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

Goal state before:
pp: case pos
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
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hab : (a ⊔ u) \ v = compress u v b
⊢ a = b

pp: case neg
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
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : ¬(Disjoint u a ∧ v ≤ a)
ha : a ∈ s ∧ a ∉ s
hab : a = compress u v b
⊢ a = b

Tactic:
·

Goal state after:
case pos
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
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hab : (a ⊔ u) \ v = compress u v b
⊢ a = b
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case pos
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
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : Disjoint u b ∧ v ≤ b
hb : b ∈ s ∧ (b ⊔ u) \ v ∉ s
hab : (a ⊔ u) \ v = (b ⊔ u) \ v
⊢ a = b

pp: case neg
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
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : ¬(Disjoint u b ∧ v ≤ b)
hb : b ∈ s ∧ b ∉ s
hab : (a ⊔ u) \ v = b
⊢ a = b

Tactic:
·

Goal state after:
case pos
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
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : Disjoint u b ∧ v ≤ b
hb : b ∈ s ∧ (b ⊔ u) \ v ∉ s
hab : (a ⊔ u) \ v = (b ⊔ u) \ v
⊢ a = b
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
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
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : ¬(Disjoint u b ∧ v ≤ b)
hb : b ∈ s ∧ b ∉ s
hab : (a ⊔ u) \ v = b
⊢ a = b

Tactic:
·

Goal state after:
case neg
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
a b : α
has : Disjoint u a ∧ v ≤ a
ha : a ∈ s ∧ (a ⊔ u) \ v ∉ s
hbs : ¬(Disjoint u b ∧ v ≤ b)
hb : b ∈ s ∧ b ∉ s
hab : (a ⊔ u) \ v = b
⊢ a = b
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
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
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : ¬(Disjoint u a ∧ v ≤ a)
ha : a ∈ s ∧ a ∉ s
hab : a = compress u v b
⊢ a = b

Tactic:
·

Goal state after:
case neg
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
a b : α
hb : b ∈ s ∧ compress u v b ∉ s
has : ¬(Disjoint u a ∧ v ≤ a)
ha : a ∈ s ∧ a ∉ s
hab : a = compress u v b
⊢ a = b
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 simpa using h𝒜.compls

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

Goal state before:
pp: case pos
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
h : compress u v a ∈ s
⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a

pp: case neg
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
h : compress u v a ∉ s
⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a

Tactic:
·

Goal state after:
case pos
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
h : compress u v a ∈ s
⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
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
h : compress u v a ∉ s
⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a

Tactic:
·

Goal state after:
case neg
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
h : compress u v a ∉ s
⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case inl
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
left✝ : a ∈ s
ha : compress u v a ∈ s
⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a

pp: case inr
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 ∧ compress u v (compress u v b) ∈ s ∨
    compress u v (compress u v b) ∉ s ∧ ∃ b_1 ∈ s, compress u v b_1 = compress u v (compress u v b)

Tactic:
·

Goal state after:
case inl
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
left✝ : a ∈ s
ha : compress u v a ∈ s
⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compress u v a
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case inr
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 ∧ compress u v (compress u v b) ∈ s ∨
    compress u v (compress u v b) ∉ s ∧ ∃ b_1 ∈ s, compress u v b_1 = compress u v (compress u v b)

Tactic:
·

Goal state after:
case inr
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 ∧ compress u v (compress u v b) ∈ s ∨
    compress u v (compress u v b) ∉ s ∧ ∃ b_1 ∈ s, compress u v b_1 = compress u v (compress u v b)
Module: Mathlib.Order.Irreducible

Goal state before:
pp: case pos
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
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b

pp: case neg
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
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b

Tactic:
·

Goal state after:
case pos
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
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
Module: Mathlib.Order.Irreducible

Goal state before:
pp: case neg.inl
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
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b

pp: case neg.inr
α : Type u_2
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : WellFoundedLT α
b c : α
ih : ∀ y < b ⊔ c, ∃ s, s.sup id = y ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
hb : b < b ⊔ c
hc : c < b ⊔ c
⊢ ∃ s, s.sup id = b ⊔ c ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b

Tactic:
·

Goal state after:
case neg.inl
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
⊢ ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case mp
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜) → ∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k

pp: case mpr
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k) → ∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜) → ∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case mpr
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k) → ∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ s ∈ 𝒜, t ⊆ s ∧ #(s \ t) = k) → ∃ u, #u = k ∧ Disjoint t u ∧ t ∪ u ∈ 𝒜
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∈ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ lowerClosure ↑𝒜
hℬ : s ∉ lowerClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedSup s) + #((𝒜 ⊼ ℬ).truncatedSup s) = #(𝒜.truncatedSup s) + #(ℬ.truncatedSup s)
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case inl
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
h : a ∈ s ∧ compress u v a ∈ s
⊢ u ≤ a

pp: case inr
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
b : α
hb : b ∈ s
hba : compress u v b = a
⊢ u ≤ a

Tactic:
·

Goal state after:
case inl
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
h : a ∈ s ∧ compress u v a ∈ s
⊢ u ≤ a
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case pos
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
b : α
hb : b ∈ s
h : Disjoint u b ∧ v ≤ b
hba : (b ⊔ u) \ v = a
⊢ u ≤ a

pp: case neg
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
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ u ≤ a

Tactic:
·

Goal state after:
case pos
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
b : α
hb : b ∈ s
h : Disjoint u b ∧ v ≤ b
hba : (b ⊔ u) \ v = a
⊢ u ≤ a
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
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
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ u ≤ a

Tactic:
·

Goal state after:
case neg
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
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ u ≤ a
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∈ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)
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 simp_rw [← sup_eq_left, sup_inf_left]; exact @h _ _

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

Goal state before:
pp: case inl
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
h : a ∈ s ∧ compress u v a ∈ s
⊢ Disjoint v a

pp: case inr
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
b : α
hb : b ∈ s
hba : compress u v b = a
⊢ Disjoint v a

Tactic:
·

Goal state after:
case inl
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
h : a ∈ s ∧ compress u v a ∈ s
⊢ Disjoint v a
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
k : ℕ
ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜
t : Finset α
⊢ (∃ a ∈ t, ∃ u, #u = k ∧ (u ⊆ t ∧ a ∉ u) ∧ t \ insert a u ∈ 𝒜) →
    ∃ u, (∃ a t, a ∉ t ∧ insert a t = u ∧ #t = k) ∧ u ⊆ t ∧ t \ u ∈ 𝒜

pp: case succ.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
k : ℕ
ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜
t : Finset α
⊢ (∃ u, (∃ a t, a ∉ t ∧ insert a t = u ∧ #t = k) ∧ u ⊆ t ∧ t \ u ∈ 𝒜) →
    ∃ a ∈ t, ∃ u, #u = k ∧ (u ⊆ t ∧ a ∉ u) ∧ t \ insert a u ∈ 𝒜

Tactic:
·

Goal state after:
case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
k : ℕ
ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜
t : Finset α
⊢ (∃ a ∈ t, ∃ u, #u = k ∧ (u ⊆ t ∧ a ∉ u) ∧ t \ insert a u ∈ 𝒜) →
    ∃ u, (∃ a t, a ∉ t ∧ insert a t = u ∧ #t = k) ∧ u ⊆ t ∧ t \ u ∈ 𝒜
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
a : α
hat : a ∈ t
u : Finset α
ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[#u] 𝒜 ↔ ∃ u_1, #u_1 = #u ∧ u_1 ⊆ t ∧ t \ u_1 ∈ 𝒜
htu : t \ insert a u ∈ 𝒜
hut : u ⊆ t
hau : a ∉ u
⊢ ∃ u_1, (∃ a t, a ∉ t ∧ insert a t = u_1 ∧ #t = #u) ∧ u_1 ⊆ t ∧ t \ u_1 ∈ 𝒜

Tactic:
exact ⟨_, ⟨_, _, hau, rfl, rfl⟩, insert_subset hat hut, htu⟩

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

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∈ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 ℬ : Finset (Finset α)
s : Finset α
h𝒜 : s ∉ upperClosure ↑𝒜
hℬ : s ∉ upperClosure ↑ℬ
⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case pos
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
b : α
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
hba : (b ⊔ u) \ v = a
⊢ Disjoint v a

pp: case neg
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
b : α
hb : b ∈ s
h✝ : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ Disjoint v a

Tactic:
·

Goal state after:
case pos
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
b : α
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
hba : (b ⊔ u) \ v = a
⊢ Disjoint v a
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: case refine_1
α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
hs : InfIrred s
⊢ ∃ a, Ici a = s

pp: case refine_2
α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
⊢ (∃ a, Ici a = s) → InfIrred s

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
hs : InfIrred s
⊢ ∃ a, Ici a = s
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: case refine_2
α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
⊢ (∃ a, Ici a = s) → InfIrred s

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝¹ : PartialOrder α
s : UpperSet α
inst✝ : Finite α
⊢ (∃ a, Ici a = s) → InfIrred s