-- Mathlib/Combinatorics/SetFamily/Compression/UV.lean
theorem le_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : u ≤ a

Proof state:
case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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:
cases ne_of_mem_of_not_mem hb ha hba

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
h : a ∈ 𝓒 u v s
ha : a ∉ s
⊢ u ≤ a

Tactic:
rw [mem_compression] at h

Result:
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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
⊢ u ≤ a

Proof state:
case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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

Tactic:
rw [← hba, le_sdiff]

Result:
case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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 ≤ b ⊔ u ∧ Disjoint u v

Proof state:
case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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 ≤ b ⊔ u ∧ Disjoint u v

Tactic:
exact ⟨le_sup_right, h.1.mono_right h.2⟩

Result:
no goals

Proof state:
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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
⊢ u ≤ a

Tactic:
obtain h | ⟨-, b, hb, hba⟩ := h

Result:
case inl
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
h : a ∈ s ∧ compress u v a ∈ s
⊢ u ≤ a

case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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

Proof state:
case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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:
unfold compress at hba

Result:
case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
b : α
hb : b ∈ s
hba : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \ v else b) = a
⊢ u ≤ a

Proof state:
case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
b : α
hb : b ∈ s
hba : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \ v else b) = a
⊢ u ≤ a

Tactic:
split_ifs at hba with h

Result:
case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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

case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
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

Proof state:
case inl
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
ha : a ∉ s
h : a ∈ s ∧ compress u v a ∈ s
⊢ u ≤ a

Tactic:
cases ha h.1

Result:
no goals