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

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x else 0) =
    ↑s.multinomial * s.prod fun r m ↦ a r ^ m

pp: case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ∀ b ∈ a.support.piAntidiag n,
    b ≠ ⇑s →
      (if s = indicator a.support fun i x ↦ b i then ↑(Nat.multinomial a.support b) * ∏ x ∈ a.support, a x ^ b x
        else 0) =
        0

pp: case h₁
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ⇑s ∉ a.support.piAntidiag n →
    (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x
      else 0) =
      0

Tactic:
·

Goal state after:
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x else 0) =
    ↑s.multinomial * s.prod fun r m ↦ a r ^ m
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs' : s.support ⊆ a.support
⊢ ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x = ↑s.multinomial * s.prod fun r m ↦ a r ^ m

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs' : ¬s.support ⊆ a.support
⊢ 0 = ↑s.multinomial * s.prod fun r m ↦ a r ^ m

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs' : s.support ⊆ a.support
⊢ ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x = ↑s.multinomial * s.prod fun r m ↦ a r ^ m
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs' : ¬s.support ⊆ a.support
⊢ 0 = ↑s.multinomial * s.prod fun r m ↦ a r ^ m

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs' : ¬s.support ⊆ a.support
⊢ 0 = ↑s.multinomial * s.prod fun r m ↦ a r ^ m
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ∀ b ∈ a.support.piAntidiag n,
    b ≠ ⇑s →
      (if s = indicator a.support fun i x ↦ b i then ↑(Nat.multinomial a.support b) * ∏ x ∈ a.support, a x ^ b x
        else 0) =
        0

pp: case h₁
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ⇑s ∉ a.support.piAntidiag n →
    (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x
      else 0) =
      0

Tactic:
·

Goal state after:
case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ∀ b ∈ a.support.piAntidiag n,
    b ≠ ⇑s →
      (if s = indicator a.support fun i x ↦ b i then ↑(Nat.multinomial a.support b) * ∏ x ∈ a.support, a x ^ b x
        else 0) =
        0
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₁
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ⇑s ∉ a.support.piAntidiag n →
    (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x
      else 0) =
      0

Tactic:
·

Goal state after:
case h₁
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ ⇑s ∉ a.support.piAntidiag n →
    (if s = indicator a.support fun i x ↦ s i then ↑(Nat.multinomial a.support ⇑s) * ∏ x ∈ a.support, a x ^ s x
      else 0) =
      0
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₁.inl
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs'' : s.support ⊆ a.support
hs' : ¬a.support.sum ⇑s = n
⊢ False

pp: case h₁.inr
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs'' : s.support ⊆ a.support
hs' : ¬∀ (i : σ), s i ≠ 0 → i ∈ a.support
⊢ False

Tactic:
·

Goal state after:
case h₁.inl
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs'' : s.support ⊆ a.support
hs' : ¬a.support.sum ⇑s = n
⊢ False
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e_a
α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ (∑ i ∈ s, f i)! = (∑ i ∈ s, g i)!

pp: case e_a
α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ ∏ i ∈ s, (f i)! = ∏ i ∈ s, (g i)!

Tactic:
·

Goal state after:
case e_a
α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ (∑ i ∈ s, f i)! = (∑ i ∈ s, g i)!
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e_a
α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ ∏ i ∈ s, (f i)! = ∏ i ∈ s, (g i)!

Tactic:
·

Goal state after:
case e_a
α : Type u_1
s : Finset α
f g : α → ℕ
h : ∀ a ∈ s, f a = g a
⊢ ∏ i ∈ s, (f i)! = ∏ i ∈ s, (g i)!
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: 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
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := ⋯
μ' : Measure (ι → ℝ) := ⋯
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this✝ : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
this : ∀ J ∈ B', ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)
⊢ μ' (⋃ J ∈ B', ↑J) ≤ μ' U

Tactic:
·

Goal state after:
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
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := ⋯
μ' : Measure (ι → ℝ) := ⋯
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this✝ : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
this : ∀ J ∈ B', ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)
⊢ μ' (⋃ J ∈ B', ↑J) ≤ μ' U
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₁.inr
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs'' : s.support ⊆ a.support
hs' : ¬∀ (i : σ), s i ≠ 0 → i ∈ a.support
⊢ False

Tactic:
·

Goal state after:
case h₁.inr
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
hs'' : s.support ⊆ a.support
hs' : ¬∀ (i : σ), s i ≠ 0 → i ∈ a.support
⊢ False
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case pos
α : Type u_1
s : Finset α
a : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∈ s
⊢ n ! = ∏ x ∈ s, (Pi.single a n x)!

pp: case neg
α : Type u_1
s : Finset α
a : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∉ s
⊢ 0! = ∏ x ∈ s, (Pi.single a n x)!

Tactic:
·

Goal state after:
case pos
α : Type u_1
s : Finset α
a : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∈ s
⊢ n ! = ∏ x ∈ s, (Pi.single a n x)!
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case neg
α : Type u_1
s : Finset α
a : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∉ s
⊢ 0! = ∏ x ∈ s, (Pi.single a n x)!

Tactic:
·

Goal state after:
case neg
α : Type u_1
s : Finset α
a : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∉ s
⊢ 0! = ∏ x ∈ s, (Pi.single a n x)!
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∈ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∉ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∈ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∉ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∉ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s

pp: case mpr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s

Tactic:
·

Goal state after:
case mp
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
s : σ →₀ ℕ
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ coeff s ((linearCombination R X) a ^ n) = ↑s.multinomial * s.prod fun r m ↦ a r ^ m

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
s : σ →₀ ℕ
n : ℕ
hs : ¬(s.sum fun x m ↦ m) = n
⊢ coeff s ((linearCombination R X) a ^ n) = 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
s : σ →₀ ℕ
n : ℕ
hs : (s.sum fun x m ↦ m) = n
⊢ coeff s ((linearCombination R X) a ^ n) = ↑s.multinomial * s.prod fun r m ↦ a r ^ m
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
s : σ →₀ ℕ
n : ℕ
hs : ¬(s.sum fun x m ↦ m) = n
⊢ coeff s ((linearCombination R X) a ^ n) = 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
s : σ →₀ ℕ
n : ℕ
hs : ¬(s.sum fun x m ↦ m) = n
⊢ coeff s ((linearCombination R X) a ^ n) = 0
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
d : Fin 2 →₀ ℕ
n : ℕ
this : (d.sum fun x m ↦ m) = d 0 + d 1
hd : d 0 + d 1 = n
⊢ d.multinomial = n.choose (d 0)

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
d : Fin 2 →₀ ℕ
n : ℕ
this : (d.sum fun x m ↦ m) = d 0 + d 1
hd : ¬d 0 + d 1 = n
⊢ 0 = 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
d : Fin 2 →₀ ℕ
n : ℕ
this : (d.sum fun x m ↦ m) = d 0 + d 1
hd : d 0 + d 1 = n
⊢ d.multinomial = n.choose (d 0)
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
d : Fin 2 →₀ ℕ
n : ℕ
this : (d.sum fun x m ↦ m) = d 0 + d 1
hd : ¬d 0 + d 1 = n
⊢ 0 = 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommSemiring R
d : Fin 2 →₀ ℕ
n : ℕ
this : (d.sum fun x m ↦ m) = d 0 + d 1
hd : ¬d 0 + d 1 = n
⊢ 0 = 0
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : ∀ (m : σ →₀ ℕ), coeff m ((monomial j) s) = coeff m ((monomial i) r * x)
hi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0
hi : i ≤ j
hj : s = r * coeff (j - i) x
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s

pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : ∀ (m : σ →₀ ℕ), coeff m ((monomial j) s) = coeff m ((monomial i) r * x)
hi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0
hi : ¬i ≤ j
hj : s = 0
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : ∀ (m : σ →₀ ℕ), coeff m ((monomial j) s) = coeff m ((monomial i) r * x)
hi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0
hi : i ≤ j
hj : s = r * coeff (j - i) x
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : ∀ (m : σ →₀ ℕ), coeff m ((monomial j) s) = coeff m ((monomial i) r * x)
hi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0
hi : ¬i ≤ j
hj : s = 0
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s

Tactic:
·

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : ∀ (m : σ →₀ ℕ), coeff m ((monomial j) s) = coeff m ((monomial i) r * x)
hi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0
hi : ¬i ≤ j
hj : s = 0
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mpr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s

Tactic:
·

Goal state after:
case mpr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ (∑ i ∈ f.support, f i)! = (∑ i ∈ s, f i)!

pp: case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ ∏ i ∈ f.support, (f i)! = ∏ i ∈ s, (f i)!

Tactic:
·

Goal state after:
case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ (∑ i ∈ f.support, f i)! = (∑ i ∈ s, f i)!
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mpr.inl
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r : R
i j : σ →₀ ℕ
d : R
h : r * d = 0
⊢ (monomial i) r ∣ (monomial j) (r * d)

pp: case mpr.inr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r : R
i j : σ →₀ ℕ
hij : i ≤ j
d : R
⊢ (monomial i) r ∣ (monomial j) (r * d)

Tactic:
·

Goal state after:
case mpr.inl
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r : R
i j : σ →₀ ℕ
d : R
h : r * d = 0
⊢ (monomial i) r ∣ (monomial j) (r * d)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mpr.inr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r : R
i j : σ →₀ ℕ
hij : i ≤ j
d : R
⊢ (monomial i) r ∣ (monomial j) (r * d)

Tactic:
·

Goal state after:
case mpr.inr
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r : R
i j : σ →₀ ℕ
hij : i ≤ j
d : R
⊢ (monomial i) r ∣ (monomial j) (r * d)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ ∏ i ∈ f.support, (f i)! = ∏ i ∈ s, (f i)!

Tactic:
·

Goal state after:
case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ ∏ i ∈ f.support, (f i)! = ∏ i ∈ s, (f i)!
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ ∏ i ∈ f.support, (f i)! = ∏ i ∈ s, (f i)!

Tactic:
rw [Finset.prod_subset h]

Goal state after:
case e_a
α : Type u_1
f : α →₀ ℕ
s : Finset α
h : f.support ⊆ s
⊢ ∀ x ∈ s, x ∉ f.support → (f x)! = 1
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ
⊢ X i ∣ X j ↔ i = j

Tactic:
refine monomial_one_dvd_monomial_one.trans ?_

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ
⊢ Finsupp.single i 1 ≤ Finsupp.single j 1 ↔ i = j
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case pos
α : Type u_1
a : α
f : α →₀ ℕ
h : a ∈ f.support
⊢ Nat.multinomial f.support ⇑f =
    (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)

pp: case neg
α : Type u_1
a : α
f : α →₀ ℕ
h : a ∉ f.support
⊢ Nat.multinomial f.support ⇑f =
    (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)

Tactic:
·

Goal state after:
case pos
α : Type u_1
a : α
f : α →₀ ℕ
h : a ∈ f.support
⊢ Nat.multinomial f.support ⇑f =
    (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h✝ : X i ∣ p - r
h : r = p.modMonomial (Finsupp.single i 1)
n : σ →₀ ℕ
⊢ n ∈ r.support → n i = 0

pp: case refine_2
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h : X i ∣ p - r
hr : ∀ n ∈ r.support, n i = 0
⊢ r = p.modMonomial (Finsupp.single i 1)

Tactic:
·

Goal state after:
case refine_1
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h✝ : X i ∣ p - r
h : r = p.modMonomial (Finsupp.single i 1)
n : σ →₀ ℕ
⊢ n ∈ r.support → n i = 0
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e'_3.e'_5.e'_1
α : Type u_1
inst✝ : DecidableEq α
a : α
m : Multiset α
⊢ m.card = (toFinsupp m).sum fun x ↦ id

pp: case e'_3.e'_6
α : Type u_1
inst✝ : DecidableEq α
a : α
m : Multiset α
⊢ toFinsupp (filter (fun x ↦ a ≠ x) m) = (toFinsupp m).update a 0

Tactic:
·

Goal state after:
case e'_3.e'_5.e'_1
α : Type u_1
inst✝ : DecidableEq α
a : α
m : Multiset α
⊢ m.card = (toFinsupp m).sum fun x ↦ id
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case refine_2
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h : X i ∣ p - r
hr : ∀ n ∈ r.support, n i = 0
⊢ r = p.modMonomial (Finsupp.single i 1)

Tactic:
·

Goal state after:
case refine_2
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
i : σ
p r : MvPolynomial σ R
h : X i ∣ p - r
hr : ∀ n ∈ r.support, n i = 0
⊢ r = p.modMonomial (Finsupp.single i 1)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case e'_3.e'_6
α : Type u_1
inst✝ : DecidableEq α
a : α
m : Multiset α
⊢ toFinsupp (filter (fun x ↦ a ≠ x) m) = (toFinsupp m).update a 0

Tactic:
·

Goal state after:
case e'_3.e'_6
α : Type u_1
inst✝ : DecidableEq α
a : α
m : Multiset α
⊢ toFinsupp (filter (fun x ↦ a ≠ x) m) = (toFinsupp m).update a 0
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
⊢ X i ∣ p * q → X i ∣ p ∨ X i ∣ q

pp: case mpr
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
⊢ X i ∣ p ∨ X i ∣ q → X i ∣ p * q

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
⊢ X i ∣ p * q → X i ∣ p ∨ X i ∣ q
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case pos
α : Type u_1
inst✝ : DecidableEq α
a✝ : α
m : Multiset α
a : α
h : a✝ ≠ a
⊢ count a m = Function.update (⇑(toFinsupp m)) a✝ 0 a

pp: case neg
α : Type u_1
inst✝ : DecidableEq α
a✝ : α
m : Multiset α
a : α
h : ¬a✝ ≠ a
⊢ 0 = Function.update (⇑(toFinsupp m)) a✝ 0 a

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : DecidableEq α
a✝ : α
m : Multiset α
a : α
h : a✝ ≠ a
⊢ count a m = Function.update (⇑(toFinsupp m)) a✝ 0 a
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case neg
α : Type u_1
inst✝ : DecidableEq α
a✝ : α
m : Multiset α
a : α
h : ¬a✝ ≠ a
⊢ 0 = Function.update (⇑(toFinsupp m)) a✝ 0 a

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : DecidableEq α
a✝ : α
m : Multiset α
a : α
h : ¬a✝ ≠ a
⊢ 0 = Function.update (⇑(toFinsupp m)) a✝ 0 a
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 ∈ (p.modMonomial (Finsupp.single i 1) * q.modMonomial (Finsupp.single i 1)).support, n i = 0

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
⊢ X i ∣ p * q - p.modMonomial (Finsupp.single i 1) * q.modMonomial (Finsupp.single i 1)

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 ∈ (p.modMonomial (Finsupp.single i 1) * q.modMonomial (Finsupp.single i 1)).support, n i = 0