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

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ p ∉ maximalConsistentTheory consisT ↔ maximalConsistentTheory consisT ⊢! ~p
Tactic:
by_cases hp : p ∈ maximalConsistentTheory consisT <;> simp[hp]
State after:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ ¬maximalConsistentTheory consisT ⊢! ~p

case neg
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ maximalConsistentTheory consisT ⊢! ~p

State before:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ ¬maximalConsistentTheory consisT ⊢! ~p
Tactic:
intro bnp
State after:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ False

State before:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ False
Tactic:
have : Inconsistent (maximalConsistentTheory consisT) :=
  System.inconsistent_of_provable (by prover [mem_maximalConsistentTheory_iff.mp hp, bnp])
State after:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False

State before:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False
Tactic:
have := this.not_con
State after:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False

State before:
case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
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 : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ maximalConsistentTheory consisT ⊢! ⊥
Tactic:
prover [mem_maximalConsistentTheory_iff.mp hp, bnp]
State after:
no goals

State before:
case neg
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ maximalConsistentTheory consisT ⊢! ~p
Tactic:
exact mem_maximalConsistentTheory_iff.mp
  (by simpa [hp] using mem_or_neg_mem_maximalConsistentTheory (consisT := consisT) p)
State after:
no goals

State before:
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ ~p ∈ maximalConsistentTheory consisT
Tactic:
simpa [hp] using mem_or_neg_mem_maximalConsistentTheory (consisT := consisT) p
State after:
no goals