mathlib-initiative/mathlib-tactics huggingface dataset:
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.truncatedInf s ⊓ ℬ.truncatedInf s) + #(𝒜.truncatedInf s ⊔ 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.truncatedInf s ⊓ ℬ.truncatedInf s) + #(𝒜.truncatedInf s ⊔ ℬ.truncatedInf s) =
    #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)
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✝ t : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t✝ = #t + 1 ↔ #t✝ = #t + 1

pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t✝ t : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t ≤ #t✝

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t✝ t : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t ≤ #t✝
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 ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
ha : a ∉ s
⊢ Disjoint v a

Tactic:
]

Goal state after:
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 ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
ha : a ∉ s
⊢ Disjoint v 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.truncatedInf s) + #⊥ = #(𝒜.truncatedInf s) + #⊥

Tactic:
]

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
⊢ #⊥ + #(ℬ.truncatedInf s) = #⊥ + #(ℬ.truncatedInf s)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a : α
s t : UpperSet α
hst : s ⊓ t = Ici a
this : a ∈ s ⊓ t
⊢ s = Ici a ∨ t = Ici a

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
a : α
s t : UpperSet α
hst : s ⊓ t = Ici a
this : a ∈ s ⊓ t
⊢ s = Ici a ∨ t = Ici a
Module: Mathlib.Order.Birkhoff

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
a : α
s t : LowerSet α
hst : s ⊔ t = Iic a
this : a ∈ s ⊔ t
⊢ s = Iic a ∨ t = Iic a

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
a : α
s t : LowerSet α
hst : s ⊔ t = Iic a
this : a ∈ s ⊔ t
⊢ s = Iic a ∨ t = Iic 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
⊢ #⊥ + #⊥ = #⊥ + #⊥

Tactic:
]

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
b : α
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
hba : (b ⊔ u) \ v = a
⊢ Disjoint v ((b ⊔ u) \ v)

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 ((b ⊔ u) \ v)
Module: Mathlib.Combinatorics.SetFamily.Shadow

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

Tactic:
]

Goal state after:
case succ.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
a : α
u : Finset α
hau : a ∉ u
ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[#u] 𝒜 ↔ ∃ u_1, #u_1 = #u ∧ u_1 ⊆ t ∧ t \ u_1 ∈ 𝒜
hut : a ∈ t ∧ u ⊆ t
htu : t \ insert a u ∈ 𝒜
⊢ ∃ a ∈ t, ∃ u_1, #u_1 = #u ∧ (u_1 ⊆ t ∧ a ∉ u_1) ∧ t \ insert a u_1 ∈ 𝒜
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 ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
ha : a ∉ s
⊢ (a ⊔ v) \ u ∈ s

Tactic:
]

Goal state after:
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 ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
ha : a ∉ s
⊢ (a ⊔ v) \ u ∈ s
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
⊢ (∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜) ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #(t \ s) = k

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t : Finset α
k : ℕ
⊢ (∃ u, #u = k ∧ u ⊆ t ∧ t \ u ∈ 𝒜) ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ #(t \ s) = k
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)
⊢ univ.sum ?m.48 =
    ∑ x,
      (↑(#(𝒜.truncatedSup x)) / ((↑(card α) - ↑(#x)) * ↑((card α).choose #x)) +
        ↑(#(ℬ.truncatedSup x)) / ((↑(card α) - ↑(#x)) * ↑((card α).choose #x)))

pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ Finset α → ℚ

pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∀ s ∈ univ,
    ↑(#((𝒜 ∪ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) +
        ↑(#((𝒜 ⊼ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) =
      ?m.48 s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∀ s ∈ univ,
    ↑(#((𝒜 ∪ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) +
        ↑(#((𝒜 ⊼ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) =
      ↑(#(𝒜.truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) +
        ↑(#(ℬ.truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s))
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
⊢ #u = #u

Tactic:
]

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 }
a : { a // SupIrred a }
⊢ a ∈ ↑{ carrier := {b | ↑b ≤ (↑s).toFinset.sup Subtype.val}, lower' := ⋯ } ↔ a ∈ ↑s

Tactic:
refine ⟨fun ha ↦ ?_, fun ha ↦ ?_⟩

Goal state after:
case refine_1
α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : OrderBot α
s : LowerSet { a // SupIrred a }
a : { a // SupIrred a }
ha : a ∈ ↑{ carrier := {b | ↑b ≤ (↑s).toFinset.sup Subtype.val}, lower' := ⋯ }
⊢ a ∈ ↑s

case refine_2
α : Type u_1
inst✝³ : DistribLattice α
inst✝² : Fintype α
inst✝¹ : DecidablePred SupIrred
inst✝ : OrderBot α
s : LowerSet { a // SupIrred a }
a : { a // SupIrred a }
ha : a ∈ ↑s
⊢ a ∈ ↑{ carrier := {b | ↑b ≤ (↑s).toFinset.sup Subtype.val}, lower' := ⋯ }
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
⊢ s ∈ 𝒜

Tactic:
]

Goal state after:
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
⊢ s ∈ 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
⊢ b ∈ s

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
⊢ b ∈ s
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 : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t✝ = #t + k ↔ #t✝ = #t + k

pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t✝ : Finset α
k : ℕ
t : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t ≤ #t✝

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
t✝ : Finset α
k : ℕ
t : Finset α
x✝ : t ∈ 𝒜
hst : t ⊆ t✝
⊢ #t ≤ #t✝
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
this : Subsingleton (OrderBot α)
⊢ birkhoffSet a = ↑(OrderIso.lowerSetSupIrred a)

Tactic:
simp +instances [birkhoffSet, this.allEq]

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)
⊢ univ.sum ?m.48 =
    ∑ x,
      (↑(#(𝒜.truncatedInf x)) / (↑(#x) * ↑((card α).choose #x)) +
        ↑(#(ℬ.truncatedInf x)) / (↑(#x) * ↑((card α).choose #x)))

pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ Finset α → ℚ

pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∀ s ∈ univ,
    ↑(#((𝒜 ∪ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
        ↑(#((𝒜 ⊻ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) =
      ?m.48 s

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∀ s ∈ univ,
    ↑(#((𝒜 ∪ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
        ↑(#((𝒜 ⊻ ℬ).truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) =
      ↑(#(𝒜.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
        ↑(#(ℬ.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s))
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
A : Finset α
hA : A ∈ 𝒜
i : α
hi : i ∉ A
h : insert i A ∈ ↑(∂⁺ 𝒜)
⊢ r + 1 = r + 1

Tactic:
]

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 ∧ (a ⊔ u) \ v ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
hva : v ≤ a
hua : Disjoint u a
⊢ (a ⊔ u) \ v ∈ s

Tactic:
]

Goal state after:
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 ∧ (a ⊔ u) \ v ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
hva : v ≤ a
hua : Disjoint u a
⊢ (a ⊔ u) \ v ∈ s
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case zero
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ 𝒜
hst : t ⊆ s
hcard : #t + 0 = #s
⊢ t ∈ ∂⁺ ^[0] 𝒜

Tactic:
]

Goal state after:
case zero
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ 𝒜
hst : t ⊆ s
hcard : #t + 0 = #s
⊢ t ∈ ∂⁺ ^[0] 𝒜
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
hs : s ∈ 𝒜
⊢ (↑((card α).choose #s))⁻¹ = (↑((card α).choose #s))⁻¹

pp: case ha
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
hs : s ∈ 𝒜
⊢ ↑(#s) ≠ 0

Tactic:
]

Goal state after:
case ha
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
hs : s ∈ 𝒜
⊢ ↑(#s) ≠ 0
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
this : u = ⊥
⊢ u = ⊥

Tactic:
]

Goal state after:
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
this : u = ⊥
⊢ u = ⊥
Module: Mathlib.Order.Booleanisation

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

Tactic:
]

Goal state after:
Module: Mathlib.Order.Booleanisation

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

Tactic:
]

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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
⊢ u \ v ≤ (compress u v b ⊔ u) \ v

Tactic:
]

Goal state after:
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
⊢ u \ v ≤ (compress u v b ⊔ u) \ v
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a✝ b✝ a b c : α
⊢ b \ a ⊔ c \ a ≤ b \ a ⊔ c \ a

Tactic:
]

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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
⊢ Disjoint v v

Tactic:
]

Goal state after:
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
⊢ Disjoint v v
Module: Mathlib.Combinatorics.SetFamily.Shadow

Goal state before:
pp: case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ ∂⁺ 𝒜
hts : t ⊆ s
hcardst : #t + k = #s
u : Finset α
hu : u ∈ 𝒜
hut : u ⊆ t
hcardtu : #t = #u + 1
⊢ #u + (k + 1) = #u + k + 1

Tactic:
]

Goal state after:
case succ.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ ∂⁺ 𝒜
hts : t ⊆ s
hcardst : #t + k = #s
u : Finset α
hu : u ∈ 𝒜
hut : u ⊆ t
hcardtu : #t = #u + 1
⊢ #u + (k + 1) = #u + k + 1
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a✝ b✝ a b c : α
⊢ a \ b ⊔ a \ c ≤ a \ b ⊔ a \ c

Tactic:
]

Goal state after:
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a✝ b✝ a b c : α
⊢ a \ b ⊔ a ⊓ c ≤ a \ b ⊔ a ⊓ c

Tactic:
]

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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
⊢ Disjoint v ((compress u v b ⊔ u) \ v)

Tactic:
]

Goal state after:
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
⊢ Disjoint v ((compress u v b ⊔ u) \ v)
Module: Mathlib.Order.Booleanisation

Goal state before:
pp: α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
a✝ b✝ a b c : α
⊢ a ⊓ b ⊔ a \ c ≤ a ⊓ b ⊔ a \ c

Tactic:
]

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

Goal state before:
pp: case succ.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ 𝒜
hts : t ⊆ s
hcard : #t + (k + 1) = #s
u : Finset α
htu : t ⊆ u
hus : u ⊆ s
hu : #u = #t + 1
⊢ #t + k + 1 = #t + (k + 1)

Tactic:
]

Goal state after:
case succ.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
k : ℕ
𝒜 : Finset (Finset α)
s t : Finset α
ht : t ∈ 𝒜
hts : t ⊆ s
hcard : #t + (k + 1) = #s
u : Finset α
htu : t ⊆ u
hus : u ⊆ s
hu : #u = #t + 1
⊢ #t + k + 1 = #t + (k + 1)
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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
hv : v = ⊥
⊢ b ∈ s

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
hva : v ≤ compress u v b
hua : Disjoint u (compress u v b)
left✝ : compress u v b ∉ s
hu : u = ⊥
hv : v = ⊥
⊢ b ∈ s
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 ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
⊢ a ∈ s

Tactic:
]

Goal state after:
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 ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
⊢ a ∈ 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
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hva : v = ⊥
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
h : (b ⊔ u) \ v = a
⊢ a ∈ s

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
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hva : v = ⊥
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
h : (b ⊔ u) \ v = a
⊢ a ∈ s
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : Nonempty α
hs : s ≠ univ
this :
  ∀ (t : Finset α),
    (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
      if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) else 0
⊢ ∑ j ∈ range (#s + 1), ∑ t ∈ powersetCard j s, (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
    ∑ i ∈ range (#s + 1), ↑((#s).choose i) * (↑(card α) - ↑(#s)) / ((↑(card α) - ↑i) * ↑((card α).choose i))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : Nonempty α
hs : s ≠ univ
this :
  ∀ (t : Finset α),
    (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
      if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) else 0
⊢ ∑ j ∈ range (#s + 1), ∑ t ∈ powersetCard j s, (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
    ∑ i ∈ range (#s + 1), ↑((#s).choose i) * (↑(card α) - ↑(#s)) / ((↑(card α) - ↑i) * ↑((card α).choose i))
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
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hva : v = ⊥
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
h : (b ⊔ u) \ v = a
⊢ b ∈ s

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
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hva : v = ⊥
hb : b ∈ s
h✝ : Disjoint u b ∧ v ≤ b
h : (b ⊔ u) \ v = a
⊢ b ∈ s
Module: Mathlib.Combinatorics.SetFamily.AhlswedeZhang

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : Nonempty α
hs : s ≠ univ
this :
  ∀ (t : Finset α),
    (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
      if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) else 0
n : ℕ
x✝ : n ∈ range (#s + 1)
⊢ ∑ t ∈ powersetCard n s, (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
    (#s).choose n • ((↑(card α) - ↑(#s)) / ((↑(card α) - ↑n) * ↑((card α).choose n)))

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
s : Finset α
inst✝ : Nonempty α
hs : s ≠ univ
this :
  ∀ (t : Finset α),
    (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
      if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) else 0
n : ℕ
x✝ : n ∈ range (#s + 1)
⊢ ∑ t ∈ powersetCard n s, (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =
    (#s).choose n • ((↑(card α) - ↑(#s)) / ((↑(card α) - ↑n) * ↑((card α).choose n)))
Module: Mathlib.Combinatorics.SetFamily.HarrisKleitman

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
a : α
h : IsLowerSet ↑𝒜
s : Finset α
⊢ insert a s ∈ 𝒜 ∧ a ∉ s → s ∈ 𝒜 ∧ a ∉ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
a : α
h : IsLowerSet ↑𝒜
s : Finset α
⊢ insert a s ∈ 𝒜 ∧ a ∉ s → s ∈ 𝒜 ∧ a ∉ 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
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hb : b ∈ s
h✝ : ¬(Disjoint u b ∧ v ≤ b)
h : b = a
⊢ b ∈ s

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
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hb : b ∈ s
h✝ : ¬(Disjoint u b ∧ v ≤ b)
h : b = a
⊢ b ∈ 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
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hb : b ∈ s
h✝ : ¬(Disjoint u b ∧ v ≤ b)
h : b = a
⊢ a ∈ s

Tactic:
rwa [← h]

Goal state after: