mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
hn : ¬n i = 0
⊢ ∑ x ∈ Finset.antidiagonal n,
      coeff x.1 (p.modMonomial (Finsupp.single i 1)) * coeff x.2 (q.modMonomial (Finsupp.single i 1)) =
    0

Tactic:
]

Goal state after:
case mp
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
hn : ¬n i = 0
⊢ ∑ x ∈ Finset.antidiagonal n,
      coeff x.1 (p.modMonomial (Finsupp.single i 1)) * coeff x.2 (q.modMonomial (Finsupp.single i 1)) =
    0
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
⊢ (f a + ∑ x ∈ s, f x) ^ n =
    ∑ i ∈ antidiagonal n,
      ∑ x ∈ map (addRightEmbedding fun t ↦ if t = a then i.1 else 0) (s.piAntidiag i.2),
        ↑(multinomial (cons a s has) x) * (cons a s has).noncommProd (fun i ↦ f i ^ x i) ⋯

Tactic:
]

Goal state after:
case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
⊢ (f a + ∑ x ∈ s, f x) ^ n =
    ∑ i ∈ antidiagonal n,
      ∑ x ∈ map (addRightEmbedding fun t ↦ if t = a then i.1 else 0) (s.piAntidiag i.2),
        ↑(multinomial (cons a s has) x) * (cons a s has).noncommProd (fun i ↦ f i ^ x i) ⋯
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case hc
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hc : ContinuousOn f (Box.Icc I)
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
⊢ μ ((Box.Icc I)ᶜ ∩ Box.Icc I) = 0

pp: case hc
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hc : ContinuousOn f (Box.Icc I)
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
⊢ MeasurableSet (Box.Icc I)ᶜ

Tactic:
]

Goal state after:
case hc
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hc : ContinuousOn f (Box.Icc I)
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
⊢ μ ((Box.Icc I)ᶜ ∩ Box.Icc I) = 0

case hc
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hc : ContinuousOn f (Box.Icc I)
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
⊢ MeasurableSet (Box.Icc I)ᶜ
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
⊢ (f a + ∑ x ∈ s, f x) ^ n =
    ∑ x ∈ antidiagonal n,
      ∑ g ∈ s.piAntidiag x.2, ↑(n.choose x.1) * ↑(multinomial s g) * (f a ^ x.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)

Tactic:
]

Goal state after:
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
⊢ (f a + ∑ x ∈ s, f x) ^ n =
    ∑ x ∈ antidiagonal n,
      ∑ g ∈ s.piAntidiag x.2, ↑(n.choose x.1) * ↑(multinomial s g) * (f a ^ x.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp.inl
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.1 i = 0
⊢ 0 = 0

pp: case mp.inl.h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.1 i = 0
⊢ Finsupp.single i 1 ≤ x.1

Tactic:
]

Goal state after:
case mp.inl.h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.1 i = 0
⊢ Finsupp.single i 1 ≤ x.1
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
⊢ ∑ m ∈ range (n + 1), f a ^ m * (∑ i ∈ s, f i) ^ (n - m) * ↑(n.choose m) =
    ∑ x ∈ range (n + 1),
      ∑ x_1 ∈ s.piAntidiag (n - x),
        ↑(n.choose x) * ↑(multinomial s x_1) * (f a ^ x * s.noncommProd (fun i ↦ f i ^ x_1 i) ⋯)

Tactic:
]

Goal state after:
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
⊢ ∑ m ∈ range (n + 1), f a ^ m * (∑ i ∈ s, f i) ^ (n - m) * ↑(n.choose m) =
    ∑ x ∈ range (n + 1),
      ∑ x_1 ∈ s.piAntidiag (n - x),
        ↑(n.choose x) * ↑(multinomial s x_1) * (f a ^ x * s.noncommProd (fun i ↦ f i ^ x_1 i) ⋯)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp.inr
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.2 i = 0
⊢ 0 = 0

pp: case mp.inr.h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.2 i = 0
⊢ Finsupp.single i 1 ≤ x.2

Tactic:
]

Goal state after:
case mp.inr.h
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
a✝ : Nontrivial R
h : X i ∣ p * q
hp : p.modMonomial (Finsupp.single i 1) + X i * p.divMonomial (Finsupp.single i 1) = p
hq : q.modMonomial (Finsupp.single i 1) + X i * q.divMonomial (Finsupp.single i 1) = q
n : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = n
hn : ¬x.2 i = 0
⊢ Finsupp.single i 1 ≤ x.2
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
i : ℕ
x✝¹ : i ∈ range (n + 1)
g : α → ℕ
x✝ : g ∈ s.piAntidiag (n - i)
⊢ ↑(n.choose i) * (↑(multinomial s g) * (f a ^ i * s.noncommProd (fun i ↦ f i ^ g i) ⋯)) =
    ↑(n.choose i) * (↑(multinomial s g) * (f a ^ i * s.noncommProd (fun i ↦ f i ^ g i) ⋯))

Tactic:
]

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((f a + p.1 + s.sum f).choose (f a + p.1)) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ (f a + p.1) * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)

Tactic:
]

Goal state after:
case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((f a + p.1 + s.sum f).choose (f a + p.1)) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ (f a + p.1) * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
i : σ
inst✝¹ : IsCancelMulZero R
inst✝ : Nontrivial R
h : ∃ b, X i * b = 1 ∧ b * X i = 1
⊢ False

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
i : σ
inst✝¹ : IsCancelMulZero R
inst✝ : Nontrivial R
h : ∃ b, X i * b = 1 ∧ b * X i = 1
⊢ False
Module: Mathlib.Algebra.MvPolynomial.Nilpotent

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
P✝ : MvPolynomial σ R
inst✝¹ : Finite σ
this : Fintype σ
α β : Type u_1
inst✝ : Fintype β
e : α ≃ β
h₁ : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)
P : MvPolynomial β R
⊢ (∀ (b : β →₀ ℕ), IsNilpotent (coeff ((Finsupp.equivCongrLeft e).symm b) ((rename ⇑e.symm) P))) ↔
    ∀ (i : β →₀ ℕ), IsNilpotent (coeff i P)

Tactic:
]

Goal state after:
case refine_1
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
P✝ : MvPolynomial σ R
inst✝¹ : Finite σ
this : Fintype σ
α β : Type u_1
inst✝ : Fintype β
e : α ≃ β
h₁ : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)
P : MvPolynomial β R
⊢ (∀ (b : β →₀ ℕ), IsNilpotent (coeff ((Finsupp.equivCongrLeft e).symm b) ((rename ⇑e.symm) P))) ↔
    ∀ (i : β →₀ ℕ), IsNilpotent (coeff i P)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
i : σ
inst✝¹ : IsCancelMulZero R
inst✝ : Nontrivial R
h : IsUnit (X i)
⊢ False

Tactic:
rw [isUnit_iff_exists] at h

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
i : σ
inst✝¹ : IsCancelMulZero R
inst✝ : Nontrivial R
h : ∃ b, X i * b = 1 ∧ b * X i = 1
⊢ False
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = p * (X i * r)
this : X i ∣ p ∨ X i ∣ X i * r
⊢ q = p * r

Tactic:
← X_mul_cancel_left_iff (i := i),

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = p * (X i * r)
this : X i ∣ p ∨ X i ∣ X i * r
⊢ X i * q = X i * (p * r)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = p * (X i * r)
this : X i ∣ p ∨ X i ∣ X i * r
⊢ X i * (p * r) = X i * (p * r)

Tactic:
]

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((p.1 + p.2).choose p.1) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)

Tactic:
]

Goal state after:
case cons
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑((p.1 + p.2).choose p.1) * ↑(multinomial s (f + fun i ↦ if i = a then p.1 else 0)) *
      (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ (f i + if i = a then p.1 else 0)) ⋯) =
    ↑(n.choose p.1) * ↑(multinomial s f) * (f✝ a ^ p.1 * s.noncommProd (fun i ↦ f✝ i ^ f i) ⋯)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = p * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q

Tactic:
]

Goal state after:
case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : X i * q = p * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : q = p.divMonomial (Finsupp.single i 1) * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q

Tactic:
]

Goal state after:
case mp.g
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
r : MvPolynomial σ R
hp : q = p.divMonomial (Finsupp.single i 1) * r
this : X i ∣ p ∨ X i ∣ r
hip : p.modMonomial (Finsupp.single i 1) = 0
⊢ p.divMonomial (Finsupp.single i 1) ∣ q
Module: Mathlib.Algebra.MvPolynomial.Nilpotent

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
f : Fin n → σ
hf : Function.Injective f
P : MvPolynomial (Fin n) R
⊢ (∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)) ↔ ∀ (i : σ →₀ ℕ), IsNilpotent (coeff i ((rename f) P))

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
f : Fin n → σ
hf : Function.Injective f
P : MvPolynomial (Fin n) R
⊢ (∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)) ↔ ∀ (i : σ →₀ ℕ), IsNilpotent (coeff i ((rename f) P))
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
hi : X i ∣ p
hq : p.divMonomial (Finsupp.single i 1) ∣ q
this : p = X i * p.divMonomial (Finsupp.single i 1)
⊢ X i * p.divMonomial (Finsupp.single i 1) ∣ X i * q

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
i : σ
p q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
hi : X i ∣ p
hq : p.divMonomial (Finsupp.single i 1) ∣ q
this : p = X i * p.divMonomial (Finsupp.single i 1)
⊢ X i * p.divMonomial (Finsupp.single i 1) ∣ X i * q
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑(n.choose p.1) = ↑(n.choose p.1)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.Nilpotent

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
P : MvPolynomial (Fin n) R
f : Fin n ↪ σ
H✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)
i : σ →₀ ℕ
H : i ∉ Set.range (Finsupp.embDomain f)
⊢ IsNilpotent 0

pp: case neg.h
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
P : MvPolynomial (Fin n) R
f : Fin n ↪ σ
H✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)
i : σ →₀ ℕ
H : i ∉ Set.range (Finsupp.embDomain f)
⊢ ∀ (u : Fin n →₀ ℕ), Finsupp.mapDomain (⇑f) u = i → coeff u P = 0

Tactic:
]

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
P : MvPolynomial (Fin n) R
f : Fin n ↪ σ
H✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)
i : σ →₀ ℕ
H : i ∉ Set.range (Finsupp.embDomain f)
⊢ IsNilpotent 0

case neg.h
σ : Type u_1
R : Type u_2
inst✝ : CommRing R
n : ℕ
P : MvPolynomial (Fin n) R
f : Fin n ↪ σ
H✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)
i : σ →₀ ℕ
H : i ∉ Set.range (Finsupp.embDomain f)
⊢ ∀ (u : Fin n →₀ ℕ), Finsupp.mapDomain (⇑f) u = i → coeff u P = 0
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case inr.zero
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p✝ q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n✝ : σ →₀ ℕ
hR : Nontrivial R
n : σ →₀ ℕ
hn : n = 0
p : MvPolynomial σ R
⊢ ∀ (q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r

Tactic:
]

Goal state after:
case inr.zero
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p✝ q : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n✝ : σ →₀ ℕ
hR : Nontrivial R
n : σ →₀ ℕ
hn : n = 0
p : MvPolynomial σ R
⊢ ∀ (q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p✝ q✝ : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n✝ : σ →₀ ℕ
hR : Nontrivial R
d : ℕ
hd :
  ∀ (n : σ →₀ ℕ),
    Finsupp.degree n = d →
      ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r
n : σ →₀ ℕ
hn : Finsupp.degree n = d + 1
p q : MvPolynomial σ R
hp : p ∣ (monomial n) 1 * q
⊢ n ≠ 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝¹ : CommRing R
p✝ q✝ : MvPolynomial σ R
inst✝ : IsCancelMulZero R
n✝ : σ →₀ ℕ
hR : Nontrivial R
d : ℕ
hd :
  ∀ (n : σ →₀ ℕ),
    Finsupp.degree n = d →
      ∀ (p q : MvPolynomial σ R), p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r
n : σ →₀ ℕ
hn : Finsupp.degree n = d + 1
p q : MvPolynomial σ R
hp : p ∣ (monomial n) 1 * q
⊢ n ≠ 0
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ↑(multinomial s ?m.522) = ↑(multinomial s f)

pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ∀ a_1 ∈ s, (f + fun i ↦ if i = a then p.1 else 0) a_1 = ?m.522 a_1

pp: α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ α → ℕ

Tactic:
]

Goal state after:
case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ∀ a_1 ∈ s, (f + fun i ↦ if i = a then p.1 else 0) a_1 = f a_1