mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
this :
  ∀ {x : ℕ},
    ∀ s ∈ powersetCard x univ,
      ↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) / ((↑(card α) - ↑x) * ↑((card α).choose x))
n : ℕ
hn : n ∈ range (card α + 1)
⊢ ↑((card α).choose n) / ((↑(card α) - ↑n) * ↑((card α).choose n)) = (↑(card α) - ↑n)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∈ A
⊢ n ∈ A

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∈ B
⊢ (fun x1 x2 ↦ x1 + x2) 0 n = n

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
⊢ #(image f (sA.disjSum sB)) ≤ #(Ioo 0 n)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
⊢ ∑ s, ↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹ + 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
⊢ #(Ioo 0 n) < n

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : SemilatticeSup α
s t : Finset α
a : α
inst✝ : DecidableEq α
⊢ a ∈ lowerClosure ↑(s ∪ t) ↔ a ∈ lowerClosure ↑s ∨ a ∈ lowerClosure ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeSup α
inst✝¹ : DecidableLE α
inst✝ : OrderTop α
a : α
⊢ a ∉ lowerClosure ↑∅

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
⊢ n ≤ #(sA.disjSum sB)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeSup α
inst✝¹ : DecidableLE α
inst✝ : OrderTop α
b a : α
⊢ {b}.truncatedSup a = if a ≤ b then b else ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeSup α
s : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : OrderTop α
⊢ a ≤ s.truncatedSup a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeSup α
inst✝⁴ : SemilatticeSup β
inst✝³ : BoundedOrder β
inst✝² : DecidableLE α
inst✝¹ : OrderTop α
inst✝ : DecidableLE β
e : α ≃o β
s : Finset α
a : α
⊢ e a ∈ lowerClosure ↑(map e.toEmbedding s) ↔ a ∈ lowerClosure ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeSup α
inst✝⁴ : SemilatticeSup β
inst✝³ : BoundedOrder β
inst✝² : DecidableLE α
inst✝¹ : OrderTop α
inst✝ : DecidableLE β
e : α ≃o β
s : Finset α
a : α
⊢ e (s.truncatedSup a) = (map e.toEmbedding s).truncatedSup (e a)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeSup α
s : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : OrderTop α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) ↑s
ha : a ∈ s
⊢ s.truncatedSup a = a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeSup α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : OrderTop α
inst✝ : DecidableEq α
hs : a ∈ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t
⊢ (s ∪ t).truncatedSup a = s.truncatedSup a ⊔ t.truncatedSup a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
a b : ℕ
left✝ : Sum.inl a ≠ Sum.inr b
hxy : f (Sum.inl a) = f (Sum.inr b)
ha : (0 < a ∧ a ≤ n) ∧ a ∈ A
hb : (0 < b ∧ b ≤ n) ∧ b ∈ B
⊢ a ∈ A

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : SemilatticeSup α
s t : Finset α
c : α
⊢ c ∈ s ⊻ t ↔ ∃ a ∈ s, ∃ b ∈ t, a ⊔ b = c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeSup α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : OrderTop α
inst✝ : DecidableEq α
hs : a ∈ lowerClosure ↑s
ht : a ∉ lowerClosure ↑t
⊢ (s ∪ t).truncatedSup a = s.truncatedSup a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝⁵ : DecidableEq α
inst✝⁴ : DecidableEq β
inst✝³ : SemilatticeSup α
inst✝² : SemilatticeSup β
inst✝¹ : FunLike F α β
inst✝ : SupHomClass F α β
f : F
hf : Injective ⇑f
s t : Finset α
⊢ map { toFun := ⇑f, inj' := hf } (s ⊻ t) = map { toFun := ⇑f, inj' := hf } s ⊻ map { toFun := ⇑f, inj' := hf } t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
a b : ℕ
left✝ : Sum.inl a ≠ Sum.inr b
hxy : f (Sum.inl a) = f (Sum.inr b)
ha : (0 < a ∧ a ≤ n) ∧ a ∈ A
hb : (0 < b ∧ b ≤ n) ∧ b ∈ B
⊢ b ∈ B

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeSup α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : OrderTop α
inst✝ : DecidableEq α
hs : a ∉ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t
⊢ (s ∪ t).truncatedSup a = t.truncatedSup a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : SemilatticeSup α
s : Finset α
⊢ s ⊻ s = s ↔ SupClosed ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : SemilatticeSup α
inst✝ : Fintype α
⊢ univ ⊻ univ = univ

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : SemilatticeSup α
inst✝ : DecidableLE α
s t : Finset α
a : α
⊢ {b ∈ s ⊻ t | b ≤ a} = {b ∈ s | b ≤ a} ⊻ {b ∈ t | b ≤ a}

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : SemilatticeInf α
s t : Finset α
c : α
⊢ c ∈ s ⊼ t ↔ ∃ a ∈ s, ∃ b ∈ t, a ⊓ b = c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
a b : ℕ
left✝ : Sum.inl a ≠ Sum.inr b
hxy : f (Sum.inl a) = f (Sum.inr b)
ha : (0 < a ∧ a ≤ n) ∧ a ∈ A
hb : (0 < b ∧ b ≤ n) ∧ b ∈ B
⊢ (fun x1 x2 ↦ x1 + x2) a b = n

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝⁵ : DecidableEq α
inst✝⁴ : DecidableEq β
inst✝³ : SemilatticeInf α
inst✝² : SemilatticeInf β
inst✝¹ : FunLike F α β
inst✝ : InfHomClass F α β
f : F
hf : Injective ⇑f
s t : Finset α
⊢ map { toFun := ⇑f, inj' := hf } (s ⊼ t) = map { toFun := ⇑f, inj' := hf } s ⊼ map { toFun := ⇑f, inj' := hf } t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : SemilatticeInf α
s : Finset α
⊢ s ⊼ s = s ↔ InfClosed ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : SemilatticeInf α
inst✝ : Fintype α
⊢ univ ⊼ univ = univ

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : SemilatticeInf α
s t : Finset α
a : α
inst✝ : DecidableEq α
⊢ a ∈ upperClosure ↑(s ∪ t) ↔ a ∈ upperClosure ↑s ∨ a ∈ upperClosure ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝² : DecidableEq α
inst✝¹ : SemilatticeInf α
inst✝ : DecidableLE α
s t : Finset α
a : α
⊢ {b ∈ s ⊼ t | a ≤ b} = {b ∈ s | a ≤ b} ⊼ {b ∈ t | a ≤ b}

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
b a : ℕ
left✝ : Sum.inr b ≠ Sum.inl a
hxy : f (Sum.inr b) = f (Sum.inl a)
ha : (0 < b ∧ b ≤ n) ∧ b ∈ B
hb : (0 < a ∧ a ≤ n) ∧ a ∈ A
⊢ a ∈ A

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s t : Finset α
⊢ (s ∪ t).powerset = s.powerset ⊻ t.powerset

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeInf α
s : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
⊢ s.truncatedInf a ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeInf α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
a : α
⊢ a ∉ upperClosure ↑∅

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
b a : ℕ
left✝ : Sum.inr b ≠ Sum.inl a
hxy : f (Sum.inr b) = f (Sum.inl a)
ha : (0 < b ∧ b ≤ n) ∧ b ∈ B
hb : (0 < a ∧ a ≤ n) ∧ a ∈ A
⊢ b ∈ B

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s t : Finset α
⊢ (s ∩ t).powerset = s.powerset ⊼ t.powerset

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Finset α
⊢ s.powerset ⊻ s.powerset = s.powerset

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Finset α
⊢ s.powerset ⊼ s.powerset = s.powerset

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
c : α
⊢ c ∈ s ○ t ↔ ∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeInf α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
b a : α
⊢ {b}.truncatedInf a = if b ≤ a then b else ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : SemilatticeInf β
inst✝³ : BoundedOrder β
inst✝² : DecidableLE β
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
e : α ≃o β
s : Finset α
a : α
⊢ e a ∈ upperClosure ↑(map e.toEmbedding s) ↔ a ∈ upperClosure ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
⊢ s ○ t ⊆ s ⊻ t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
m : ℕ
n : ℕ := m + 1
hnA : n ∉ A
hnB : n ∉ B
f : ℕ ⊕ ℕ → ℕ :=
  fun x ↦
    match x with
    | Sum.inl x => x
    | Sum.inr y => n - y
sA : Finset ℕ := {a ∈ Ioc 0 n | a ∈ A}
sB : Finset ℕ := {b ∈ Ioc 0 n | b ∈ B}
hc : #(image f (sA.disjSum sB)) < #(sA.disjSum sB)
b a : ℕ
left✝ : Sum.inr b ≠ Sum.inl a
hxy : f (Sum.inr b) = f (Sum.inl a)
ha : (0 < b ∧ b ≤ n) ∧ b ∈ B
hb : (0 < a ∧ a ≤ n) ∧ a ∈ A
⊢ (fun x1 x2 ↦ x1 + x2) a b = n

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
p : α → Prop
⊢ (∀ c ∈ s ○ t, p c) ↔ ∀ a ∈ s, ∀ b ∈ t, Disjoint a b → p (a ⊔ b)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
⊢ (s ○ t).Nonempty → s.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
⊢ (s ○ t).Nonempty → t.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
t : Finset α
⊢ ∅ ○ t = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s : Finset α
⊢ s ○ ∅ = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
a b : α
⊢ {a} ○ {b} = if Disjoint a b then {a ⊔ b} else ∅

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s₁ s₂ t : Finset α
⊢ (s₁ ∪ s₂) ○ t = s₁ ○ t ∪ s₂ ○ t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.Schnirelmann

Goal state before:
pp: A B : Set ℕ
inst✝¹ : DecidablePred fun x ↦ x ∈ A
inst✝ : DecidablePred fun x ↦ x ∈ B
hA : 0 ∈ A
hB : 0 ∈ B
h : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
⊢ A + B = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t₁ t₂ : Finset α
⊢ s ○ (t₁ ∪ t₂) = s ○ t₁ ∪ s ○ t₂

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s₁ s₂ t : Finset α
⊢ (s₁ ∩ s₂) ○ t ⊆ s₁ ○ t ∩ s₂ ○ t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t₁ t₂ : Finset α
⊢ s ○ (t₁ ∩ t₂) ⊆ s ○ t₁ ∩ s ○ t₂

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : SemilatticeInf α
inst✝⁴ : SemilatticeInf β
inst✝³ : BoundedOrder β
inst✝² : DecidableLE β
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
e : α ≃o β
s : Finset α
a : α
⊢ e (s.truncatedInf a) = (map e.toEmbedding s).truncatedInf (e a)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t : Finset α
⊢ s ○ t = t ○ s

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
u v a : α
ha : a ∈ {x | Disjoint u x ∧ v ≤ x}
b : α
hb : b ∈ {x | Disjoint u x ∧ v ≤ x}
hab : (fun x ↦ (x ⊔ u) \ v) a = (fun x ↦ (x ⊔ u) \ v) b
⊢ ((a ⊔ u) \ v) \ u ⊔ v = ((b ⊔ u) \ v) \ u ⊔ v

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
u v : α
⊢ Set.InjOn (fun x ↦ (x ⊔ u) \ v) {x | Disjoint u x ∧ v ≤ x}

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
u v a : α
hva : Disjoint v a
hua : u ≤ a
⊢ compress u v ((a ⊔ v) \ u) = a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : SemilatticeInf α
s : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) ↑s
ha : a ∈ s
⊢ s.truncatedInf a = a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
u a : α
⊢ compress u u a = a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∈ upperClosure ↑s
ht : a ∈ upperClosure ↑t
⊢ (s ∪ t).truncatedInf a = s.truncatedInf a ⊓ t.truncatedInf a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
a b : α
⊢ compress (a \ b) (b \ a) b = a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
⊢ ∀ (s t u : Finset α), s ○ t ○ u = s ○ (t ○ u)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∈ upperClosure ↑s
ht : a ∉ upperClosure ↑t
⊢ (s ∪ t).truncatedInf a = s.truncatedInf a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∉ upperClosure ↑s
ht : a ∈ upperClosure ↑t
⊢ (t ∪ s).truncatedInf a = t.truncatedInf a

Tactic:
truncatedInf_union_left ht hs

Goal state after:
α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∉ upperClosure ↑s
ht : a ∈ upperClosure ↑t
⊢ t.truncatedInf a = t.truncatedInf a
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t u : Finset α
⊢ s ○ (t ○ u) = t ○ (s ○ u)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∉ upperClosure ↑s
ht : a ∈ upperClosure ↑t
⊢ (s ∪ t).truncatedInf a = t.truncatedInf a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t u : Finset α
⊢ s ○ t ○ u = s ○ u ○ t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝³ : DecidableEq α
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t u v : Finset α
⊢ s ○ t ○ (u ○ v) = s ○ u ○ (t ○ v)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : GeneralizedBooleanAlgebra α
s t : Finset α
c : α
⊢ c ∈ s \\ t ↔ ∃ a ∈ s, ∃ b ∈ t, a \ b = c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableLE α
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∉ upperClosure ↑s
ht : a ∉ upperClosure ↑t
⊢ a ∉ upperClosure ↑(s ∪ t)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
s : Finset α
a : α
⊢ a ∈ sᶜˢ ↔ aᶜ ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
s : Finset α
inst✝ : DecidableEq α
⊢ image compl s = sᶜˢ

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : DecidableEq α
s t : Finset α
a : α
⊢ a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
s : Finset α
p : α → Prop
⊢ (∃ a ∈ sᶜˢ, p a) ↔ ∃ a ∈ s, p aᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableLE α
u v a : α
⊢ compress u v (compress u v a) = compress u v a

Tactic:
by

Goal state after:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
s : Finset α
⊢ sᶜˢᶜˢ = s

Tactic:
by

Goal state after: