mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Concept

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
f : ι → Concept α β r
⊢ (⨅ i, f i).intent = upperPolar r (⋂ i, (f i).extent)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : CompleteLattice α
hs : CountableSupClosed s
A : Set α
hA_c : A.Countable
hA_ne : A.Nonempty
hA : ∀ a ∈ A, a ∈ s
⊢ sSup A ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : SemilatticeSup α
hs : CountableSupClosed s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ {a, b} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a : α
c : DedekindCut α
⊢ principal a < c ↔ ∃ b ∈ c.left, a < b

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : SemilatticeSup α
hs : CountableSupClosed s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ {a, b}.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : SemilatticeSup α
hs : CountableSupClosed s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
⊢ {a, b}.Countable

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : PartialOrder α
x : α
s : Set α
hs_subset : s ⊆ {x}
hs_ne : s.Nonempty
x✝ : s.Countable
y : α
hy : IsLUB s y
⊢ s = {x}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : PartialOrder α
x : α
s : Set α
hs_subset : s ⊆ {x}
hs_ne : s.Nonempty
x✝ : s.Countable
y : α
hy : IsLUB s y
⊢ y ∈ {x}

Tactic:
have h_eq : s = {x} := by
      ext y
      simp_all only [subset_singleton_iff, mem_singleton_iff]
      refine ⟨hs_subset y, ?_⟩
      rintro rfl
      obtain ⟨z, hzs⟩ := hs_ne
      rwa [hs_subset z hzs] at hzs

Goal state after:
α : Type u_2
inst✝ : PartialOrder α
x : α
s : Set α
hs_subset : s ⊆ {x}
hs_ne : s.Nonempty
x✝ : s.Countable
y : α
hy : IsLUB s y
h_eq : s = {x}
⊢ y ∈ {x}
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : PartialOrder α
x : α
s : Set α
hs_subset : s ⊆ {x}
hs_ne : s.Nonempty
x✝ : s.Countable
y : α
hy : IsLUB s y
⊢ y ∈ {x}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : LE α
x✝⁵ : Set α
x✝⁴ : x✝⁵ ⊆ univ
x✝³ : x✝⁵.Nonempty
x✝² : x✝⁵.Countable
x✝¹ : α
x✝ : IsLUB x✝⁵ x✝¹
⊢ x✝¹ ∈ univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : LE α
x✝⁵ : Set α
x✝⁴ : x✝⁵ ⊆ ∅
x✝³ : x✝⁵.Nonempty
x✝² : x✝⁵.Countable
x✝¹ : α
x✝ : IsLUB x✝⁵ x✝¹
⊢ x✝¹ ∈ ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a : α
c : DedekindCut α
⊢ c < principal a ↔ ∃ b ∈ c.right, b < a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
S : Set (Set α)
inst✝ : LE α
hS : ∀ s ∈ S, CountableSupClosed s
A : Set α
hAS : A ⊆ ⋂₀ S
hA_ne : A.Nonempty
hAc : A.Countable
x : α
hx : IsLUB A x
⊢ x ∈ ⋂₀ S

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
s : Set α
inst✝¹ : Preorder α
inst✝ : Preorder β
t : Set β
hs : CountableSupClosed s
ht : CountableSupClosed t
A : Set (α × β)
hAst : A ⊆ s ×ˢ t
hA_ne : A.Nonempty
hAc : A.Countable
x : α
y : β
hxy : IsLUB (Prod.fst '' A) (x, y).1 ∧ IsLUB (Prod.snd '' A) (x, y).2
⊢ Prod.fst '' A ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
s : Set α
inst✝¹ : Preorder α
inst✝ : Preorder β
t : Set β
hs : CountableSupClosed s
ht : CountableSupClosed t
A : Set (α × β)
hAst : A ⊆ s ×ˢ t
hA_ne : A.Nonempty
hAc : A.Countable
x : α
y : β
hxy : IsLUB (Prod.fst '' A) (x, y).1 ∧ IsLUB (Prod.snd '' A) (x, y).2
⊢ (Prod.fst '' A).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Order.Extension.Linear

Goal state before:
pp: α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
S : Set (α → α → Prop) := {s | IsPartialOrder α s}
⊢ ∀ c ⊆ S, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
s : Set α
inst✝¹ : Preorder α
inst✝ : Preorder β
t : Set β
hs : CountableSupClosed s
ht : CountableSupClosed t
A : Set (α × β)
hAst : A ⊆ s ×ˢ t
hA_ne : A.Nonempty
hAc : A.Countable
x : α
y : β
hxy : IsLUB (Prod.fst '' A) (x, y).1 ∧ IsLUB (Prod.snd '' A) (x, y).2
⊢ Prod.snd '' A ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
s : Set α
inst✝¹ : Preorder α
inst✝ : Preorder β
t : Set β
hs : CountableSupClosed s
ht : CountableSupClosed t
A : Set (α × β)
hAst : A ⊆ s ×ˢ t
hA_ne : A.Nonempty
hAc : A.Countable
x : α
y : β
hxy : IsLUB (Prod.fst '' A) (x, y).1 ∧ IsLUB (Prod.snd '' A) (x, y).2
⊢ (Prod.snd '' A).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
s : Set α
inst✝¹ : Preorder α
inst✝ : Preorder β
t : Set β
hs : CountableSupClosed s
ht : CountableSupClosed t
A : Set (α × β)
hAst : A ⊆ s ×ˢ t
hA_ne : A.Nonempty
hAc : A.Countable
⊢ ∀ (x : α × β), IsLUB A x → x ∈ s ×ˢ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.Extension.Linear

Goal state before:
pp: α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
S : Set (α → α → Prop) := {s | IsPartialOrder α s}
hS : ∀ c ⊆ S, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub
s : α → α → Prop
hrs : r ≤ s
hs : Maximal (fun x ↦ x ∈ S) s
this : IsPartialOrder α s
x y : α
h : ¬s x y ∧ ¬s y x
⊢ False

Tactic:
let s' x' y' := s x' y' ∨ s x' x ∧ s y y'

Goal state after:
α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
S : Set (α → α → Prop) := {s | IsPartialOrder α s}
hS : ∀ c ⊆ S, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub
s : α → α → Prop
hrs : r ≤ s
hs : Maximal (fun x ↦ x ∈ S) s
this : IsPartialOrder α s
x y : α
h : ¬s x y ∧ ¬s y x
s' : α → α → Prop := fun x' y' ↦ s x' y' ∨ s x' x ∧ s y y'
⊢ False
Module: Mathlib.SetTheory.Ordinal.Rank

Goal state before:
pp: α : Type u
a : α
r : α → α → Prop
h : Acc r a
⊢ h.rank = ⨆ b, Order.succ ⋯.rank

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s : Set α
x : α
hxs : x ∈ s
⊢ {x} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s : Set α
x : α
hxs : x ∈ s
⊢ {x}.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s : Set α
x : α
hxs : x ∈ s
⊢ {x}.Countable

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Rank

Goal state before:
pp: α : Type u
a b : α
r : α → α → Prop
hb : Acc r b
h : r a b
⊢ Order.succ ⋯.rank ≤ hb.rank

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s : Set α
x : α
hxs : x ∈ s
⊢ IsLUB {x} x

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s A : Set α
hA_ne : A.Nonempty
hAc : A.Countable
x : α
hx : IsLUB A x
B : ⦃a : α⦄ → a ∈ A → Set α
hB : ∀ ⦃a : α⦄ (a_1 : a ∈ A), B a_1 ⊆ s
hB_ne : ∀ ⦃a : α⦄ (a_1 : a ∈ A), (B a_1).Nonempty
hBc : ∀ ⦃a : α⦄ (a_1 : a ∈ A), (B a_1).Countable
hB_lub : ∀ ⦃a : α⦄ (a_1 : a ∈ A), IsLUB (B a_1) a
⊢ ⋃ a, B ⋯ ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Rank

Goal state before:
pp: α : Type u
a : α
r : α → α → Prop
o : Ordinal.{u}
ha : Acc r a
ho : o ≤ ha.rank
⊢ ∃ b, ∃ (hb : Acc r b), hb.rank = o

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Rank

Goal state before:
pp: α : Type u
a : α
r : α → α → Prop
hwf : IsWellFounded α r
o : Ordinal.{u}
h : o ≤ rank r a
⊢ o ∈ Set.range (rank r)

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Ordinal.Rank

Goal state before:
pp: α : Type u
r : α → α → Prop
inst✝ : IsWellOrder α r
⊢ rank r = ⇑(Ordinal.typein r).toRelEmbedding

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
S : Set (Set α)
s✝ t : Set α
a b : α
inst✝ : Preorder α
s : Set α
⊢ CountableSupClosed {a | ∃ A, ∃ (_ : A ⊆ s) (_ : A.Nonempty) (_ : A.Countable), IsLUB A a}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
⊢ CountableSupClosed (⋂₀ {t | s ⊆ t ∧ CountableSupClosed t})

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
this : CountableSupClosed (⋂₀ {t | s ⊆ t ∧ CountableSupClosed t})
⊢ s ⊆ ⋂₀ {t | s ⊆ t ∧ CountableSupClosed t}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
this : CountableSupClosed (⋂₀ {t | s ⊆ t ∧ CountableSupClosed t})
⊢ CountableSupClosed (⋂₀ {t | s ⊆ t ∧ CountableSupClosed t})

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
⊢ countableSupClosure s = ⋂₀ {t | s ⊆ t ∧ CountableSupClosed t}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a : α
inst✝ : Preorder α
⊢ a ∈ countableSupClosure s ↔ ∃ A, ∃ (_ : A ⊆ s) (_ : A.Nonempty) (_ : A.Countable), IsLUB A a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
⊢ countableSupClosure Set.univ = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
⊢ countableSupClosure ∅ = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
a : α
ha : a ∈ upperBounds s
b : α
t : Set α
hts : t ⊆ s
ht_ne : t.Nonempty
ht_c : t.Countable
ht_lub : IsLUB t b
⊢ b ≤ a

Tactic:
have hat : a ∈ upperBounds t := fun x hx ↦ ha (hts hx)

Goal state after:
α : Type u_2
inst✝ : Preorder α
s : Set α
a : α
ha : a ∈ upperBounds s
b : α
t : Set α
hts : t ⊆ s
ht_ne : t.Nonempty
ht_c : t.Countable
ht_lub : IsLUB t b
hat : a ∈ upperBounds t
⊢ b ≤ a
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : Preorder α
s : Set α
⊢ upperBounds s ⊆ upperBounds (countableSupClosure s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a : α
inst✝ : Preorder α
⊢ IsLUB (countableSupClosure s) a ↔ IsLUB s a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
t : Set β
a : α
b : β
u : Set α
hu : u ⊆ s
hu_ne : u.Nonempty
hu_c : u.Countable
hu_lub : IsLUB u a
v : Set β
hv : v ⊆ t
hv_ne : v.Nonempty
hv_c : v.Countable
hv_lub : IsLUB v b
⊢ u ×ˢ v ⊆ s ×ˢ t

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
t : Set β
a : α
b : β
u : Set α
hu : u ⊆ s
hu_ne : u.Nonempty
hu_c : u.Countable
hu_lub : IsLUB u a
v : Set β
hv : v ⊆ t
hv_ne : v.Nonempty
hv_c : v.Countable
hv_lub : IsLUB v b
⊢ (u ×ˢ v).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
t : Set β
⊢ countableSupClosure s ×ˢ countableSupClosure t ⊆ countableSupClosure (s ×ˢ t)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a : α
inst✝ : CompleteLattice α
this : countableSupClosure s = {a | ∃ t, (∀ (n : ℕ), t n ∈ s) ∧ ⨆ n, t n = a}
⊢ a ∈ countableSupClosure s ↔ ∃ t, (∀ (n : ℕ), t n ∈ s) ∧ ⨆ n, t n = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a : α
inst✝ : CompleteLattice α
⊢ CountableSupClosed {a | ∃ t, (∀ (n : ℕ), t n ∈ s) ∧ ⨆ n, t n = a}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a✝ : α
inst✝ : CompleteLattice α
h_csc : CountableSupClosed {a | ∃ t, (∀ (n : ℕ), t n ∈ s) ∧ ⨆ n, t n = a}
a : α
ha : a ∈ s
⊢ (∀ (n : ℕ), (fun x ↦ a) n ∈ s) ∧ ⨆ n, (fun x ↦ a) n = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Extension.Linear

Goal state before:
pp: α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
⊢ ∃ s, IsLinearOrder α s ∧ r ≤ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
a : α
inst✝ : CompleteLattice α
⊢ a ∈ countableSupClosure s ↔ ∃ t, (∀ (n : ℕ), t n ∈ s) ∧ ⨆ n, t n = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
inst✝ : PartialOrder α
x : α
⊢ countableSupClosure {x} = {x}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: case refine_1
α : Type u_2
s : Set α
inst✝ : Order.Coframe α
hs : SupClosed s
t : ℕ → α
ht : ∀ (n : ℕ), t n ∈ s
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ s
n : ℕ
⊢ (fun n ↦ t (Nat.unpair n).1 ⊔ u (Nat.unpair n).2) n ∈ s

Tactic:
simp only

Goal state after:
case refine_1
α : Type u_2
s : Set α
inst✝ : Order.Coframe α
hs : SupClosed s
t : ℕ → α
ht : ∀ (n : ℕ), t n ∈ s
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ s
n : ℕ
⊢ t (Nat.unpair n).1 ⊔ u (Nat.unpair n).2 ∈ s
Module: Mathlib.Order.CountableSupClosed

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : Order.Coframe α
hs : SupClosed s
⊢ SupClosed (countableInfClosure s)

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
⊢ ∀ (p : Nat.Primes), B ↑p =ᵐ[μ] ∅ ∨ B ↑p =ᵐ[μ] univ

Tactic:
by

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

Goal state before:
pp: α : Type u
l : Filter α
⊢ ∀ (S : Set (Set α)), #↑S < ℵ₀ → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ l : Filter α
inst✝ : CountableInterFilter l
S : Set (Set α)
hS : #↑S < ℵ_ 1
a : ∀ s ∈ S, s ∈ l
⊢ ⋂₀ S ∈ l

Tactic:
by

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

Goal state before:
pp: α : Type u
l : Filter α
x✝ : CardinalInterFilter l (ℵ_ 1)
⊢ CountableInterFilter l

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
s : ι → Set α
hic : #ι < c
⊢ ⋂ i, s i ∈ l ↔ ∀ (i : ι), s i ∈ l

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s : (i : ι) → i ∈ S → Set α
⊢ ⋂ i, ⋂ (hi : i ∈ S), s i hi ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
p : α → ι → Prop
hic : #ι < c
⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p x i) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p x i

Tactic:
by

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

Goal state before:
pp: α β : Type u
f : β → Filter α
fs : List β
t : Set (List α)
⊢ t ∈ traverse f fs ↔ ∃ us, Forall₂ (fun b s ↦ s ∈ f b) fs us ∧ sequence us ⊆ t

Tactic:
constructor
  · induction fs generalizing t with
    | nil =>
      simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,
        singleton_subset_iff, traverse_nil]
    | cons b fs ih =>
      intro ht
      rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩
      rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hwu⟩
      rcases ih v hv with ⟨us, hus, hu⟩
      exact ⟨w::us, Forall₂.cons hw hus, (Set.seq_mono hwu hu).trans ht⟩
  · rintro ⟨us, hus, hs⟩
    exact mem_of_superset (mem_traverse _ _ hus) hs

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

Goal state before:
pp: α β : Type u
f : β → Filter α
fs : List β
t : Set (List α)
⊢ t ∈ traverse f fs ↔ ∃ us, Forall₂ (fun b s ↦ s ∈ f b) fs us ∧ sequence us ⊆ t

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
p : α → (i : ι) → i ∈ S → Prop
⊢ (∀ᶠ (x : α) in l, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᶠ (x : α) in l, p x i hi

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s t : (i : ι) → i ∈ S → Set α
h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi
⊢ ⋃ i, ⋃ (h : i ∈ S), s i h ≤ᶠ[l] ⋃ i, ⋃ (h : i ∈ S), t i h

Tactic:
by

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

Goal state before:
pp: ι α : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
s t : (i : ι) → i ∈ S → Set α
h : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi
⊢ ⋂ i, ⋂ (h : i ∈ S), s i h ≤ᶠ[l] ⋂ i, ⋂ (h : i ∈ S), t i h

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.WellApproximable

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n ↦ approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p ↦ blimsup X atTop fun n ↦ 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∧ B ↑p =ᵐ[μ] ∅ → E =ᵐ[μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᵐ[μ] ∅ ∨ A ↑p =ᵐ[μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᵐ[μ] ∅ ∨ B ↑p =ᵐ[μ] univ
⊢ ∀ (p : Nat.Primes), u p +ᵥ C ↑p = C ↑p

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝ : CardinalInterFilter l✝ c
l : Set (Set α)
hc : 2 < c
hl : ∀ (S : Set (Set α)), #↑S < c → S ⊆ l → ⋂₀ S ∈ l
h_mono : ∀ (s t : Set α), s ∈ l → s ⊆ t → t ∈ l
s t : Set α
hs : s ∈ l
ht : t ∈ l
⊢ #↑{t} + 1 = 2

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝ : CardinalInterFilter l✝ c
l : Set (Set α)
hc : 2 < c
hl : ∀ (S : Set (Set α)), #↑S < c → S ⊆ l → ⋂₀ S ∈ l
h_mono : ∀ (s t : Set α), s ∈ l → s ⊆ t → t ∈ l
s t : Set α
hs : s ∈ l
ht : t ∈ l
⊢ #↑{s, t} ≤ 2

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝ : CardinalInterFilter l✝ c
l : Set (Set α)
hc : 2 < c
hl : ∀ (S : Set (Set α)), #↑S < c → S ⊆ l → ⋂₀ S ∈ l
h_mono : ∀ (s t : Set α), s ∈ l → s ⊆ t → t ∈ l
s t : Set α
hs : s ∈ l
ht : t ∈ l
⊢ ⋂₀ {s, t} ∈ l

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝ : CardinalInterFilter l✝ c
l : Set (Set α)
hc : 2 < c
hUnion : ∀ (S : Set (Set α)), #↑S < c → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l
hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l
⊢ Filter α

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
⊢ CardinalInterFilter ⊥ c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
⊢ CardinalInterFilter ⊤ c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝¹ : CardinalInterFilter l✝ c
l : Filter β
inst✝ : CardinalInterFilter l c
f : α → β
⊢ CardinalInterFilter (comap f l) c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l✝ : Filter α
inst✝¹ : CardinalInterFilter l✝ c
l : Filter α
inst✝ : CardinalInterFilter l c
f : α → β
⊢ CardinalInterFilter (map f l) c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝² : CardinalInterFilter l c
l₁ l₂ : Filter α
inst✝¹ : CardinalInterFilter l₁ c
inst✝ : CardinalInterFilter l₂ c
⊢ CardinalInterFilter (l₁ ⊓ l₂) c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝² : CardinalInterFilter l c
l₁ l₂ : Filter α
c₁ c₂ : Cardinal.{u}
inst✝¹ : CardinalInterFilter l₁ c₁
inst✝ : CardinalInterFilter l₂ c₂
⊢ CardinalInterFilter (l₁ ⊓ l₂) (min c₁ c₂)

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝² : CardinalInterFilter l c
l₁ l₂ : Filter α
inst✝¹ : CardinalInterFilter l₁ c
inst✝ : CardinalInterFilter l₂ c
⊢ CardinalInterFilter (l₁ ⊔ l₂) c

Tactic:
by

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

Goal state before:
pp: ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝² : CardinalInterFilter l c
l₁ l₂ : Filter α
c₁ c₂ : Cardinal.{u}
inst✝¹ : CardinalInterFilter l₁ c₁
inst✝ : CardinalInterFilter l₂ c₂
⊢ CardinalInterFilter (l₁ ⊔ l₂) (min c₁ c₂)

Tactic:
by

Goal state after: