Theorem: LO.Modal.Standard.Theory.not_provable_falsum (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
T : Theory α
consisT : (L)-Consistent T
⊢ T *⊬[L]! ⊥
Tactic:
by_contra hC
State after:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
⊢ False

State before:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
⊢ False
Tactic:
obtain ⟨Γ, hΓ₁, hΓ₂⟩ := Context.provable_iff.mp $ hC
State after:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
⊢ False

State before:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
⊢ False
Tactic:
have : L ⊬! List.conj' Γ ⟶ ⊥ := consisT hΓ₁
State after:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
this : L ⊬! Γ.conj' ⟶ ⊥
⊢ False

State before:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
this : L ⊬! Γ.conj' ⟶ ⊥
⊢ False
Tactic:
have : L ⊢! List.conj' Γ ⟶ ⊥ := implyLeft_conj_eq_conj'!.mp hΓ₂
State after:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
this✝ : L ⊬! Γ.conj' ⟶ ⊥
this : L ⊢! Γ.conj' ⟶ ⊥
⊢ False

State before:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
consisT : (L)-Consistent T
hC : { ctx := T } ⊢! ⊥
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ T
hΓ₂ : Γ ⊢[L]! ⊥
this✝ : L ⊬! Γ.conj' ⟶ ⊥
this : L ⊢! Γ.conj' ⟶ ⊥
⊢ False
Tactic:
contradiction
State after:
no goals