-/

/- warning: polynomial.sum_def -> Polynomial.sum_def is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (f : Nat -> R -> S), Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p f) (Finset.sum.{u2, 0} S Nat _inst_2 (Polynomial.support.{u1} R _inst_1 p) (fun (n : Nat) => f n (Polynomial.coeff.{u1} R _inst_1 p n)))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (f : Nat -> R -> S), Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p f) (Finset.sum.{u1, 0} S Nat _inst_2 (Polynomial.support.{u2} R _inst_1 p) (fun (n : Nat) => f n (Polynomial.coeff.{u2} R _inst_1 p n)))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_def Polynomial.sum_defₓ'. -/
theorem sum_def {S : Type _} [AddCommMonoid S] (p : R[X]) (f : ℕ → R → S) :
    p.Sum f = ∑ n in p.support, f n (p.coeff n) :=
  rfl
#align polynomial.sum_def Polynomial.sum_def

/- warning: polynomial.sum_eq_of_subset -> Polynomial.sum_eq_of_subset is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u2} S (f i (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))) -> (forall (s : Finset.{0} Nat), (HasSubset.Subset.{0} (Finset.{0} Nat) (Finset.hasSubset.{0} Nat) (Polynomial.support.{u1} R _inst_1 p) s) -> (Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p f) (Finset.sum.{u2, 0} S Nat _inst_2 s (fun (n : Nat) => f n (Polynomial.coeff.{u1} R _inst_1 p n)))))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u1} S (f i (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))) -> (forall (s : Finset.{0} Nat), (HasSubset.Subset.{0} (Finset.{0} Nat) (Finset.instHasSubsetFinset.{0} Nat) (Polynomial.support.{u2} R _inst_1 p) s) -> (Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p f) (Finset.sum.{u1, 0} S Nat _inst_2 s (fun (n : Nat) => f n (Polynomial.coeff.{u2} R _inst_1 p n)))))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_eq_of_subset Polynomial.sum_eq_of_subsetₓ'. -/
theorem sum_eq_of_subset {S : Type _} [AddCommMonoid S] (p : R[X]) (f : ℕ → R → S)
    (hf : ∀ i, f i 0 = 0) (s : Finset ℕ) (hs : p.support ⊆ s) :
    p.Sum f = ∑ n in s, f n (p.coeff n) :=
  by
  apply Finset.sum_subset hs fun n hn h'n => _
  rw [not_mem_support_iff] at h'n
  simp [h'n, hf]
#align polynomial.sum_eq_of_subset Polynomial.sum_eq_of_subset

/- warning: polynomial.mul_eq_sum_sum -> Polynomial.mul_eq_sum_sum is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q) (Finset.sum.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (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)))) (Polynomial.support.{u1} R _inst_1 p) (fun (i : Nat) => Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) q (fun (j : Nat) (a : R) => 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 (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) i j)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (Polynomial.coeff.{u1} R _inst_1 p i) a))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {p : Polynomial.{u1} R _inst_1} {q : Polynomial.{u1} R _inst_1}, Eq.{succ u1} (Polynomial.{u1} R _inst_1) (HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) p q) (Finset.sum.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (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)))) (Polynomial.support.{u1} R _inst_1 p) (fun (i : Nat) => Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) q (fun (j : Nat) (a : R) => 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 (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) i j)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (Polynomial.coeff.{u1} R _inst_1 p i) a))))
Case conversion may be inaccurate. Consider using '#align polynomial.mul_eq_sum_sum Polynomial.mul_eq_sum_sumₓ'. -/
/-- Expressing the product of two polynomials as a double sum. -/
theorem mul_eq_sum_sum :
    p * q = ∑ i in p.support, q.Sum fun j a => (monomial (i + j)) (p.coeff i * a) :=
  by
  apply to_finsupp_injective
  rcases p with ⟨⟩; rcases q with ⟨⟩
  simp [support, Sum, coeff, to_finsupp_sum]
  rfl
#align polynomial.mul_eq_sum_sum Polynomial.mul_eq_sum_sum

/- warning: polynomial.sum_zero_index -> Polynomial.sum_zero_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (f : Nat -> R -> S), Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 (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)))) f) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (f : Nat -> R -> S), Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 (OfNat.ofNat.{u2} (Polynomial.{u2} R _inst_1) 0 (Zero.toOfNat0.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1))) f) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_zero_index Polynomial.sum_zero_indexₓ'. -/
@[simp]
theorem sum_zero_index {S : Type _} [AddCommMonoid S] (f : ℕ → R → S) : (0 : R[X]).Sum f = 0 := by
  simp [Sum]
#align polynomial.sum_zero_index Polynomial.sum_zero_index

/- warning: polynomial.sum_monomial_index -> Polynomial.sum_monomial_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (n : Nat) (a : R) (f : Nat -> R -> S), (Eq.{succ u2} S (f n (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))) -> (Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 (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) f) (f n a))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (n : Nat) (a : R) (f : Nat -> R -> S), (Eq.{succ u1} S (f n (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))) -> (Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u2, u2, u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R _inst_1) (Polynomial.module.{u2, u2} R _inst_1 R _inst_1 (Semiring.toModule.{u2} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u2} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u2, u2} R R R (Polynomial.{u2} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R _inst_1) (Polynomial.module.{u2, u2} R _inst_1 R _inst_1 (Semiring.toModule.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Polynomial.monomial.{u2} R _inst_1 n) a) f) (f n a))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_monomial_index Polynomial.sum_monomial_indexₓ'. -/
@[simp]
theorem sum_monomial_index {S : Type _} [AddCommMonoid S] (n : ℕ) (a : R) (f : ℕ → R → S)
    (hf : f n 0 = 0) : (monomial n a : R[X]).Sum f = f n a :=
  by
  by_cases h : a = 0
  · simp [h, hf]
  · simp [Sum, support_monomial, h, coeff_monomial]
#align polynomial.sum_monomial_index Polynomial.sum_monomial_index

/- warning: polynomial.sum_C_index -> Polynomial.sum_C_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {a : R} {β : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} β] {f : Nat -> R -> β}, (Eq.{succ u2} β (f (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)))))))) (OfNat.ofNat.{u2} β 0 (OfNat.mk.{u2} β 0 (Zero.zero.{u2} β (AddZeroClass.toHasZero.{u2} β (AddMonoid.toAddZeroClass.{u2} β (AddCommMonoid.toAddMonoid.{u2} β _inst_2))))))) -> (Eq.{succ u2} β (Polynomial.sum.{u1, u2} R _inst_1 β _inst_2 (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) f) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) a))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {a : R} {β : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} β] {f : Nat -> R -> β}, (Eq.{succ u1} β (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} β 0 (Zero.toOfNat0.{u1} β (AddMonoid.toZero.{u1} β (AddCommMonoid.toAddMonoid.{u1} β _inst_2))))) -> (Eq.{succ u1} β (Polynomial.sum.{u2, u1} R _inst_1 β _inst_2 (FunLike.coe.{succ u2, 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))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u2} R _inst_1) _x) (MulHomClass.toFunLike.{u2, u2, 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))) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, 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))) R (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, 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))) 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)) (RingHom.instRingHomClassRingHom.{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) f) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) a))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_C_index Polynomial.sum_C_indexₓ'. -/
@[simp]
theorem sum_C_index {a} {β} [AddCommMonoid β] {f : ℕ → R → β} (h : f 0 0 = 0) :
    (C a).Sum f = f 0 a :=
  sum_monomial_index 0 a f h
#align polynomial.sum_C_index Polynomial.sum_C_index

/- warning: polynomial.sum_X_index -> Polynomial.sum_X_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] {f : Nat -> R -> S}, (Eq.{succ u2} S (f (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))) -> (Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 (Polynomial.X.{u1} R _inst_1) f) (f (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] {f : Nat -> R -> S}, (Eq.{succ u1} S (f (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))) -> (Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 (Polynomial.X.{u2} R _inst_1) f) (f (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (Semiring.toOne.{u2} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_X_index Polynomial.sum_X_indexₓ'. -/
-- the assumption `hf` is only necessary when the ring is trivial
@[simp]
theorem sum_X_index {S : Type _} [AddCommMonoid S] {f : ℕ → R → S} (hf : f 1 0 = 0) :
    (X : R[X]).Sum f = f 1 1 :=
  sum_monomial_index 1 1 f hf
#align polynomial.sum_X_index Polynomial.sum_X_index

/- warning: polynomial.sum_add_index -> Polynomial.sum_add_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (q : Polynomial.{u1} R _inst_1) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u2} S (f i (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))) -> (forall (a : Nat) (b₁ : R) (b₂ : R), Eq.{succ u2} S (f a (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) b₁ b₂)) (HAdd.hAdd.{u2, u2, u2} S S S (instHAdd.{u2} S (AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))) (f a b₁) (f a b₂))) -> (Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 (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) f) (HAdd.hAdd.{u2, u2, u2} S S S (instHAdd.{u2} S (AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 q f)))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (q : Polynomial.{u2} R _inst_1) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u1} S (f i (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))) -> (forall (a : Nat) (b₁ : R) (b₂ : R), Eq.{succ u1} S (f a (HAdd.hAdd.{u2, u2, u2} R R R (instHAdd.{u2} R (Distrib.toAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) b₁ b₂)) (HAdd.hAdd.{u1, u1, u1} S S S (instHAdd.{u1} S (AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))) (f a b₁) (f a b₂))) -> (Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 (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 q) f) (HAdd.hAdd.{u1, u1, u1} S S S (instHAdd.{u1} S (AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 q f)))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_add_index Polynomial.sum_add_indexₓ'. -/
theorem sum_add_index {S : Type _} [AddCommMonoid S] (p q : R[X]) (f : ℕ → R → S)
    (hf : ∀ i, f i 0 = 0) (h_add : ∀ a b₁ b₂, f a (b₁ + b₂) = f a b₁ + f a b₂) :
    (p + q).Sum f = p.Sum f + q.Sum f := by
  rcases p with ⟨⟩; rcases q with ⟨⟩
  simp only [← of_finsupp_add, Sum, support, coeff, Pi.add_apply, coe_add]
  exact Finsupp.sum_add_index' hf h_add
#align polynomial.sum_add_index Polynomial.sum_add_index

/- warning: polynomial.sum_add' -> Polynomial.sum_add' is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (f : Nat -> R -> S) (g : Nat -> R -> S), Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Nat -> R -> S) (Nat -> R -> S) (Nat -> R -> S) (instHAdd.{max u1 u2} (Nat -> R -> S) (Pi.instAdd.{0, max u1 u2} Nat (fun (ᾰ : Nat) => R -> S) (fun (i : Nat) => Pi.instAdd.{u1, u2} R (fun (ᾰ : R) => S) (fun (i : R) => AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))))) f g)) (HAdd.hAdd.{u2, u2, u2} S S S (instHAdd.{u2} S (AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p g))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (f : Nat -> R -> S) (g : Nat -> R -> S), Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Nat -> R -> S) (Nat -> R -> S) (Nat -> R -> S) (instHAdd.{max u2 u1} (Nat -> R -> S) (Pi.instAdd.{0, max u2 u1} Nat (fun (ᾰ : Nat) => R -> S) (fun (i : Nat) => Pi.instAdd.{u2, u1} R (fun (ᾰ : R) => S) (fun (i : R) => AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))))) f g)) (HAdd.hAdd.{u1, u1, u1} S S S (instHAdd.{u1} S (AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p g))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_add' Polynomial.sum_add'ₓ'. -/
theorem sum_add' {S : Type _} [AddCommMonoid S] (p : R[X]) (f g : ℕ → R → S) :
    p.Sum (f + g) = p.Sum f + p.Sum g := by simp [sum_def, Finset.sum_add_distrib]
#align polynomial.sum_add' Polynomial.sum_add'

/- warning: polynomial.sum_add -> Polynomial.sum_add is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (f : Nat -> R -> S) (g : Nat -> R -> S), Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p (fun (n : Nat) (x : R) => HAdd.hAdd.{u2, u2, u2} S S S (instHAdd.{u2} S (AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))) (f n x) (g n x))) (HAdd.hAdd.{u2, u2, u2} S S S (instHAdd.{u2} S (AddZeroClass.toHasAdd.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2)))) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p g))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (f : Nat -> R -> S) (g : Nat -> R -> S), Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p (fun (n : Nat) (x : R) => HAdd.hAdd.{u1, u1, u1} S S S (instHAdd.{u1} S (AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))) (f n x) (g n x))) (HAdd.hAdd.{u1, u1, u1} S S S (instHAdd.{u1} S (AddZeroClass.toAdd.{u1} S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2)))) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p f) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p g))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_add Polynomial.sum_addₓ'. -/
theorem sum_add {S : Type _} [AddCommMonoid S] (p : R[X]) (f g : ℕ → R → S) :
    (p.Sum fun n x => f n x + g n x) = p.Sum f + p.Sum g :=
  sum_add' _ _ _
#align polynomial.sum_add Polynomial.sum_add

/- warning: polynomial.sum_smul_index -> Polynomial.sum_smul_index is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : AddCommMonoid.{u2} S] (p : Polynomial.{u1} R _inst_1) (b : R) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u2} S (f i (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S _inst_2))))))) -> (Eq.{succ u2} S (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 (SMul.smul.{u1, u1} R (Polynomial.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u1, u1} R (Polynomial.{u1} R _inst_1) (Polynomial.zero.{u1} R _inst_1) (Polynomial.smulZeroClass.{u1, u1} R _inst_1 R (SMulWithZero.toSmulZeroClass.{u1, u1} R R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (MulZeroClass.toSMulWithZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))) b p) f) (Polynomial.sum.{u1, u2} R _inst_1 S _inst_2 p (fun (n : Nat) (a : R) => f n (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) b a))))
but is expected to have type
  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : AddCommMonoid.{u1} S] (p : Polynomial.{u2} R _inst_1) (b : R) (f : Nat -> R -> S), (forall (i : Nat), Eq.{succ u1} S (f i (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))))) (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S _inst_2))))) -> (Eq.{succ u1} S (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 (HSMul.hSMul.{u2, u2, u2} R (Polynomial.{u2} R _inst_1) (Polynomial.{u2} R _inst_1) (instHSMul.{u2, u2} R (Polynomial.{u2} R _inst_1) (SMulZeroClass.toSMul.{u2, u2} R (Polynomial.{u2} R _inst_1) (Polynomial.zero.{u2} R _inst_1) (Polynomial.smulZeroClass.{u2, u2} R _inst_1 R (SMulWithZero.toSMulZeroClass.{u2, u2} R R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (MulZeroClass.toSMulWithZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) b p) f) (Polynomial.sum.{u2, u1} R _inst_1 S _inst_2 p (fun (n : Nat) (a : R) => f n (HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) b a))))
Case conversion may be inaccurate. Consider using '#align polynomial.sum_smul_index Polynomial.sum_smul_indexₓ'. -/
theorem sum_smul_index {S : Type _} [AddCommMonoid S] (p : R[X]) (b : R) (f : ℕ → R → S)
    (hf : ∀ i, f i 0 = 0) : (b • p).Sum f = p.Sum fun n a => f n (b * a) :=
  by
  rcases p with ⟨⟩
  simpa [Sum, support, coeff] using Finsupp.sum_smul_index hf
#align polynomial.sum_smul_index Polynomial.sum_smul_index

/- warning: polynomial.sum_monomial_eq -> Polynomial.sum_monomial_eq is a dubious translation:
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) (Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) p (fun (n : Nat) (a : R) => 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)) 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) (Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) p (fun (n : Nat) (a : R) => 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)) p
Case conversion may be inaccurate. Consider using '#align polynomial.sum_monomial_eq Polynomial.sum_monomial_eqₓ'. -/
theorem sum_monomial_eq : ∀ p : R[X], (p.Sum fun n a => monomial n a) = p
  | ⟨p⟩ => (ofFinsupp_sum _ _).symm.trans (congr_arg _ <| Finsupp.sum_single _)
#align polynomial.sum_monomial_eq Polynomial.sum_monomial_eq

/- warning: polynomial.sum_C_mul_X_pow_eq -> Polynomial.sum_C_mul_X_pow_eq is a dubious translation:
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) (Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) p (fun (n : Nat) (a : R) => HMul.hMul.{u1, u1, u1} (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.mul'.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) 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) (Polynomial.sum.{u1, u1} R _inst_1 (Polynomial.{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)))) p (fun (n : Nat) (a : R) => HMul.hMul.{u1, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.{u1} R _inst_1) (Polynomial.{u1} R _inst_1) (instHMul.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) a) (Polynomial.mul'.{u1} R _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R _inst_1) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))))) (Polynomial.C.{u1} R _inst_1) a) (HPow.hPow.{u1, 0, u1} (Polynomial.{u1} R _inst_1) Nat (Polynomial.{u1} R _inst_1) (instHPow.{u1, 0} (Polynomial.{u1} R _inst_1) Nat (Monoid.Pow.{u1} (Polynomial.{u1} R _inst_1) (MonoidWithZero.toMonoid.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toMonoidWithZero.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))))) (Polynomial.X.{u1} R _inst_1) n))) p
Case conversion may be inaccurate. Consider using '#align polynomial.sum_C_mul_X_pow_eq Polynomial.sum_C_mul_X_pow_eqₓ'. -/
theorem sum_C_mul_X_pow_eq (p : R[X]) : (p.Sum fun n a => C a * X ^ n) = p := by
  simp_rw [C_mul_X_pow_eq_monomial, sum_monomial_eq]
#align polynomial.sum_C_mul_X_pow_eq Polynomial.sum_C_mul_X_pow_eq

#print Polynomial.erase /-
/-- `erase p n` is the polynomial `p` in which the `X^n` term has been erased. -/
irreducible_def erase (n : ℕ) : R[X] → R[X]
  | ⟨p⟩ => ⟨p.eraseₓ n⟩
#align polynomial.erase Polynomial.erase
-/

/- warning: polynomial.to_finsupp_erase -> Polynomial.toFinsupp_erase is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p)) (Finsupp.erase.{0, u1} Nat R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) n (Polynomial.toFinsupp.{u1} R _inst_1 p))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{succ u1} (AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (Polynomial.toFinsupp.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p)) (Finsupp.erase.{0, u1} Nat R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) n (Polynomial.toFinsupp.{u1} R _inst_1 p))
Case conversion may be inaccurate. Consider using '#align polynomial.to_finsupp_erase Polynomial.toFinsupp_eraseₓ'. -/
@[simp]
theorem toFinsupp_erase (p : R[X]) (n : ℕ) : toFinsupp (p.eraseₓ n) = p.toFinsupp.eraseₓ n :=
  by
  rcases p with ⟨⟩
  simp only [erase]
#align polynomial.to_finsupp_erase Polynomial.toFinsupp_erase

/- warning: polynomial.of_finsupp_erase -> Polynomial.ofFinsupp_erase is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (n : Nat), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (Finsupp.erase.{0, u1} Nat R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) n p)) (Polynomial.erase.{u1} R _inst_1 n (Polynomial.ofFinsupp.{u1} R _inst_1 p))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : AddMonoidAlgebra.{u1, 0} R Nat _inst_1) (n : Nat), Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.ofFinsupp.{u1} R _inst_1 (Finsupp.erase.{0, u1} Nat R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) n p)) (Polynomial.erase.{u1} R _inst_1 n (Polynomial.ofFinsupp.{u1} R _inst_1 p))
Case conversion may be inaccurate. Consider using '#align polynomial.of_finsupp_erase Polynomial.ofFinsupp_eraseₓ'. -/
@[simp]
theorem ofFinsupp_erase (p : AddMonoidAlgebra R ℕ) (n : ℕ) :
    (⟨p.eraseₓ n⟩ : R[X]) = (⟨p⟩ : R[X]).eraseₓ n :=
  by
  rcases p with ⟨⟩
  simp only [erase]
#align polynomial.of_finsupp_erase Polynomial.ofFinsupp_erase

/- warning: polynomial.support_erase -> Polynomial.support_erase is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{1} (Finset.{0} Nat) (Polynomial.support.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p)) (Finset.erase.{0} Nat (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (Polynomial.support.{u1} R _inst_1 p) n)
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{1} (Finset.{0} Nat) (Polynomial.support.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p)) (Finset.erase.{0} Nat (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) (Polynomial.support.{u1} R _inst_1 p) n)
Case conversion may be inaccurate. Consider using '#align polynomial.support_erase Polynomial.support_eraseₓ'. -/
@[simp]
theorem support_erase (p : R[X]) (n : ℕ) : support (p.eraseₓ n) = (support p).eraseₓ n :=
  by
  rcases p with ⟨⟩
  simp only [support, erase, support_erase]
#align polynomial.support_erase Polynomial.support_erase

/- warning: polynomial.monomial_add_erase -> Polynomial.monomial_add_erase is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.monomial_add_erase Polynomial.monomial_add_eraseₓ'. -/
theorem monomial_add_erase (p : R[X]) (n : ℕ) : monomial n (coeff p n) + p.eraseₓ n = p :=
  toFinsupp_injective <| by
    rcases p with ⟨⟩
    rw [to_finsupp_add, to_finsupp_monomial, to_finsupp_erase, coeff]
    exact Finsupp.single_add_erase _ _
#align polynomial.monomial_add_erase Polynomial.monomial_add_erase

/- warning: polynomial.coeff_erase -> Polynomial.coeff_erase is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (i : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p) i) (ite.{succ u1} R (Eq.{1} Nat i n) (Nat.decidableEq i n) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))) (Polynomial.coeff.{u1} R _inst_1 p i))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (i : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p) i) (ite.{succ u1} R (Eq.{1} Nat i n) (instDecidableEqNat i n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (Polynomial.coeff.{u1} R _inst_1 p i))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_erase Polynomial.coeff_eraseₓ'. -/
theorem coeff_erase (p : R[X]) (n i : ℕ) : (p.eraseₓ n).coeff i = if i = n then 0 else p.coeff i :=
  by
  rcases p with ⟨⟩
  simp only [erase, coeff]
  convert rfl
#align polynomial.coeff_erase Polynomial.coeff_erase

#print Polynomial.erase_zero /-
@[simp]
theorem erase_zero (n : ℕ) : (0 : R[X]).eraseₓ n = 0 :=
  toFinsupp_injective <| by simp
#align polynomial.erase_zero Polynomial.erase_zero
-/

/- warning: polynomial.erase_monomial -> Polynomial.erase_monomial 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] {n : Nat} {a : R}, Eq.{succ u1} (Polynomial.{u1} R _inst_1) (Polynomial.erase.{u1} R _inst_1 n (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)) (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.erase_monomial Polynomial.erase_monomialₓ'. -/
@[simp]
theorem erase_monomial {n : ℕ} {a : R} : erase n (monomial n a) = 0 :=
  toFinsupp_injective <| by simp
#align polynomial.erase_monomial Polynomial.erase_monomial

/- warning: polynomial.erase_same -> Polynomial.erase_same is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p) n) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.erase.{u1} R _inst_1 n p) n) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1))))
Case conversion may be inaccurate. Consider using '#align polynomial.erase_same Polynomial.erase_sameₓ'. -/
@[simp]
theorem erase_same (p : R[X]) (n : ℕ) : coeff (p.eraseₓ n) n = 0 := by simp [coeff_erase]
#align polynomial.erase_same Polynomial.erase_same

#print Polynomial.erase_ne /-
@[simp]
theorem erase_ne (p : R[X]) (n i : ℕ) (h : i ≠ n) : coeff (p.eraseₓ n) i = coeff p i := by
  simp [coeff_erase, h]
#align polynomial.erase_ne Polynomial.erase_ne
-/

section Update

#print Polynomial.update /-
/-- Replace the coefficient of a `p : R[X]` at a given degree `n : ℕ`
by a given value `a : R`. If `a = 0`, this is equal to `p.erase n`
If `p.nat_degree < n` and `a ≠ 0`, this increases the degree to `n`.  -/
def update (p : R[X]) (n : ℕ) (a : R) : R[X] :=
  Polynomial.ofFinsupp (p.toFinsupp.update n a)
#align polynomial.update Polynomial.update
-/

/- warning: polynomial.coeff_update -> Polynomial.coeff_update is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (a : R), Eq.{succ u1} (Nat -> R) (Polynomial.coeff.{u1} R _inst_1 (Polynomial.update.{u1} R _inst_1 p n a)) (Function.update.{1, succ u1} Nat (fun (ᾰ : Nat) => R) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) (Polynomial.coeff.{u1} R _inst_1 p) n a)
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (a : R), Eq.{succ u1} (Nat -> R) (Polynomial.coeff.{u1} R _inst_1 (Polynomial.update.{u1} R _inst_1 p n a)) (Function.update.{1, succ u1} Nat (fun (ᾰ : Nat) => R) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) (Polynomial.coeff.{u1} R _inst_1 p) n a)
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_update Polynomial.coeff_updateₓ'. -/
theorem coeff_update (p : R[X]) (n : ℕ) (a : R) :
    (p.update n a).coeff = Function.update p.coeff n a :=
  by
  ext
  cases p
  simp only [coeff, update, Function.update_apply, coe_update]
#align polynomial.coeff_update Polynomial.coeff_update

/- warning: polynomial.coeff_update_apply -> Polynomial.coeff_update_apply is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (a : R) (i : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.update.{u1} R _inst_1 p n a) i) (ite.{succ u1} R (Eq.{1} Nat i n) (Nat.decidableEq i n) a (Polynomial.coeff.{u1} R _inst_1 p i))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] (p : Polynomial.{u1} R _inst_1) (n : Nat) (a : R) (i : Nat), Eq.{succ u1} R (Polynomial.coeff.{u1} R _inst_1 (Polynomial.update.{u1} R _inst_1 p n a) i) (ite.{succ u1} R (Eq.{1} Nat i n) (instDecidableEqNat i n) a (Polynomial.coeff.{u1} R _inst_1 p i))
Case conversion may be inaccurate. Consider using '#align polynomial.coeff_update_apply Polynomial.coeff_update_applyₓ'. -/
theorem coeff_update_apply (p : R[X]) (n : ℕ) (a : R) (i : ℕ) :
    (p.update n a).coeff i = if i = n then a else p.coeff i := by
  rw [coeff_update, Function.update_apply]
#align polynomial.coeff_update_apply Polynomial.coeff_update_apply

#print Polynomial.coeff_update_same /-
@[simp]
theorem coeff_update_same (p : R[X]) (n : ℕ) (a : R) : (p.update n a).coeff n = a := by
  rw [p.coeff_update_apply, if_pos rfl]
#align polynomial.coeff_update_same Polynomial.coeff_update_same
-/

#print Polynomial.coeff_update_ne /-
theorem coeff_update_ne (p : R[X]) {n : ℕ} (a : R) {i : ℕ} (h : i ≠ n) :
    (p.update n a).coeff i = p.coeff i := by rw [p.coeff_update_apply, if_neg h]
#align polynomial.coeff_update_ne Polynomial.coeff_update_ne
-/

/- warning: polynomial.update_zero_eq_erase -> Polynomial.update_zero_eq_erase is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.update_zero_eq_erase Polynomial.update_zero_eq_eraseₓ'. -/
@[simp]
theorem update_zero_eq_erase (p : R[X]) (n : ℕ) : p.update n 0 = p.eraseₓ n :=
  by
  ext
  rw [coeff_update_apply, coeff_erase]
#align polynomial.update_zero_eq_erase Polynomial.update_zero_eq_erase

/- warning: polynomial.support_update -> Polynomial.support_update is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.support_update Polynomial.support_updateₓ'. -/
theorem support_update (p : R[X]) (n : ℕ) (a : R) [Decidable (a = 0)] :
    support (p.update n a) = if a = 0 then p.support.eraseₓ n else insert n p.support := by
  classical
    cases p
    simp only [support, update, support_update]
    congr
#align polynomial.support_update Polynomial.support_update

/- warning: polynomial.support_update_zero -> Polynomial.support_update_zero 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] (p : Polynomial.{u1} R _inst_1) (n : Nat), Eq.{1} (Finset.{0} Nat) (Polynomial.support.{u1} R _inst_1 (Polynomial.update.{u1} R _inst_1 p n (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))))) (Finset.erase.{0} Nat (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) (Polynomial.support.{u1} R _inst_1 p) n)
Case conversion may be inaccurate. Consider using '#align polynomial.support_update_zero Polynomial.support_update_zeroₓ'. -/
theorem support_update_zero (p : R[X]) (n : ℕ) : support (p.update n 0) = p.support.eraseₓ n := by
  rw [update_zero_eq_erase, support_erase]
#align polynomial.support_update_zero Polynomial.support_update_zero

/- warning: polynomial.support_update_ne_zero -> Polynomial.support_update_ne_zero is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align polynomial.support_update_ne_zero Polynomial.support_update_ne_zeroₓ'. -/
theorem support_update_ne_zero (p : R[X]) (n : ℕ) {a : R} (ha : a ≠ 0) :
    support (p.update n a) = insert n p.support := by classical rw [support_update, if_neg ha]
#align polynomial.support_update_ne_zero Polynomial.support_update_ne_zero

end Update

end Semiring

section CommSemiring

variable [CommSemiring R]

instance : CommSemiring R[X] :=
  Function.Injective.commSemiring toFinsupp toFinsupp_injective toFinsupp_zero toFinsupp_one
    toFinsupp_add toFinsupp_mul (fun _ _ => toFinsupp_smul _ _) toFinsupp_pow fun _ => rfl

end CommSemiring

section Ring

variable [Ring R]

instance : IntCast R[X] :=
  ⟨fun n => ofFinsupp n⟩

instance : Ring R[X] :=
  Function.Injective.ring toFinsupp toFinsupp_injective toFinsupp_zero toFinsupp_one toFinsupp_add
    toFinsupp_mul toFinsupp_neg toFinsupp_sub (fun _ _ => toFinsupp_smul _ _)
    (fun _ _ => toFinsupp_smul _ _) toFinsupp_pow (fun _ => rfl) fun _ => rfl

/- warning: polynomial.coeff_neg -> Polynomial.coeff_neg is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align polynomial.coeff_neg Polynomial.coeff_negₓ'. -/
@[simp]
theorem coeff_neg (p : R[X]) (n : ℕ) : coeff (-p) n = -coeff p n :=
  by
  rcases p with ⟨⟩
  rw [← of_finsupp_neg, coeff, coeff, Finsupp.neg_apply]
#align polynomial.coeff_neg Polynomial.coeff_neg

/- warning: polynomial.coeff_sub -> Polynomial.coeff_sub is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align polynomial.coeff_sub Polynomial.coeff_subₓ'. -/
@[simp]
theorem coeff_sub (p q : R[X]) (n : ℕ) : coeff (p - q) n = coeff p n - coeff q n :=
  by
  rcases p with ⟨⟩
  rcases q with ⟨⟩
  rw [← of_finsupp_sub, coeff, coeff, coeff, Finsupp.sub_apply]
#align polynomial.coeff_sub Polynomial.coeff_sub

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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R (Ring.toSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Polynomial.monomial.{u1} R (Ring.toSemiring.{u1} R _inst_1) n) a))
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] (n : Nat) (a : R), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) a)) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R (Ring.toSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R (Ring.toSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Polynomial.monomial.{u1} R (Ring.toSemiring.{u1} R _inst_1) n) (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) a)) (Neg.neg.{u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) a) (Polynomial.neg'.{u1} R _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R (Ring.toSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.module.{u1, u1} R (Ring.toSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Polynomial.monomial.{u1} R (Ring.toSemiring.{u1} R _inst_1) n) a))
Case conversion may be inaccurate. Consider using '#align polynomial.monomial_neg Polynomial.monomial_negₓ'. -/
@[simp]
theorem monomial_neg (n : ℕ) (a : R) : monomial n (-a) = -monomial n a := by
  rw [eq_neg_iff_add_eq_zero, ← monomial_add, neg_add_self, monomial_zero_right]
#align polynomial.monomial_neg Polynomial.monomial_neg

#print Polynomial.support_neg /-
@[simp]
theorem support_neg {p : R[X]} : (-p).support = p.support :=
  by
  rcases p with ⟨⟩
  rw [← of_finsupp_neg, support, support, Finsupp.support_neg]
#align polynomial.support_neg Polynomial.support_neg
-/

/- warning: polynomial.C_eq_int_cast -> Polynomial.C_eq_int_cast is a dubious translation:
lean 3 declaration is
  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] (n : Int), Eq.{succ u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) => R -> (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R _inst_1)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int R (HasLiftT.mk.{1, succ u1} Int R (CoeTCₓ.coe.{1, succ u1} Int R (Int.castCoe.{u1} R (AddGroupWithOne.toHasIntCast.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasLiftT.mk.{1, succ u1} Int (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (CoeTCₓ.coe.{1, succ u1} Int (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Int.castCoe.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.intCast.{u1} R _inst_1)))) n)
but is expected to have type
  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] (n : Int), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Int.cast.{u1} R (Ring.toIntCast.{u1} R _inst_1) n)) (FunLike.coe.{succ u1, succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) _x) (MulHomClass.toFunLike.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.instRingHomClassRingHom.{u1, u1} R (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Polynomial.semiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))))))) (Polynomial.C.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Int.cast.{u1} R (Ring.toIntCast.{u1} R _inst_1) n)) (Int.cast.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Int.cast.{u1} R (Ring.toIntCast.{u1} R _inst_1) n)) (Polynomial.intCast.{u1} R _inst_1) n)
Case conversion may be inaccurate. Consider using '#align polynomial.C_eq_int_cast Polynomial.C_eq_int_castₓ'. -/
@[simp]
theorem C_eq_int_cast (n : ℤ) : C (n : R) = n :=
  map_intCast C n
#align polynomial.C_eq_int_cast Polynomial.C_eq_int_cast

end Ring

instance [CommRing R] : CommRing R[X] :=
  Function.Injective.commRing toFinsupp toFinsupp_injective toFinsupp_zero toFinsupp_one
    toFinsupp_add toFinsupp_mul toFinsupp_neg toFinsupp_sub (fun _ _ => toFinsupp_smul _ _)
    (fun _ _ => toFinsupp_smul _ _) toFinsupp_pow (fun _ => rfl) fun _ => rfl

section NonzeroSemiring

variable [Semiring R] [Nontrivial R]

instance : Nontrivial R[X] :=
  by
  have h : Nontrivial (AddMonoidAlgebra R ℕ) := by infer_instance
  rcases h.exists_pair_ne with ⟨x, y, hxy⟩
  refine' ⟨⟨⟨x⟩, ⟨y⟩, _⟩⟩
  simp [hxy]

#print Polynomial.X_ne_zero /-
theorem X_ne_zero : (X : R[X]) ≠ 0 :=
  mt (congr_arg fun p => coeff p 1) (by simp)
#align polynomial.X_ne_zero Polynomial.X_ne_zero
-/

end NonzeroSemiring

#print Polynomial.nontrivial_iff /-
@[simp]
theorem nontrivial_iff [Semiring R] : Nontrivial R[X] ↔ Nontrivial R :=
  ⟨fun h =>
    let ⟨r, s, hrs⟩ := @exists_pair_ne _ h
    Nontrivial.of_polynomial_ne hrs,
    fun h => @Polynomial.nontrivial _ _ h⟩
#align polynomial.nontrivial_iff Polynomial.nontrivial_iff
-/

section repr

variable [Semiring R]

open Classical

instance [Repr R] : Repr R[X] :=
  ⟨fun p =>
    if p = 0 then "0"
    else
      (p.support.sort (· ≤ ·)).foldr
        (fun n a =>
          (a ++ if a = "" then "" else " + ") ++
            if n = 0 then "C (" ++ repr (coeff p n) ++ ")"
            else
              if n = 1 then if coeff p n = 1 then "X" else "C (" ++ repr (coeff p n) ++ ") * X"
              else
                if coeff p n = 1 then "X ^ " ++ repr n
                else "C (" ++ repr (coeff p n) ++ ") * X ^ " ++ repr n)
        ""⟩

end repr

end Polynomial