Theorem: LO.Modal.Standard.Theory.unprovable_iff_insert_neg_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
T : Theory α
p : Formula α
⊢ T *⊬[L]! p ↔ (L)-Consistent insert (~p) T
Tactic:
constructor
State after:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ T *⊬[L]! p → (L)-Consistent insert (~p) T

case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ (L)-Consistent insert (~p) T → T *⊬[L]! p

State before:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ T *⊬[L]! p → (L)-Consistent insert (~p) T

case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ (L)-Consistent insert (~p) T → T *⊬[L]! p
Tactic:
. contrapose; simp [not_not]; apply provable_iff_insert_neg_not_Lconsistent.mpr;
State after:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ (L)-Consistent insert (~p) T → T *⊬[L]! p

State before:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ (L)-Consistent insert (~p) T → T *⊬[L]! p
Tactic:
. contrapose; simp [not_not]; apply provable_iff_insert_neg_not_Lconsistent.mp;
State after:
no goals

State before:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ T *⊬[L]! p → (L)-Consistent insert (~p) T
Tactic:
contrapose
State after:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬(L)-Consistent insert (~p) T → ¬T *⊬[L]! p

State before:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬(L)-Consistent insert (~p) T → ¬T *⊬[L]! p
Tactic:
simp [not_not]
State after:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬(L)-Consistent insert (p ⟶ ⊥) T → { ctx := T } ⊢! p

State before:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬(L)-Consistent insert (p ⟶ ⊥) T → { ctx := T } ⊢! p
Tactic:
apply provable_iff_insert_neg_not_Lconsistent.mpr
State after:
no goals

State before:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ (L)-Consistent insert (~p) T → T *⊬[L]! p
Tactic:
contrapose
State after:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬T *⊬[L]! p → ¬(L)-Consistent insert (~p) T

State before:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ ¬T *⊬[L]! p → ¬(L)-Consistent insert (~p) T
Tactic:
simp [not_not]
State after:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ { ctx := T } ⊢! p → ¬(L)-Consistent insert (p ⟶ ⊥) T

State before:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
T : Theory α
p : Formula α
⊢ { ctx := T } ⊢! p → ¬(L)-Consistent insert (p ⟶ ⊥) T
Tactic:
apply provable_iff_insert_neg_not_Lconsistent.mp
State after:
no goals