text:
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser

! This file was ported from Lean 3 source module data.mv_polynomial.division
! leanprover-community/mathlib commit 72c366d0475675f1309d3027d3d7d47ee4423951
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Algebra.MonoidAlgebra.Division
import Mathbin.Data.MvPolynomial.Basic

/-!
# Division of `mv_polynomial` by monomials

## Main definitions

* `mv_polynomial.div_monomial x s`: divides `x` by the monomial `mv_polynomial.monomial 1 s`
* `mv_polynomial.mod_monomial x s`: the remainder upon dividing `x` by the monomial
  `mv_polynomial.monomial 1 s`.

## Main results

* `mv_polynomial.div_monomial_add_mod_monomial`, `mv_polynomial.mod_monomial_add_div_monomial`:
  `div_monomial` and `mod_monomial` are well-behaved as quotient and remainder operators.

## Implementation notes

Where possible, the results in this file should be first proved in the generality of
`add_monoid_algebra`, and then the versions specialized to `mv_polynomial` proved in terms of these.

-/

variable {σ R : Type _} [CommSemiring R]

namespace MvPolynomial

section CopiedDeclarations

/-! Please ensure the declarations in this section are direct translations of `add_monoid_algebra`
results. -/

#print MvPolynomial.divMonomial /-
/-- Divide by `monomial 1 s`, discarding terms not divisible by this. -/
noncomputable def divMonomial (p : MvPolynomial σ R) (s : σ →₀ ℕ) : MvPolynomial σ R :=
  AddMonoidAlgebra.divOf p s
#align mv_polynomial.div_monomial MvPolynomial.divMonomial
-/

-- mathport name: «expr /ᵐᵒⁿᵒᵐⁱᵃˡ »
local infixl:70 " /ᵐᵒⁿᵒᵐⁱᵃˡ " => divMonomial

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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.coeff_div_monomial MvPolynomial.coeff_divMonomialₓ'. -/
@[simp]
theorem coeff_divMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) (s' : σ →₀ ℕ) :
    coeff s' (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) = coeff (s + s') x :=
  rfl
#align mv_polynomial.coeff_div_monomial MvPolynomial.coeff_divMonomial

/- warning: mv_polynomial.support_div_monomial -> MvPolynomial.support_divMonomial is a dubious translation:
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Nat.linearOrderedCommMonoidWithZero)) (Finsupp.add.{u2, 0} σ Nat (AddMonoid.toAddZeroClass.{0} Nat Nat.addMonoid))) x._@.Mathlib.Algebra.Group.Defs._hyg.2622 x._@.Mathlib.Algebra.Group.Defs._hyg.2624) s) (Finset.toSet.{u2} (Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (MvPolynomial.support.{u1, u2} R σ _inst_1 x)))))
Case conversion may be inaccurate. Consider using '#align mv_polynomial.support_div_monomial MvPolynomial.support_divMonomialₓ'. -/
@[simp]
theorem support_divMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    (x /ᵐᵒⁿᵒᵐⁱᵃˡ s).support = x.support.Preimage _ ((add_right_injective s).InjOn _) :=
  rfl
#align mv_polynomial.support_div_monomial MvPolynomial.support_divMonomial

/- warning: mv_polynomial.zero_div_monomial -> MvPolynomial.zero_divMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.zero_div_monomial MvPolynomial.zero_divMonomialₓ'. -/
@[simp]
theorem zero_divMonomial (s : σ →₀ ℕ) : (0 : MvPolynomial σ R) /ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 :=
  AddMonoidAlgebra.zero_divOf _
#align mv_polynomial.zero_div_monomial MvPolynomial.zero_divMonomial

/- warning: mv_polynomial.div_monomial_zero -> MvPolynomial.divMonomial_zero is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_zero MvPolynomial.divMonomial_zeroₓ'. -/
theorem divMonomial_zero (x : MvPolynomial σ R) : x /ᵐᵒⁿᵒᵐⁱᵃˡ 0 = x :=
  x.divOf_zero
#align mv_polynomial.div_monomial_zero MvPolynomial.divMonomial_zero

/- warning: mv_polynomial.add_div_monomial -> MvPolynomial.add_divMonomial is a dubious translation:
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  forall {σ : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] (x : MvPolynomial.{u1, u2} σ R _inst_1) (y : MvPolynomial.{u1, u2} σ R _inst_1) (s : Finsupp.{u1, 0} σ Nat Nat.hasZero), Eq.{max (succ u1) (succ u2)} (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.divMonomial.{u1, u2} σ R _inst_1 (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.{u1, u2} σ R _inst_1) (instHAdd.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (Distrib.toHasAdd.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.commSemiring.{u2, u1} R σ _inst_1))))))) x y) s) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.{u1, u2} σ R _inst_1) (instHAdd.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (Distrib.toHasAdd.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (NonUnitalNonAssocSemiring.toDistrib.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u1, u2} σ R _inst_1) (MvPolynomial.commSemiring.{u2, u1} R σ _inst_1))))))) (MvPolynomial.divMonomial.{u1, u2} σ R _inst_1 x s) (MvPolynomial.divMonomial.{u1, u2} σ R _inst_1 y s))
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.add_div_monomial MvPolynomial.add_divMonomialₓ'. -/
theorem add_divMonomial (x y : MvPolynomial σ R) (s : σ →₀ ℕ) :
    (x + y) /ᵐᵒⁿᵒᵐⁱᵃˡ s = x /ᵐᵒⁿᵒᵐⁱᵃˡ s + y /ᵐᵒⁿᵒᵐⁱᵃˡ s :=
  map_add _ _ _
#align mv_polynomial.add_div_monomial MvPolynomial.add_divMonomial

/- warning: mv_polynomial.div_monomial_add -> MvPolynomial.divMonomial_add is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_add MvPolynomial.divMonomial_addₓ'. -/
theorem divMonomial_add (a b : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x /ᵐᵒⁿᵒᵐⁱᵃˡ (a + b) = x /ᵐᵒⁿᵒᵐⁱᵃˡ a /ᵐᵒⁿᵒᵐⁱᵃˡ b :=
  x.divOf_add _ _
#align mv_polynomial.div_monomial_add MvPolynomial.divMonomial_add

/- warning: mv_polynomial.div_monomial_monomial_mul -> MvPolynomial.divMonomial_monomial_mul is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_monomial_mul MvPolynomial.divMonomial_monomial_mulₓ'. -/
@[simp]
theorem divMonomial_monomial_mul (a : σ →₀ ℕ) (x : MvPolynomial σ R) :
    monomial a 1 * x /ᵐᵒⁿᵒᵐⁱᵃˡ a = x :=
  x.of'_mul_divOf _
#align mv_polynomial.div_monomial_monomial_mul MvPolynomial.divMonomial_monomial_mul

/- warning: mv_polynomial.div_monomial_mul_monomial -> MvPolynomial.divMonomial_mul_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_mul_monomial MvPolynomial.divMonomial_mul_monomialₓ'. -/
@[simp]
theorem divMonomial_mul_monomial (a : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x * monomial a 1 /ᵐᵒⁿᵒᵐⁱᵃˡ a = x :=
  x.mul_of'_divOf _
#align mv_polynomial.div_monomial_mul_monomial MvPolynomial.divMonomial_mul_monomial

/- warning: mv_polynomial.div_monomial_monomial -> MvPolynomial.divMonomial_monomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_monomial MvPolynomial.divMonomial_monomialₓ'. -/
@[simp]
theorem divMonomial_monomial (a : σ →₀ ℕ) : monomial a 1 /ᵐᵒⁿᵒᵐⁱᵃˡ a = (1 : MvPolynomial σ R) :=
  AddMonoidAlgebra.of'_divOf _
#align mv_polynomial.div_monomial_monomial MvPolynomial.divMonomial_monomial

#print MvPolynomial.modMonomial /-
/-- The remainder upon division by `monomial 1 s`. -/
noncomputable def modMonomial (x : MvPolynomial σ R) (s : σ →₀ ℕ) : MvPolynomial σ R :=
  x.modOf s
#align mv_polynomial.mod_monomial MvPolynomial.modMonomial
-/

-- mathport name: «expr %ᵐᵒⁿᵒᵐⁱᵃˡ »
local infixl:70 " %ᵐᵒⁿᵒᵐⁱᵃˡ " => modMonomial

/- warning: mv_polynomial.coeff_mod_monomial_of_not_le -> MvPolynomial.coeff_modMonomial_of_not_le is a dubious translation:
lean 3 declaration is
  forall {σ : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {s' : Finsupp.{u1, 0} σ Nat Nat.hasZero} {s : Finsupp.{u1, 0} σ Nat Nat.hasZero} (x : MvPolynomial.{u1, u2} σ R _inst_1), (Not (LE.le.{u1} (Finsupp.{u1, 0} σ Nat Nat.hasZero) (Finsupp.hasLe.{u1, 0} σ Nat Nat.hasZero Nat.hasLe) s s')) -> (Eq.{succ u2} R (MvPolynomial.coeff.{u2, u1} R σ _inst_1 s' (MvPolynomial.modMonomial.{u1, u2} σ R _inst_1 x s)) (MvPolynomial.coeff.{u2, u1} R σ _inst_1 s' x))
but is expected to have type
  forall {σ : Type.{u2}} {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {s' : Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)} {s : Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)} (x : MvPolynomial.{u2, u1} σ R _inst_1), (Not (LE.le.{u2} (Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (Finsupp.instLEFinsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) instLENat) s s')) -> (Eq.{succ u1} R (MvPolynomial.coeff.{u1, u2} R σ _inst_1 s' (MvPolynomial.modMonomial.{u2, u1} σ R _inst_1 x s)) (MvPolynomial.coeff.{u1, u2} R σ _inst_1 s' x))
Case conversion may be inaccurate. Consider using '#align mv_polynomial.coeff_mod_monomial_of_not_le MvPolynomial.coeff_modMonomial_of_not_leₓ'. -/
@[simp]
theorem coeff_modMonomial_of_not_le {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : ¬s ≤ s') :
    coeff s' (x %ᵐᵒⁿᵒᵐⁱᵃˡ s) = coeff s' x :=
  x.modOf_apply_of_not_exists_add s s'
    (by
      rintro ⟨d, rfl⟩
      exact h le_self_add)
#align mv_polynomial.coeff_mod_monomial_of_not_le MvPolynomial.coeff_modMonomial_of_not_le

/- warning: mv_polynomial.coeff_mod_monomial_of_le -> MvPolynomial.coeff_modMonomial_of_le is a dubious translation:
lean 3 declaration is
  forall {σ : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommSemiring.{u2} R] {s' : Finsupp.{u1, 0} σ Nat Nat.hasZero} {s : Finsupp.{u1, 0} σ Nat Nat.hasZero} (x : MvPolynomial.{u1, u2} σ R _inst_1), (LE.le.{u1} (Finsupp.{u1, 0} σ Nat Nat.hasZero) (Finsupp.hasLe.{u1, 0} σ Nat Nat.hasZero Nat.hasLe) s s') -> (Eq.{succ u2} R (MvPolynomial.coeff.{u2, u1} R σ _inst_1 s' (MvPolynomial.modMonomial.{u1, u2} σ R _inst_1 x s)) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))))))))
but is expected to have type
  forall {σ : Type.{u2}} {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] {s' : Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)} {s : Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)} (x : MvPolynomial.{u2, u1} σ R _inst_1), (LE.le.{u2} (Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (Finsupp.instLEFinsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) instLENat) s s') -> (Eq.{succ u1} R (MvPolynomial.coeff.{u1, u2} R σ _inst_1 s' (MvPolynomial.modMonomial.{u2, u1} σ R _inst_1 x s)) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)))))
Case conversion may be inaccurate. Consider using '#align mv_polynomial.coeff_mod_monomial_of_le MvPolynomial.coeff_modMonomial_of_leₓ'. -/
@[simp]
theorem coeff_modMonomial_of_le {s' s : σ →₀ ℕ} (x : MvPolynomial σ R) (h : s ≤ s') :
    coeff s' (x %ᵐᵒⁿᵒᵐⁱᵃˡ s) = 0 :=
  x.modOf_apply_of_exists_add _ _ <| exists_add_of_le h
#align mv_polynomial.coeff_mod_monomial_of_le MvPolynomial.coeff_modMonomial_of_le

/- warning: mv_polynomial.monomial_mul_mod_monomial -> MvPolynomial.monomial_mul_modMonomial 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 mv_polynomial.monomial_mul_mod_monomial MvPolynomial.monomial_mul_modMonomialₓ'. -/
@[simp]
theorem monomial_mul_modMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    monomial s 1 * x %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 :=
  x.of'_mul_modOf _
#align mv_polynomial.monomial_mul_mod_monomial MvPolynomial.monomial_mul_modMonomial

/- warning: mv_polynomial.mul_monomial_mod_monomial -> MvPolynomial.mul_monomial_modMonomial is a dubious translation:
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but is expected to have type
  forall {σ : Type.{u2}} {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (s : Finsupp.{u2, 0} σ Nat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero)) (x : MvPolynomial.{u2, u1} σ R _inst_1), Eq.{max (succ u2) (succ u1)} (MvPolynomial.{u2, u1} σ R _inst_1) (MvPolynomial.modMonomial.{u2, u1} σ R _inst_1 (HMul.hMul.{max u2 u1, max u2 u1, max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => MvPolynomial.{u2, u1} σ R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MvPolynomial.{u2, u1} σ R _inst_1) (instHMul.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (NonUnitalNonAssocSemiring.toMul.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (Semiring.toNonAssocSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (CommSemiring.toSemiring.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (MvPolynomial.commSemiring.{u1, u2} R σ _inst_1)))))) x (FunLike.coe.{max (succ u2) (succ u1), succ u1, max (succ u2) (succ u1)} (LinearMap.{u1, u1, u1, max u1 u2} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) R (MvPolynomial.{u2, u1} σ R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (MvPolynomial.commSemiring.{u1, u2} R σ _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (MvPolynomial.module.{u1, u1, u2} R R σ (CommSemiring.toSemiring.{u1} R _inst_1) _inst_1 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => MvPolynomial.{u2, u1} σ R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, max u2 u1} R R R (MvPolynomial.{u2, u1} σ R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (Semiring.toNonAssocSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (CommSemiring.toSemiring.{max u1 u2} (MvPolynomial.{u2, u1} σ R _inst_1) (MvPolynomial.commSemiring.{u1, u2} R σ _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (MvPolynomial.module.{u1, u1, u2} R R σ (CommSemiring.toSemiring.{u1} R _inst_1) _inst_1 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (MvPolynomial.monomial.{u1, u2} R σ _inst_1 s) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) s) (OfNat.ofNat.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (CommMonoidWithZero.toZero.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (CommSemiring.toCommMonoidWithZero.{max u2 u1} (MvPolynomial.{u2, u1} σ R _inst_1) (MvPolynomial.commSemiring.{u1, u2} R σ _inst_1)))))
Case conversion may be inaccurate. Consider using '#align mv_polynomial.mul_monomial_mod_monomial MvPolynomial.mul_monomial_modMonomialₓ'. -/
@[simp]
theorem mul_monomial_modMonomial (s : σ →₀ ℕ) (x : MvPolynomial σ R) :
    x * monomial s 1 %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 :=
  x.mul_of'_modOf _
#align mv_polynomial.mul_monomial_mod_monomial MvPolynomial.mul_monomial_modMonomial

/- warning: mv_polynomial.monomial_mod_monomial -> MvPolynomial.monomial_modMonomial is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.monomial_mod_monomial MvPolynomial.monomial_modMonomialₓ'. -/
@[simp]
theorem monomial_modMonomial (s : σ →₀ ℕ) : monomial s (1 : R) %ᵐᵒⁿᵒᵐⁱᵃˡ s = 0 :=
  AddMonoidAlgebra.of'_modOf _
#align mv_polynomial.monomial_mod_monomial MvPolynomial.monomial_modMonomial

/- warning: mv_polynomial.div_monomial_add_mod_monomial -> MvPolynomial.divMonomial_add_modMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_add_mod_monomial MvPolynomial.divMonomial_add_modMonomialₓ'. -/
theorem divMonomial_add_modMonomial (x : MvPolynomial σ R) (s : σ →₀ ℕ) :
    monomial s 1 * (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) + x %ᵐᵒⁿᵒᵐⁱᵃˡ s = x :=
  AddMonoidAlgebra.divOf_add_modOf x s
#align mv_polynomial.div_monomial_add_mod_monomial MvPolynomial.divMonomial_add_modMonomial

/- warning: mv_polynomial.mod_monomial_add_div_monomial -> MvPolynomial.modMonomial_add_divMonomial is a dubious translation:
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(MvPolynomial.commSemiring.{u1, u2} R σ _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (MvPolynomial.module.{u1, u1, u2} R R σ (CommSemiring.toSemiring.{u1} R _inst_1) _inst_1 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (MvPolynomial.monomial.{u1, u2} R σ _inst_1 s) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MvPolynomial.divMonomial.{u2, u1} σ R _inst_1 x s))) x
Case conversion may be inaccurate. Consider using '#align mv_polynomial.mod_monomial_add_div_monomial MvPolynomial.modMonomial_add_divMonomialₓ'. -/
theorem modMonomial_add_divMonomial (x : MvPolynomial σ R) (s : σ →₀ ℕ) :
    x %ᵐᵒⁿᵒᵐⁱᵃˡ s + monomial s 1 * (x /ᵐᵒⁿᵒᵐⁱᵃˡ s) = x :=
  AddMonoidAlgebra.modOf_add_divOf x s
#align mv_polynomial.mod_monomial_add_div_monomial MvPolynomial.modMonomial_add_divMonomial

/- warning: mv_polynomial.monomial_one_dvd_iff_mod_monomial_eq_zero -> MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zero is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.monomial_one_dvd_iff_mod_monomial_eq_zero MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zeroₓ'. -/
theorem monomial_one_dvd_iff_modMonomial_eq_zero {i : σ →₀ ℕ} {x : MvPolynomial σ R} :
    monomial i (1 : R) ∣ x ↔ x %ᵐᵒⁿᵒᵐⁱᵃˡ i = 0 :=
  AddMonoidAlgebra.of'_dvd_iff_modOf_eq_zero
#align mv_polynomial.monomial_one_dvd_iff_mod_monomial_eq_zero MvPolynomial.monomial_one_dvd_iff_modMonomial_eq_zero

end CopiedDeclarations

section XLemmas

-- mathport name: «expr /ᵐᵒⁿᵒᵐⁱᵃˡ »
local infixl:70 " /ᵐᵒⁿᵒᵐⁱᵃˡ " => divMonomial

-- mathport name: «expr %ᵐᵒⁿᵒᵐⁱᵃˡ »
local infixl:70 " %ᵐᵒⁿᵒᵐⁱᵃˡ " => modMonomial

/- warning: mv_polynomial.X_mul_div_monomial -> MvPolynomial.x_mul_divMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.X_mul_div_monomial MvPolynomial.x_mul_divMonomialₓ'. -/
@[simp]
theorem x_mul_divMonomial (i : σ) (x : MvPolynomial σ R) :
    X i * x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x :=
  divMonomial_monomial_mul _ _
#align mv_polynomial.X_mul_div_monomial MvPolynomial.x_mul_divMonomial

/- warning: mv_polynomial.X_div_monomial -> MvPolynomial.x_divMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.X_div_monomial MvPolynomial.x_divMonomialₓ'. -/
@[simp]
theorem x_divMonomial (i : σ) : (X i : MvPolynomial σ R) /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 1 :=
  divMonomial_monomial (Finsupp.single i 1)
#align mv_polynomial.X_div_monomial MvPolynomial.x_divMonomial

/- warning: mv_polynomial.mul_X_div_monomial -> MvPolynomial.mul_x_divMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.mul_X_div_monomial MvPolynomial.mul_x_divMonomialₓ'. -/
@[simp]
theorem mul_x_divMonomial (x : MvPolynomial σ R) (i : σ) :
    x * X i /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x :=
  divMonomial_mul_monomial _ _
#align mv_polynomial.mul_X_div_monomial MvPolynomial.mul_x_divMonomial

/- warning: mv_polynomial.X_mul_mod_monomial -> MvPolynomial.x_mul_modMonomial is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.X_mul_mod_monomial MvPolynomial.x_mul_modMonomialₓ'. -/
@[simp]
theorem x_mul_modMonomial (i : σ) (x : MvPolynomial σ R) :
    X i * x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 :=
  monomial_mul_modMonomial _ _
#align mv_polynomial.X_mul_mod_monomial MvPolynomial.x_mul_modMonomial

/- warning: mv_polynomial.mul_X_mod_monomial -> MvPolynomial.mul_x_modMonomial 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 mv_polynomial.mul_X_mod_monomial MvPolynomial.mul_x_modMonomialₓ'. -/
@[simp]
theorem mul_x_modMonomial (x : MvPolynomial σ R) (i : σ) :
    x * X i %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 :=
  mul_monomial_modMonomial _ _
#align mv_polynomial.mul_X_mod_monomial MvPolynomial.mul_x_modMonomial

/- warning: mv_polynomial.mod_monomial_X -> MvPolynomial.modMonomial_x 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 mv_polynomial.mod_monomial_X MvPolynomial.modMonomial_xₓ'. -/
@[simp]
theorem modMonomial_x (i : σ) : (X i : MvPolynomial σ R) %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 :=
  monomial_modMonomial _
#align mv_polynomial.mod_monomial_X MvPolynomial.modMonomial_x

/- warning: mv_polynomial.div_monomial_add_mod_monomial_single -> MvPolynomial.divMonomial_add_modMonomial_single is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.div_monomial_add_mod_monomial_single MvPolynomial.divMonomial_add_modMonomial_singleₓ'. -/
theorem divMonomial_add_modMonomial_single (x : MvPolynomial σ R) (i : σ) :
    X i * (x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1) + x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = x :=
  divMonomial_add_modMonomial _ _
#align mv_polynomial.div_monomial_add_mod_monomial_single MvPolynomial.divMonomial_add_modMonomial_single

/- warning: mv_polynomial.mod_monomial_add_div_monomial_single -> MvPolynomial.modMonomial_add_divMonomial_single is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.mod_monomial_add_div_monomial_single MvPolynomial.modMonomial_add_divMonomial_singleₓ'. -/
theorem modMonomial_add_divMonomial_single (x : MvPolynomial σ R) (i : σ) :
    x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 + X i * (x /ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1) = x :=
  modMonomial_add_divMonomial _ _
#align mv_polynomial.mod_monomial_add_div_monomial_single MvPolynomial.modMonomial_add_divMonomial_single

/- warning: mv_polynomial.X_dvd_iff_mod_monomial_eq_zero -> MvPolynomial.x_dvd_iff_modMonomial_eq_zero is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.X_dvd_iff_mod_monomial_eq_zero MvPolynomial.x_dvd_iff_modMonomial_eq_zeroₓ'. -/
theorem x_dvd_iff_modMonomial_eq_zero {i : σ} {x : MvPolynomial σ R} :
    X i ∣ x ↔ x %ᵐᵒⁿᵒᵐⁱᵃˡ Finsupp.single i 1 = 0 :=
  monomial_one_dvd_iff_modMonomial_eq_zero
#align mv_polynomial.X_dvd_iff_mod_monomial_eq_zero MvPolynomial.x_dvd_iff_modMonomial_eq_zero

end XLemmas

/-! ### Some results about dvd (`∣`) on `monomial` and `X` -/

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Case conversion may be inaccurate. Consider using '#align mv_polynomial.monomial_dvd_monomial MvPolynomial.monomial_dvd_monomialₓ'. -/
theorem monomial_dvd_monomial {r s : R} {i j : σ →₀ ℕ} :
    monomial i r ∣ monomial j s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s :=
  by
  constructor
  · rintro ⟨x, hx⟩
    rw [MvPolynomial.ext_iff] at hx
    have hj := hx j
    have hi := hx i
    classical
      simp_rw [coeff_monomial, if_pos] at hj hi
      simp_rw [coeff_monomial_mul', if_pos] at hi hj
      split_ifs  at hi hj with hi hi
      · exact ⟨Or.inr hi, _, hj⟩
      · exact ⟨Or.inl hj, hj.symm ▸ dvd_zero _⟩
  · rintro ⟨h | hij, d, rfl⟩
    · simp_rw [h, monomial_zero, dvd_zero]
    · refine' ⟨monomial (j - i) d, _⟩
      rw [monomial_mul, add_tsub_cancel_of_le hij]
#align mv_polynomial.monomial_dvd_monomial MvPolynomial.monomial_dvd_monomial

/- warning: mv_polynomial.monomial_one_dvd_monomial_one -> MvPolynomial.monomial_one_dvd_monomial_one is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align mv_polynomial.monomial_one_dvd_monomial_one MvPolynomial.monomial_one_dvd_monomial_oneₓ'. -/
@[simp]
theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : σ →₀ ℕ} :
    monomial i (1 : R) ∣ monomial j 1 ↔ i ≤ j :=
  by
  rw [monomial_dvd_monomial]
  simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]
#align mv_polynomial.monomial_one_dvd_monomial_one MvPolynomial.monomial_one_dvd_monomial_one

#print MvPolynomial.x_dvd_x /-
@[simp]
theorem x_dvd_x [Nontrivial R] {i j : σ} :
    (X i : MvPolynomial σ R) ∣ (X j : MvPolynomial σ R) ↔ i = j :=
  by
  refine' monomial_one_dvd_monomial_one.trans _
  simp_rw [Finsupp.single_le_iff, Nat.one_le_iff_ne_zero, Finsupp.single_apply_ne_zero, Ne.def,
    one_ne_zero, not_false_iff, and_true_iff]
#align mv_polynomial.X_dvd_X MvPolynomial.x_dvd_x
-/

/- warning: mv_polynomial.X_dvd_monomial -> MvPolynomial.x_dvd_monomial 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 mv_polynomial.X_dvd_monomial MvPolynomial.x_dvd_monomialₓ'. -/
@[simp]
theorem x_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
    (X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 :=
  by
  refine' monomial_dvd_monomial.trans _
  simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
#align mv_polynomial.X_dvd_monomial MvPolynomial.x_dvd_monomial

end MvPolynomial