Theorem: LO.Propositional.Classical.mem_maximalConsistentTheory_or (Logic/Propositional/Classical/Basic/Completeness.lean)

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
⊢ p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT
Tactic:
by_contra A
State after:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ False

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ False
Tactic:
have b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q := by
  simpa [not_or, not_mem_maximalConsistentTheory_iff] using A
State after:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ False

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ False
Tactic:
have : Inconsistent (maximalConsistentTheory consisT) :=
  System.inconsistent_of_provable (by prover [b.1, b.2, mem_maximalConsistentTheory_iff.mp h])
State after:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False
Tactic:
have := this.not_con
State after:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False
Tactic:
simp_all
State after:
no goals

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
Tactic:
simpa [not_or, not_mem_maximalConsistentTheory_iff] using A
State after:
no goals

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ maximalConsistentTheory consisT ⊢! ⊥
Tactic:
prover [b.1, b.2, mem_maximalConsistentTheory_iff.mp h]
State after:
no goals