Theorem: UV.compress_mem_compression
File: Mathlib/Combinatorics/SetFamily/Compression/UV.lean

tactic: rw [mem_compression]
state_before: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
ha : a ∈ s
⊢ compress u v a ∈ 𝓒 u v s
state_after: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
ha : 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: by_cases h : compress u v a ∈ s
state_before: α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
ha : 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
state_after: case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : 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

case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : 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: rw [compress_idem]
state_before: case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : 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
state_after: case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
ha : a ∈ s
h : 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

tactic: exact Or.inl ⟨h, h⟩
state_before: case pos
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
ha : a ∈ s
h : 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
state_after: no goals

tactic: exact Or.inr ⟨h, a, ha, rfl⟩
state_before: case neg
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : 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
state_after: no goals