mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
t : R
⊢ eval t (p /ₘ (X - C t) ^ rootMultiplicity t p) = (p.comp (X + C t)).trailingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: ι : Sort u_2
inst✝ : Finite ι
α : Type u_3
f : ι → Filter α
s : Set α
⊢ s ∈ ⨅ i, f i ↔ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ i, t i

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
inst✝ : ∀ (i : ι), (f i).NeBot
I : Set ι
h : I.pi s ∈ pi f
i : ι
hi : i ∈ I
⊢ s i ∈ f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n m : ℕ
p : R[X]
⊢ (⇑derivative)^[n] (p * X ^ m) =
    ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
ι' : ι → Type u_3
s : (i : ι) → ι' i → Set (α i)
p : (i : ι) → ι' i → Prop
h : ∀ (i : ι), (f i).HasBasis (p i) (s i)
⊢ (pi f).HasBasis (fun If ↦ If.1.Finite ∧ ∀ i ∈ If.1, p i (If.2 i)) fun If ↦ If.1.pi fun i ↦ s i (If.2 i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n m : ℕ
p : R[X]
hsum :
  (⇑derivative)^[n] (p * X ^ m) =
    ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))
⊢ ∀ a ∈ range (min m n).succ,
    a ∉ range n.succ → (n.choose a * m.descFactorial a) • ((⇑derivative)^[n - a] p * X ^ (m - a)) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
p0 : p ≠ 0
a : R
n : ℕ
⊢ n ≤ rootMultiplicity a p ↔ (X - C a) ^ n ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n m : ℕ
p : R[X]
hsum :
  (⇑derivative)^[n] (p * X ^ m) =
    ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))
⊢ ∀ a ∈ range n.succ,
    a ∈ range (min m n).succ →
      (n.choose a * m.descFactorial a) • ((⇑derivative)^[n - a] p * X ^ (m - a)) =
        (n.choose a * m.descFactorial a) • ((⇑derivative)^[n - a] p * X ^ (m - a))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
p : ι → Prop
s : ι → Set α
t : Set α
⊢ t ∈ ⨅ i, ⨅ (_ : p i), 𝓟 (s i) ↔ ∃ I, I.Finite ∧ (∀ i ∈ I, p i) ∧ ⋂ i ∈ I, s i ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
inst✝ : Finite ι
l : ι → Filter α
hd : Pairwise (Disjoint on l)
⊢ Pairwise fun i j ↦ ∃ s t, Disjoint ↑s ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
p0 : p ≠ 0
a : R
n : ℕ
⊢ rootMultiplicity a p ≤ n ↔ ¬(X - C a) ^ (n + 1) ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
inst✝ : Finite ι
l : ι → Filter α
hd : Pairwise (Disjoint on l)
⊢ ∃ s, (∀ (i : ι), s i ∈ l i) ∧ Pairwise (Disjoint on s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
κ : Type u_3
p : κ → Prop
s : (i : ι) → κ → Set (α i)
h : ∀ (i : ι), (f i).HasBasis p (s i)
h_dir : ∀ (I : Set ι) (k : ι → κ), I.Finite → (∀ i ∈ I, p (k i)) → ∃ k₀, p k₀ ∧ ∀ i ∈ I, s i k₀ ⊆ s i (k i)
⊢ (pi f).HasBasis (fun Ik ↦ Ik.1.Finite ∧ p Ik.2) fun Ik ↦ Ik.1.pi fun i ↦ s i Ik.2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
a : R
hzero : p + q ≠ 0
⊢ min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : Type u_3
κ : Type u_4
f : Filter α
p : κ → Prop
s : κ → Set α
h : f.HasBasis p s
⊢ (pi fun x ↦ f).HasBasis (fun Ik ↦ Ik.1.Finite ∧ p Ik.2) fun Ik ↦ Ik.1.pi fun x ↦ s Ik.2

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
s : (i : ι) → Set (α i)
⊢ (pi fun i ↦ 𝓟 (s i)) = 𝓟 (univ.pi s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
l : ι → Filter α
t : Set ι
hd : t.PairwiseDisjoint l
ht : t.Finite
⊢ ∃ s, (∀ (i : ι), s i ∈ l i) ∧ t.PairwiseDisjoint s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
s : (i : ι) → Set (α i)
t : Set ((i : ι) → α i)
⊢ (∃ i, (i.1.Finite ∧ ∀ i_1 ∈ i.1, True) ∧ (i.1.pi fun i ↦ s i) ⊆ t) ↔ ∃ I, I.Finite ∧ I.pi s ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
f : ι → Filter α
⊢ (⨅ i, f i).sets = ⋃ t, (⨅ i ∈ t, f i).sets

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
f : (i : ι) → α i
⊢ (pi fun x ↦ pure (f x)) = pure f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Sort x
f : ι → Filter α
⊢ (⨅ i, f i).sets = ⋃ t, (⨅ i ∈ t, f i.down).sets

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → α i
s : Set ((i : ι) → α i)
⊢ (s ∈ pi fun i ↦ pure (f i)) ↔ ∃ I, I.Finite ∧ ∀ (g : (i : ι) → α i), (∀ i ∈ I, g i = f i) → g ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
x : R
hpq : p * q ≠ 0
⊢ rootMultiplicity x p + rootMultiplicity x q ≤ rootMultiplicity x (p * q)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
hq : q ≠ 0
hpq : p ∣ q
x : R
⊢ rootMultiplicity x p ≤ rootMultiplicity x q

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
β : Type v
s : Finset α
f : α → Filter β
p : Subtype (Membership.mem s) → Set β
hp : ∀ (i : Subtype (Membership.mem s)), p i ∈ f ↑i
h : ⋂ i, p i ∈ ⨅ x, f ↑x
a : α
ha : a ∈ s
⊢ (fun a ↦ if h : a ∈ s then p ⟨a, h⟩ else univ) a ∈ f a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n m : ℕ
p : R[X]
hsum :
  (⇑derivative)^[n] (p * X ^ m) =
    ∑ k ∈ range n.succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))
k : ℕ
hk : k ∈ range n.succ
hk' : k ∉ range (min m n).succ
hkm : k ≤ m
⊢ n < k

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
p0 : p ≠ 0
a : R
⊢ ¬(X - C a) ^ (rootMultiplicity a p + 1) ∣ p

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
β : Type v
s : Finset α
f : α → Filter β
t : Set β
⊢ t ∈ ⨅ a ∈ s, f a ↔ ∃ p, (∀ a ∈ s, p a ∈ f a) ∧ t = ⋂ a ∈ s, p a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
⊢ pi f ⊓ 𝓟 (univ.pi s) = ⊥ ↔ ∃ i, f i ⊓ 𝓟 (s i) = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
⊢ rootMultiplicity 0 p = p.natTrailingDegree

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
⊢ q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n m : ℕ
p : R[X]
⊢ (⇑derivative)^[n] (p * X ^ m) =
    ∑ k ∈ range (min m n).succ, (n.choose k * m.descFactorial k) • ((⇑derivative)^[n - k] p * X ^ (m - k))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type w
s : Finset ι
f : ι → Set α
⊢ ⨅ i ∈ s, 𝓟 (f i) = 𝓟 (⋂ i ∈ s, f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Sort w
inst✝ : Finite ι
f : ι → Set α
⊢ ⨅ i, 𝓟 (f i) = 𝓟 (⋂ i, f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
inst✝ : ∀ (i : ι), (f i).NeBot
I : Set ι
⊢ pi f ⊓ 𝓟 (I.pi s) = ⊥ ↔ ∃ i ∈ I, f i ⊓ 𝓟 (s i) = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type w
s : Set ι
hs : s.Finite
f : ι → Set α
⊢ ⨅ i ∈ s, 𝓟 (f i) = 𝓟 (⋂ i ∈ s, f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
⊢ (pi f ⊓ 𝓟 (univ.pi s)).NeBot ↔ ∀ (i : ι), (f i ⊓ 𝓟 (s i)).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
⊢ (p /ₘ q).leadingCoeff = p.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
f : Filter α
hf : f.sets.Finite
B : Set α
⊢ ∀ U ∈ f.sets, U ∈ f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
f : Filter α
hf : f.sets.Finite
⊢ ∃ s, f = 𝓟 s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
inst✝ : Finite α
f : Filter α
⊢ f.sets ⊆ 𝒫 univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
inst✝ : ∀ (i : ι), (f i).NeBot
I : Set ι
⊢ (pi f ⊓ 𝓟 (I.pi s)).NeBot ↔ ∀ i ∈ I, (f i ⊓ 𝓟 (s i)).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Sort u_2
inst✝ : Finite ι
l : Filter α
p : ι → α → Prop
⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p i x) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p i x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n : ℕ
p : R[X]
⊢ (⇑derivative)^[n] (p * X) = (⇑derivative)^[n] p * X + n • (⇑derivative)^[n - 1] p

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
I : Set ι
hI : I.Finite
l : Filter α
p : ι → α → Prop
⊢ (∀ᶠ (x : α) in l, ∀ i ∈ I, p i x) ↔ ∀ i ∈ I, ∀ᶠ (x : α) in l, p i x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
⊢ pi f = ⊥ ↔ ∃ i, f i = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
⊢ (pi f).NeBot ↔ ∀ (i : ι), (f i).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Sort u_2
inst✝ : Finite ι
l : Filter α
p : ι → α → Prop
⊢ (∃ᶠ (x : α) in l, ∃ i, p i x) ↔ ∃ i, ∃ᶠ (x : α) in l, p i x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p : R[X]
inst✝ : IsDomain R
hi : Irreducible p
x : R
hx : p.IsRoot x
g : R[X]
hg : p = (X - C x) * g
this : IsUnit (X - C x) ∨ IsUnit g
h : IsUnit (X - C x)
h₁ : (X - C x).degree = 1
h₂ : 1 = 0
⊢ ¬1 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
I : Set ι
hI : I.Finite
l : Filter α
p : ι → α → Prop
⊢ (∃ᶠ (x : α) in l, ∃ i ∈ I, p i x) ↔ ∃ i ∈ I, ∃ᶠ (x : α) in l, p i x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
inst✝ : ∀ (i : ι), (f i).NeBot
i : ι
⊢ map (eval i) (pi f) = f i

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p : R[X]
inst✝ : IsDomain R
hi : Irreducible p
x : R
hx : p.IsRoot x
g : R[X]
hg : p = (X - C x) * g
this : IsUnit (X - C x) ∨ IsUnit g
h : IsUnit (X - C x)
⊢ p.degree = 1

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
ι : Type u_2
f : Filter ι
s : ι → Set α
t : Set α
ht : t.Finite
hs : ∀ a ∈ t, ∀ᶠ (i : ι) in f, a ∈ s i
⊢ ∀ᶠ (i : ι) in f, t ⊆ s i

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f₁ f₂ : (i : ι) → Filter (α i)
inst✝ : ∀ (i : ι), (f₁ i).NeBot
⊢ pi f₁ = pi f₂ ↔ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
f : (i : ι) → α i → β i
l : (i : ι) → Filter (β i)
⊢ (pi fun i ↦ comap (f i) (l i)) = comap (Pi.map f) (pi l)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
l : Filter α
ι : Type u_2
s : Set ι
hs : s.Finite
f g : ι → Set α
hle : ∀ i ∈ s, f i ≤ᶠ[l] g i
⊢ ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : Set ((i : ι) → α i)
⊢ s ∈ Filter.coprodᵢ f ↔ ∀ (i : ι), ∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : Set ((i : ι) → α i)
⊢ sᶜ ∈ Filter.coprodᵢ f ↔ ∀ (i : ι), (eval i '' s)ᶜ ∈ f i

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
⊢ (Filter.coprodᵢ f).NeBot ↔ (∀ (i : ι), Nonempty (α i)) ∧ ∃ d, (f d).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: α : Type u
l : Filter α
ι : Type u_2
s : Set ι
hs : s.Finite
f g : ι → Set α
hle : ∀ i ∈ s, f i ≤ᶠ[l] g i
⊢ ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
inst✝ : ∀ (i : ι), Nonempty (α i)
⊢ (Filter.coprodᵢ f).NeBot ↔ ∃ d, (f d).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p : R[X]
inst✝ : IsDomain R
hi : Irreducible p
x : R
hx : p.IsRoot x
g : R[X]
hg : p = (X - C x) * g
this : IsUnit (X - C x) ∨ IsUnit g
hgu : IsUnit g
⊢ p.degree = 1

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
⊢ Filter.coprodᵢ f = ⊥ ↔ (∃ i, IsEmpty (α i)) ∨ f = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
n : ℕ
p : R[X]
⊢ (⇑derivative)^[n] (derivative p * X) = (⇑derivative)^[n + 1] p * X + n • (⇑derivative)^[n] p

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
inst✝ : ∀ (i : ι), Nonempty (α i)
⊢ Filter.coprodᵢ f = ⊥ ↔ f = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set (α × β)
f : Filter α
g : Filter β
⊢ s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
l₁ l₂ : Filter α
⊢ (diagonal α)ᶜ ∈ l₁ ×ˢ l₂ ↔ Disjoint l₁ l₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
t : Set β
f : Filter α
s : Set (α × β)
⊢ s ∈ f ×ˢ 𝓟 t ↔ {a | ∀ b ∈ t, (a, b) ∈ s} ∈ f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
s : Set (α × β)
⊢ s ∈ f ×ˢ ⊤ ↔ {a | ∀ (b : β), (a, b) ∈ s} ∈ f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp : p.degree ≠ 0
a : R
⊢ (p /ₘ (X - C a)).leadingCoeff = p.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
p : α × β → Prop
s : Set β
⊢ (∀ᶠ (x : α × β) in f ×ˢ 𝓟 s, p x) ↔ ∀ᶠ (x : α) in f, ∀ y ∈ s, p (x, y)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
β : ι → Type u_3
m : (i : ι) → α i → β i
⊢ map (fun k i ↦ m i (k i)) (Filter.coprodᵢ f) ≤ Filter.coprodᵢ fun i ↦ map (m i) (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β × γ
b : Filter β
c : Filter γ
⊢ comap f (b ×ˢ c) = comap (Prod.fst ∘ f) b ⊓ comap (Prod.snd ∘ f) c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
f : α → β
g : γ → δ
lb : Filter β
ld : Filter δ
⊢ comap (Prod.map f g) (lb ×ˢ ld) = comap f lb ×ˢ comap g ld

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hpq : p ∣ q
h₁ : q.natDegree ≤ p.natDegree
h₂ : p.leadingCoeff = q.leadingCoeff
⊢ p = q

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
⊢ ∂ ⊤ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a : α
⊢ ∂ (￢￢a) = ∂ (￢a)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a : α
⊢ ￢∂ a = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
⊢ f ×ˢ ⊤ = comap Prod.fst f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
g : Filter β
⊢ ⊤ ×ˢ g = comap Prod.snd g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f₁ f₂ : Filter α
g : Filter β
⊢ (f₁ ⊔ f₂) ×ˢ g = f₁ ×ˢ g ⊔ f₂ ×ˢ g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
g₁ g₂ : Filter β
⊢ f ×ˢ (g₁ ⊔ g₂) = f ×ˢ g₁ ⊔ f ×ˢ g₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hpq : p ∣ q
h₁ : q.natDegree ≤ p.natDegree
h₂ : q.leadingCoeff ∣ p.leadingCoeff
r : R[X]
hr : q = p * r
u : Rˣ
hu : p.leadingCoeff * ↑u = q.leadingCoeff
⊢ p * ↑((Units.map ↑C) u) ∣ q

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a✝ b✝ : α
a b : Prop
⊢ (a ∧ b ∨ ¬(a ∧ b)) ∧ ((a ∨ b) ∨ ¬(a ∨ b)) → a ∨ ¬a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
p : α × β → Prop
⊢ (∀ᶠ (x : α × β) in f ×ˢ g, p x) ↔
    ∃ pa, (∀ᶠ (x : α) in f, pa x) ∧ ∃ pb, (∀ᶠ (y : β) in g, pb y) ∧ ∀ {x : α}, pa x → ∀ {y : β}, pb y → p (x, y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hpq : p ∣ q
h₁ : q.natDegree ≤ p.natDegree
h₂ : q.leadingCoeff ∣ p.leadingCoeff
r : R[X]
hr : q = p * r
u : Rˣ
hu : p.leadingCoeff * ↑u = q.leadingCoeff
⊢ q.natDegree ≤ (p * ↑((Units.map ↑C) u)).natDegree

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
la : Filter α
lb : Filter β
p : α × β → Prop
h : ∀ᶠ (x : α × β) in la ×ˢ lb, p x
⊢ ∀ᶠ (x : α) in la, ∀ᶠ (y : β) in lb, p (x, y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hpq : p ∣ q
h₁ : q.natDegree ≤ p.natDegree
h₂ : q.leadingCoeff ∣ p.leadingCoeff
r : R[X]
hr : q = p * r
u : Rˣ
hu : p.leadingCoeff * ↑u = q.leadingCoeff
⊢ (p * ↑((Units.map ↑C) u)).leadingCoeff = q.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
la : Filter α
lb : Filter β
p : α → β → Prop
h : ∃ᶠ (x : α) in la, ∃ᶠ (y : β) in lb, p x y
⊢ ∃ᶠ (xy : α × β) in la ×ˢ lb, p xy.1 xy.2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: K : Type u_1
inst✝ : Field K
p q : K[X]
hpq : p ∣ q
hq : q ≠ 0
h₁ : q.natDegree ≤ p.natDegree
⊢ IsUnit q.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
hp : p.Monic
hdvd : p ∣ q
hdeg : q.natDegree ≤ p.natDegree
⊢ q = C q.leadingCoeff * p

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
y : α → β
r : α → β → Prop
hy : Tendsto y f g
hr : ∀ᶠ (p : α × β) in f ×ˢ g, r p.1 p.2
⊢ ∀ᶠ (x : α) in f, r x (y x)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ a ⊓ ￢a ≤ a ⊔ (a ⊔ b)

Tactic:
{ exact le_sup_of_le_left inf_le_left }

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ a ⊓ ￢a ≤ a ⊔ ￢(a ⊔ b)

Tactic:
{ exact le_sup_of_le_left inf_le_left }

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : (α × α) × γ → Prop
h : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x
t : α × α → Prop
ht : ∀ᶠ (x : α × α) in f ×ˢ f, t x
s : γ → Prop
hs : ∀ᶠ (y : γ) in g, s y
hst : ∀ {x : α × α}, t x → ∀ {y : γ}, s y → p (x, y)
x : α × γ
hx : t (x.1, x.1) ∧ s x.2
⊢ p ((x.1, x.1), x.2)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : (α × α) × γ → Prop
⊢ (∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x) → ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.1, x.1), x.2)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: case refine_3
α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ a ⊓ ￢a ≤ a ⊔ b

Tactic:
{ exact le_sup_of_le_left inf_le_left }

Goal state after:
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: case refine_4
α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ b

Tactic:
{ exact le_sup_of_le_left inf_le_left }

Goal state after:
case refine_4
α : Type u_1
inst✝ : CoheytingAlgebra α
a b : α
⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ b
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : α × γ × γ → Prop
h : ∀ᶠ (x : α × γ × γ) in f ×ˢ g ×ˢ g, p x
t : α → Prop
ht : ∀ᶠ (x : α) in f, t x
s : γ × γ → Prop
hs : ∀ᶠ (y : γ × γ) in g ×ˢ g, s y
hst : ∀ {x : α}, t x → ∀ {y : γ × γ}, s y → p (x, y)
x : α × γ
hx : t x.1 ∧ s (x.2, x.2)
⊢ p (x.1, x.2, x.2)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : α × γ × γ → Prop
⊢ (∀ᶠ (x : α × γ × γ) in f ×ˢ g ×ˢ g, p x) → ∀ᶠ (x : α × γ) in f ×ˢ g, p (x.1, x.2, x.2)

Tactic:
by

Goal state after: