Theorem: LO.Modal.Standard.Theory.self_ParametricConsistent (Logic/Modal/Standard/Kripke/Completeness.lean)

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
⊢ (L)-Consistent Ax(L)
Tactic:
intro Γ hΓ
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
⊢ L ⊬! Γ.conj' ⟶ ⊥

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
⊢ L ⊬! Γ.conj' ⟶ ⊥
Tactic:
obtain ⟨q, hq⟩ := h.exists_unprovable
State after:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
⊢ L ⊬! Γ.conj' ⟶ ⊥

State before:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
⊢ L ⊬! Γ.conj' ⟶ ⊥
Tactic:
by_contra hC
State after:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
hC : L ⊢! Γ.conj' ⟶ ⊥
⊢ False

State before:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
hC : L ⊢! Γ.conj' ⟶ ⊥
⊢ False
Tactic:
have : L ⊢! q := imp_trans! hC efq! ⨀ (iff_provable_list_conj.mpr $ λ _ h => ⟨Deduction.maxm $ hΓ _ h⟩)
State after:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
hC : L ⊢! Γ.conj' ⟶ ⊥
this : L ⊢! q
⊢ False

State before:
case intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
h : Consistent L
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ Ax(L)
q : Formula α
hq : L ⊬! q
hC : L ⊢! Γ.conj' ⟶ ⊥
this : L ⊢! q
⊢ False
Tactic:
contradiction
State after:
no goals