mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.Finset.Sups

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
s t : Finset α
⊢ sᶜˢ ⊆ t ↔ s ⊆ tᶜˢ

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 ∈ 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✝¹ : BooleanAlgebra α
inst✝ : Fintype α
⊢ univᶜˢ = univ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
inst✝ : DecidableEq α
s t : Finset α
⊢ (s ⊼ t)ᶜˢ = sᶜˢ ⊻ tᶜˢ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
inst✝ : DecidableEq α
s t : Finset α
⊢ (s ⊻ t)ᶜˢ = sᶜˢ ⊼ tᶜˢ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
inst✝ : DecidableEq α
s t : Finset α
⊢ s ⊼ tᶜˢ = s \\ t

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
inst✝ : DecidableEq α
s t : Finset α
⊢ sᶜˢ ⊼ t = t \\ s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝¹ : BooleanAlgebra α
inst✝ : DecidableEq α
s t : Finset α
⊢ s \\ tᶜˢ = s ⊼ t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : DecidableEq α
s t : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder 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.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v : α
inst✝ : DecidableEq α
⊢ Set.InjOn (compress u v) ↑({a ∈ s | compress u v a ∉ s})

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
n : ℕ
h𝒜 : Set.Sized n ↑𝒜
s : Finset α
hs : s ∈ 𝒜
⊢ #sᶜ = Fintype.card α - n

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
n : ℕ
hn : n ≤ Fintype.card α
h𝒜 : Set.Sized n ↑𝒜ᶜˢ
⊢ Set.Sized (Fintype.card α - n) ↑𝒜

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
⊢ a ∈ 𝓒 u v s ↔ a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
inst✝ : DecidableEq α
u : α
s : Finset α
⊢ 𝓒 u u s = s

Tactic:
by

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

Goal state before:
pp: α : Type u_4
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
n : ℕ
hn : n ≤ Fintype.card α
h𝒜 : Set.Sized (Fintype.card α - n) ↑𝒜
⊢ Set.Sized n ↑𝒜ᶜˢ

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : SemilatticeSup α
a : α
⊢ ¬SupIrred a ↔ IsMin a ∨ ∃ b c, b ⊔ c = a ∧ b < a ∧ c < a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : SemilatticeSup α
a : α
⊢ ¬SupPrime a ↔ IsMin a ∨ ∃ b c, a ≤ b ⊔ c ∧ ¬a ≤ b ∧ ¬a ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : SemilatticeSup α
a : α
h : ∀ ⦃b c : α⦄, a ≤ b ⊔ c → a ≤ b ∨ a ≤ c
b c : α
ha : b ⊔ c = a
⊢ b = a ∨ c = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeSup α
a : α
inst✝ : OrderBot α
ha : SupIrred a
⊢ a ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeSup α
a : α
inst✝ : OrderBot α
ha : SupPrime a
⊢ a ≠ ⊥

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
k : ℕ
⊢ ∂^[k] ∅ = ∅

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
⊢ ∂ {{a}} = {∅}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∈ s
⊢ compress u v a ∈ 𝓒 u v s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝¹ : SemilatticeSup α
a : α
inst✝ : OrderBot α
s : Finset ι
f : ι → α
ha : SupIrred a
h : s.sup f = a
⊢ ∃ i ∈ s, f i = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂ 𝒜 ↔ ∃ s ∈ 𝒜, ∃ a ∈ s, s.erase a = t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : DecidableEq α
s t : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder 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.Shadow

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂ 𝒜 ↔ ∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝¹ : SemilatticeSup α
a : α
inst✝ : OrderBot α
s : Finset ι
f : ι → α
ha : SupPrime a
⊢ a ≤ s.sup f ↔ ∃ i ∈ s, a ≤ f i

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂ 𝒜 ↔ ∃ a ∉ t, insert a t ∈ 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : WellFoundedLT α
a : α
ih : ∀ y < a, ∃ s, s.sup id = y ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
ha : SupIrred a
⊢ {a}.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ {a} → SupIrred b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v : α
inst✝ : DecidableEq α
b : α
hb : b ∈ s
left✝ : compress u v b ∉ s
⊢ compress u v (compress u v b) ∉ s

Tactic:
rwa [compress_idem]

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v : α
inst✝ : DecidableEq α
b : α
hb : b ∈ s
left✝ : compress u v b ∉ s
⊢ compress u v (compress u v b) ∉ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂ 𝒜 ↔ ∃ s ∈ 𝒜, t ⊆ s ∧ #s = #t + 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∈ 𝓒 u v s
⊢ compress u v a ∈ 𝓒 u v s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : DecidableEq α
s t : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
ha : a ∉ lowerClosure ↑s ⊓ lowerClosure ↑t
⊢ a ∉ lowerClosure ↑(s ⊼ t)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂^[k] 𝒜 ↔ ∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : WellFoundedLT α
a : α
ih : ∀ y < a, ∃ s, s.sup id = y ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
ha : IsMin a
⊢ ∅.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ ∅ → SupIrred b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : DecidableEq α
s t : Finset α
a : α
inst✝¹ : DecidableLE α
inst✝ : BoundedOrder α
ha : a ∉ upperClosure ↑s ⊔ upperClosure ↑t
⊢ a ∉ upperClosure ↑(s ⊻ t)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t u : Finset α
htu : Disjoint t u
hsuA : t ∪ u ∈ 𝒜
⊢ #((t ∪ u) \ t) = #u

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂^[k] 𝒜 ↔ ∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
inst✝ : DecidableEq α
u v : α
s : Finset α
⊢ 𝓒 u v (𝓒 u v s) = 𝓒 u v s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂^[k] 𝒜 ↔ ∃ s ∈ 𝒜, t ⊆ s ∧ #s = #t + k

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
⊢ Set.Sized (r - 1) ↑(∂ 𝒜)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : WellFoundedLT α
a : α
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
inst✝ : DecidableEq α
u v : α
s : Finset α
⊢ #(𝓒 u v s) = #s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
k r : ℕ
h𝒜 : Set.Sized r ↑𝒜
⊢ Set.Sized (r - k) ↑(∂^[k] 𝒜)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : SemilatticeInf α
a : α
h : ∀ ⦃b c : α⦄, b ⊓ c ≤ a → b ≤ a ∨ c ≤ a
b c : α
ha : b ⊓ c = a
⊢ b = a ∨ c = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeInf α
a : α
inst✝ : OrderTop α
ha : InfIrred a
⊢ a ≠ ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
r : ℕ
h : ∅ ∉ 𝒜
⊢ Set.Sized r ↑(∂ 𝒜) ↔ Set.Sized (r + 1) ↑𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeInf α
a : α
inst✝ : OrderTop α
ha : InfPrime a
⊢ a ≠ ⊤

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂⁺ 𝒜 ↔ ∃ s ∈ 𝒜, ∃ a ∉ s, insert a s = t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂⁺ 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #(t \ s) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : DistribLattice α
a : α
h : ∀ ⦃b c : α⦄, b ⊔ c = a → b = a ∨ c = a
b c : α
⊢ a ≤ b ⊔ c → a ≤ b ∨ a ≤ c

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂⁺ 𝒜 ↔ ∃ a ∈ t, t.erase a ∈ 𝒜

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
⊢ t ∈ ∂⁺ 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #t = #s + 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
h : a ∈ 𝓒 u v s
ha : a ∉ s
⊢ u ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : DistribLattice α
a : α
h : ∀ ⦃b c : α⦄, b ⊓ c = a → b = a ∨ c = a
b c : α
⊢ b ⊓ c ≤ a → b ≤ a ∨ c ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : LinearOrder α
a : α
⊢ ∀ ⦃b c : α⦄, a ≤ max b c → a ≤ b ∨ a ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : LinearOrder α
a : α
⊢ ∀ ⦃b c : α⦄, min b c ≤ a → b ≤ a ∨ c ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : LinearOrder α
a x✝¹ x✝ : α
⊢ max x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Irreducible

Goal state before:
pp: α : Type u_2
inst✝ : LinearOrder α
a x✝¹ x✝ : α
⊢ min x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a : α
⊢ InfIrred (Ici a)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
⊢ InfIrred s ↔ ∃ a, Ici a = s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a : α
⊢ SupIrred (Iic a)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
s : LowerSet α
inst✝ : Finite α
⊢ SupIrred s ↔ ∃ a, Iic a = s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
x✝ : α
⊢ ∀ ⦃a₂ : α⦄, (fun a ↦ ⟨LowerSet.Iic a, ⋯⟩) x✝ = (fun a ↦ ⟨LowerSet.Iic a, ⋯⟩) a₂ → x✝ = a₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
⊢ ∀ {a b : α},
    { toFun := fun a ↦ ⟨LowerSet.Iic a, ⋯⟩, inj' := ⋯ } a ≤ { toFun := fun a ↦ ⟨LowerSet.Iic a, ⋯⟩, inj' := ⋯ } b ↔
      a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
x✝ : α
⊢ ∀ ⦃a₂ : α⦄, (fun a ↦ ⟨UpperSet.Ici a, ⋯⟩) x✝ = (fun a ↦ ⟨UpperSet.Ici a, ⋯⟩) a₂ → x✝ = a₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
⊢ ∀ {a b : α},
    { toFun := fun a ↦ ⟨UpperSet.Ici a, ⋯⟩, inj' := ⋯ } a ≤ { toFun := fun a ↦ ⟨UpperSet.Ici a, ⋯⟩, inj' := ⋯ } b ↔
      a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
inst✝ : Finite α
⊢ Surjective ⇑supIrredLowerSet

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
inst✝ : Finite α
⊢ Surjective ⇑infIrredUpperSet

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
h : a ∈ 𝓒 u v s
ha : a ∉ s
⊢ Disjoint v a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeSup α
inst✝² : OrderBot α
inst✝¹ : Finite α
s : { s // SupIrred s }
inst✝ : Fintype ↥↑s
⊢ supIrredLowerSet.symm s = (↑↑s).toFinset.sup id

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t u : Finset α
hut : u ⊆ t
htu : t \ u ∈ 𝒜
⊢ #(t \ (t \ u)) = #u

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : SemilatticeInf α
inst✝² : OrderTop α
inst✝¹ : Finite α
s : { s // InfIrred s }
inst✝ : Fintype ↥↑s
⊢ infIrredUpperSet.symm s = (↑↑s).toFinset.inf id

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : OrderBot α
a : α
⊢ (fun s ↦ (↑s).toFinset.sup Subtype.val) ((fun a ↦ { carrier := {b | ↑b ≤ a}, lower' := ⋯ }) a) = a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t s : Finset α
hs : s ∈ 𝒜
hst : s ⊆ t
⊢ t \ (t \ s) ∈ 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : OrderBot α
s : LowerSet { a // SupIrred a }
⊢ (fun a ↦ { carrier := {b | ↑b ≤ a}, lower' := ⋯ }) ((fun s ↦ (↑s).toFinset.sup Subtype.val) s) = s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : Fintype α
inst✝ : DecidablePred SupIrred
⊢ α ↪o Set { a // SupIrred a }

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : Fintype α
inst✝ : DecidablePred SupIrred
⊢ α ↪o Finset { a // SupIrred a }

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #(t \ s) = k

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ supSum (𝒜 ∪ ℬ) + supSum (𝒜 ⊼ ℬ) = supSum 𝒜 + supSum ℬ

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : Fintype α
inst✝ : DecidablePred SupIrred
a : α
⊢ ↑(birkhoffFinset a) = birkhoffSet a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : Fintype α
inst✝ : DecidablePred SupIrred
a b : α
⊢ birkhoffSet (a ⊔ b) = birkhoffSet a ∪ birkhoffSet b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #t = #s + k

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝² : DistribLattice α
inst✝¹ : Fintype α
inst✝ : DecidablePred SupIrred
a b : α
⊢ birkhoffSet (a ⊓ b) = birkhoffSet a ∩ birkhoffSet b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableLE α
s : Finset α
u v a : α
inst✝ : DecidableEq α
h : a ∈ 𝓒 u v s
ha : a ∉ s
⊢ (a ⊔ v) \ u ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : OrderBot α
a : α
⊢ birkhoffSet a = ↑(OrderIso.lowerSetSupIrred a)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : DecidableEq α
a b : α
⊢ birkhoffFinset (a ⊔ b) = birkhoffFinset a ∪ birkhoffFinset b

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
⊢ Set.Sized (r + 1) ↑(∂⁺ 𝒜)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ infSum (𝒜 ∪ ℬ) + infSum (𝒜 ⊻ ℬ) = infSum 𝒜 + infSum ℬ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
h𝒜₀ : ∅ ∉ 𝒜
s : Finset α
⊢ 0 ≤ ↑(#(𝒜.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : DecidableEq α
a b : α
⊢ birkhoffFinset (a ⊓ b) = birkhoffFinset a ∩ birkhoffFinset b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u
inst✝¹ : Finite α
inst✝ : DistribLattice α
⊢ ∃ β x x_1 f, Injective ⇑f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
h𝒜₀ : ∅ ∉ 𝒜
⊢ ∑ s ∈ 𝒜, (↑((card α).choose #s))⁻¹ ≤ infSum 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ lift a ≤ lift b → a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ comp a ≤ comp b → b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ lift a ≤ comp b → Disjoint a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ lift a < lift b → a < b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ comp a < comp b → b < a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a b : α
⊢ lift a < comp b → Disjoint a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a✝ b✝ a b : α
⊢ lift a < lift b ↔ lift a ≤ lift b ∧ ¬lift b ≤ lift a

Tactic:
by

Goal state after: