mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case h
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
              x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
              (l.bHenstock = true → x ∈ Box.Icc J) →
                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := ⋯
π : TaggedPrepartition I
c : ℝ≥0
hπδ : l.MemBaseSet I c (δ c) π
hπp : π.IsPartition
⊢ {J ∈ π.boxes | π.tag J ∉ s} ⊆ π.boxes

Tactic:
·

Goal state after:
case h
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
              x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
              (l.bHenstock = true → x ∈ Box.Icc J) →
                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := ⋯
π : TaggedPrepartition I
c : ℝ≥0
hπδ : l.MemBaseSet I c (δ c) π
hπp : π.IsPartition
⊢ {J ∈ π.boxes | π.tag J ∉ s} ⊆ π.boxes
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : MvPolynomial σ R
a : R
ha : a ≠ 0
⊢ f ∣ C a → ∃ b, b ∣ a ∧ f = C b

pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : MvPolynomial σ R
a : R
ha : a ≠ 0
⊢ (∃ b, b ∣ a ∧ f = C b) → f ∣ C a

Tactic:
·

Goal state after:
case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : MvPolynomial σ R
a : R
ha : a ≠ 0
⊢ f ∣ C a → ∃ b, b ∣ a ∧ f = C b
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : MvPolynomial σ R
a : R
ha : a ≠ 0
⊢ (∃ b, b ∣ a ∧ f = C b) → f ∣ C a

Tactic:
·

Goal state after:
case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : MvPolynomial σ R
a : R
ha : a ≠ 0
⊢ (∃ b, b ∣ a ∧ f = C b) → f ∣ C a
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : p = 0
⊢ degreeOf j (C c * p) = degreeOf j p

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ degreeOf j (C c * p) = degreeOf j p

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : p = 0
⊢ degreeOf j (C c * p) = degreeOf j p
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case pos
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : x = y
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

pp: case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : x = y
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).natDegree +
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).natDegree =
    ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).natDegree

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).leadingCoeff *
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).leadingCoeff ≠
    0

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).natDegree +
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).natDegree =
    ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).natDegree
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).leadingCoeff *
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).leadingCoeff ≠
    0

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) (C c))).leadingCoeff *
      ((optionEquivLeft R { b // b ≠ j }) ((renameEquiv R (optionSubtypeNe j).symm) p)).leadingCoeff ≠
    0
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case pos
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
k : degree (ofDegLex x) < degree (ofDegLex y)
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

pp: case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
k : ¬degree (ofDegLex x) < degree (ofDegLex y)
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
k : degree (ofDegLex x) < degree (ofDegLex y)
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
k : ¬degree (ofDegLex x) < degree (ofDegLex y)
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : ¬x = y
k : ¬degree (ofDegLex x) < degree (ofDegLex y)
⊢ x = y ∨
      degree (ofDegLex x) < degree (ofDegLex y) ∨
        degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) < toLex (ofDegLex y) ↔
    degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ (x = y ∨ toLex (ofDegLex x) < toLex (ofDegLex y))
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case inl
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) < degree (ofDegLex y)
⊢ (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y

pp: case inr
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) ≤ toLex (ofDegLex y)
⊢ (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) < degree (ofDegLex y)
⊢ (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case inr
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) ≤ toLex (ofDegLex y)
⊢ (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) ≤ toLex (ofDegLex y)
⊢ (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case inl
α : Type u_1
inst✝ : LinearOrder α
x : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) = 0
⊢ 0 < degree (ofDegLex x) ∨ 0 = degree (ofDegLex x) ∧ 0 ≤ toLex (ofDegLex x)

pp: case inr
α : Type u_1
inst✝ : LinearOrder α
x : DegLex (α →₀ ℕ)
h : 0 < degree (ofDegLex x)
⊢ 0 < degree (ofDegLex x) ∨ 0 = degree (ofDegLex x) ∧ 0 ≤ toLex (ofDegLex x)

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : LinearOrder α
x : DegLex (α →₀ ℕ)
h : degree (ofDegLex x) = 0
⊢ 0 < degree (ofDegLex x) ∨ 0 = degree (ofDegLex x) ∧ 0 ≤ toLex (ofDegLex x)
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case inr
α : Type u_1
inst✝ : LinearOrder α
x : DegLex (α →₀ ℕ)
h : 0 < degree (ofDegLex x)
⊢ 0 < degree (ofDegLex x) ∨ 0 = degree (ofDegLex x) ∧ 0 ≤ toLex (ofDegLex x)

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : LinearOrder α
x : DegLex (α →₀ ℕ)
h : 0 < degree (ofDegLex x)
⊢ 0 < degree (ofDegLex x) ∨ 0 = degree (ofDegLex x) ∧ 0 ≤ toLex (ofDegLex x)
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
⊢ (∃ r, m ≤ n ∧ r ∣ C a ∧ p = (monomial m) 1 * r) → ∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b

pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
⊢ (∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b) → ∃ r, m ≤ n ∧ r ∣ C a ∧ p = (monomial m) 1 * r

Tactic:
·

Goal state after:
case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
⊢ (∃ r, m ≤ n ∧ r ∣ C a ∧ p = (monomial m) 1 * r) → ∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case pos
α : Type u_1
σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ →₀ ℕ
h : a ≤ b
ha : degree a < degree b
⊢ degree a < degree b ∨ degree a = degree b ∧ toLex a ≤ toLex b

pp: case neg
α : Type u_1
σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ →₀ ℕ
h : a ≤ b
ha : degree b ≤ degree a
⊢ degree a < degree b ∨ degree a = degree b ∧ toLex a ≤ toLex b

Tactic:
·

Goal state after:
case pos
α : Type u_1
σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ →₀ ℕ
h : a ≤ b
ha : degree a < degree b
⊢ degree a < degree b ∨ degree a = degree b ∧ toLex a ≤ toLex b
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case neg
α : Type u_1
σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ →₀ ℕ
h : a ≤ b
ha : degree b ≤ degree a
⊢ degree a < degree b ∨ degree a = degree b ∧ toLex a ≤ toLex b

Tactic:
·

Goal state after:
case neg
α : Type u_1
σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ →₀ ℕ
h : a ≤ b
ha : degree b ≤ degree a
⊢ degree a < degree b ∨ degree a = degree b ∧ toLex a ≤ toLex b
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
⊢ (∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b) → ∃ r, m ≤ n ∧ r ∣ C a ∧ p = (monomial m) 1 * r

Tactic:
·

Goal state after:
case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
⊢ (∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b) → ∃ r, m ≤ n ∧ r ∣ C a ∧ p = (monomial m) 1 * r
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
hR : Subsingleton R
⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u

pp: case inr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
hR : Nontrivial R
⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u

Tactic:
·

Goal state after:
case inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
hR : Subsingleton R
⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
⊢ (∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b) → b ∣ r ∧ (p = C b ∨ p = b • (monomial (Finsupp.single i 1)) 1)

pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
⊢ b ∣ r ∧ (p = C b ∨ p = b • (monomial (Finsupp.single i 1)) 1) → ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b

Tactic:
·

Goal state after:
case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
⊢ (∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b) → b ∣ r ∧ (p = C b ∨ p = b • (monomial (Finsupp.single i 1)) 1)
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
⊢ m = 0 ∨ m = Finsupp.single i 1

pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
⊢ m = 0 ∨ m = Finsupp.single i 1

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
⊢ m = 0 ∨ m = Finsupp.single i 1
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
j : σ
hj : j = i
⊢ m j ≤ 0

pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
j : σ
hj : ¬j = i
⊢ m j ≤ 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
j : σ
hj : j = i
⊢ m j ≤ 0
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
j : σ
hj : ¬j = i
⊢ m j ≤ 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : m i = 0
j : σ
hj : ¬j = i
⊢ m j ≤ 0
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
⊢ m = 0 ∨ m = Finsupp.single i 1

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
⊢ m = 0 ∨ m = Finsupp.single i 1
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
j : σ
hj : j = i
⊢ (Finsupp.single i 1) j ≤ m j

pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
j : σ
hj : ¬j = i
⊢ (Finsupp.single i 1) j ≤ m j

Tactic:
·

Goal state after:
case pos
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
j : σ
hj : j = i
⊢ (Finsupp.single i 1) j ≤ m j
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
j : σ
hj : ¬j = i
⊢ (Finsupp.single i 1) j ≤ m j

Tactic:
·

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
i : σ
r : R
hr : r ≠ 0
b : R
m : σ →₀ ℕ
hmn : m ≤ Finsupp.single i 1
hb : b ∣ r
hm : ¬m i = 0
j : σ
hj : ¬j = i
⊢ (Finsupp.single i 1) j ≤ m j
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
⊢ b ∣ r ∧ (p = C b ∨ p = b • (monomial (Finsupp.single i 1)) 1) → ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b

Tactic:
·

Goal state after:
case mpr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
⊢ b ∣ r ∧ (p = C b ∨ p = b • (monomial (Finsupp.single i 1)) 1) → ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mpr.inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
hb : b ∣ r
hp : p = C b
⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b

pp: case mpr.inr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
hb : b ∣ r
hp : p = b • (monomial (Finsupp.single i 1)) 1
⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b

Tactic:
·

Goal state after:
case mpr.inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
hb : b ∣ r
hp : p = C b
⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mpr.inr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
hb : b ∣ r
hp : p = b • (monomial (Finsupp.single i 1)) 1
⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b

Tactic:
·

Goal state after:
case mpr.inr
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
i : σ
r : R
hr : r ≠ 0
b : R
hb : b ∣ r
hp : p = b • (monomial (Finsupp.single i 1)) 1
⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case mp
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ m.leadingCoeff f ≠ 0 → f ≠ 0

pp: case mpr
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ f ≠ 0 → m.leadingCoeff f ≠ 0

Tactic:
·

Goal state after:
case mp
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ m.leadingCoeff f ≠ 0 → f ≠ 0
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case mpr
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ f ≠ 0 → m.leadingCoeff f ≠ 0

Tactic:
·

Goal state after:
case mpr
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ f ≠ 0 → m.leadingCoeff f ≠ 0