url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | have : 𝓓 ⊢! (p :: Γ).conj' ⟶ Δ.disj' := implyLeft_cons_conj'!.mpr hC | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
this : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj'
⊢ False | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
this✝ : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj'
this : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | contradiction | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
this✝ : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj'
this : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | simp | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ ∀ p_1 ∈ p :: Γ, p_1 ∈ (insert p T, U).1 | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ ∀ a ∈ Γ, a = p ∨ a ∈ T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | intro q hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ ∀ a ∈ Γ, a = p ∨ a ∈ T | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
q : Formula α
hq : q ∈ Γ
⊢ q = p ∨ q ∈ T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | right | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
q : Formula α
hq : q ∈ Γ
⊢ q = p ∨ q ∈ T | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
q : Formula α
hq : q ∈ Γ
⊢ q ∈ T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | exact hΓ q hq | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
q : Formula α
hq : q ∈ Γ
⊢ q ∈ T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | intro h Γ Δ hΓ hΔ | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 ∈ (insert p T, U).1
hΔ : ∀ p_1 ∈ Δ, ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | simp_all only [Set.mem_insert_iff] | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 ∈ (insert p T, U).1
hΔ : ∀ p_1 ∈ Δ, ... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | have : 𝓓 ⊬! p ⋏ (Γ.remove p).conj' ⟶ Δ.disj' := h (by
intro q hq;
have := by simpa using hΓ q $ List.mem_of_mem_remove hq;
cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption;
) hΔ | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ ... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | by_contra hC | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | have : 𝓓 ⊢! p ⋏ (Γ.remove p).conj' ⟶ Δ.disj' := imp_trans! andComm! $ implyLeftRemoveConj' (p := p) hC | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | contradiction | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
th... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | intro q hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ ∀ p_1 ∈ L... | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q : Formula... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | have := by simpa using hΓ q $ List.mem_of_mem_remove hq; | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q : Formula... | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q : Formula... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption; | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q : Formula... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | simpa using hΓ q $ List.mem_of_mem_remove hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q : Formula... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | simpa [h] using List.mem_remove_iff.mp hq | case inl
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h✝ : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | assumption | case inr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h✝ : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 = p ∨ p_1 ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
q... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | constructor | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬(𝓓)-Consistent (insert p T, U) ↔ ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬(𝓓)-Consistent (insert p T, U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
�... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | . contrapose; push_neg; apply iff_ParametricConsistent_insert₁.mpr; | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬(𝓓)-Consistent (insert p T, U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
�... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') → ¬(𝓓)-Consistent (insert p T, U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | . contrapose; push_neg; apply iff_ParametricConsistent_insert₁.mp; | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') → ¬(𝓓)-Consistent (insert p T, U) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | contrapose | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬(𝓓)-Consistent (insert p T, U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') → ¬¬(𝓓)-Consistent (insert p T, U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | push_neg | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') → ¬¬(𝓓)-Consistent (insert p T, U) | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | apply iff_ParametricConsistent_insert₁.mpr | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | contrapose | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj') → ¬(𝓓)-Consistent (insert p T, U) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬¬(𝓓)-Consistent (insert p T, U) → ¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | push_neg | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ ¬¬(𝓓)-Consistent (insert p T, U) → ¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₁ | [50, 1] | [53, 69] | apply iff_ParametricConsistent_insert₁.mp | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | constructor | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) ↔
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
case mpr
α : Type u_1
inst✝² : D... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | . intro h Γ Δ hΓ hΔ;
by_contra hC;
have : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).disj' := h hΓ (by simp; intro q hq; right; exact hΔ q hq);
have : 𝓓 ⊢! Γ.conj' ⟶ (p :: Δ).disj' := implyRight_cons_disj'!.mpr hC;
contradiction; | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
case mpr
α : Type u_1
inst✝² : D... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj') →
(𝓓)-Consistent (T, insert p U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | . intro h Γ Δ hΓ hΔ;
simp_all;
have : 𝓓 ⊬! Γ.conj' ⟶ p ⋎ (Δ.remove p).disj' := h hΓ (by
intro q hq;
have := by simpa using hΔ q $ List.mem_of_mem_remove hq;
cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption;
);
by_contra hC;
have : 𝓓 ⊢! Γ.conj' ⟶... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj') →
(𝓓)-Consistent (T, insert p U) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | intro h Γ Δ hΓ hΔ | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | by_contra hC | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | have : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).disj' := h hΓ (by simp; intro q hq; right; exact hΔ q hq) | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
⊢ False | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
this : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).dis... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | have : 𝓓 ⊢! Γ.conj' ⟶ (p :: Δ).disj' := implyRight_cons_disj'!.mpr hC | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
this : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).dis... | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
this✝ : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).di... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | contradiction | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
this✝ : 𝓓 ⊬! Γ.conj' ⟶ (p :: Δ).di... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | simp | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
⊢ ∀ p_1 ∈ p :: Δ, p_1 ∈ (T, insert p U).2 | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
⊢ ∀ a ∈ Δ, a = p ∨ a ∈ U |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | intro q hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
⊢ ∀ a ∈ Δ, a = p ∨ a ∈ U | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
q : Formula α
hq : q ∈ Δ
⊢ q = p ∨ q ∈ U |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | right | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
q : Formula α
hq : q ∈ Δ
⊢ q = p ∨ q ∈ U | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
q : Formula α
hq : q ∈ Δ
⊢ q ∈ U |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | exact hΔ q hq | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : (𝓓)-Consistent (T, insert p U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
q : Formula α
hq : q ∈ Δ
⊢ q ∈ U | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | intro h Γ Δ hΓ hΔ | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj') →
(𝓓)-Consistent (T, insert p U) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 ∈ (T, insert p U).1
hΔ : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | simp_all | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p_1 ∈ Γ, p_1 ∈ (T, insert p U).1
hΔ : ... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | have : 𝓓 ⊬! Γ.conj' ⟶ p ⋎ (Δ.remove p).disj' := h hΓ (by
intro q hq;
have := by simpa using hΔ q $ List.mem_of_mem_remove hq;
cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption;
) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | by_contra hC | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | have : 𝓓 ⊢! Γ.conj' ⟶ p ⋎ (Δ.remove p).disj' := imp_trans! hC $ forthback_disj'_remove | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | contradiction | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | intro q hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | have := by simpa using hΔ q $ List.mem_of_mem_remove hq; | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption; | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | simpa using hΔ q $ List.mem_of_mem_remove hq | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ∨ p_1 ∈ U
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | simpa [h] using List.mem_remove_iff.mp hq | case inl
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h✝ : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₂ | [55, 1] | [73, 19] | assumption | case inr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
h✝ : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! Γ.conj' ⟶ p ⋎ Δ.disj'
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p_1 ∈ Δ, p_1 = p ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | constructor | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬(𝓓)-Consistent (T, insert p U) ↔ ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬(𝓓)-Consistent (T, insert p U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : In... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | . contrapose; push_neg; apply iff_ParametricConsistent_insert₂.mpr; | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬(𝓓)-Consistent (T, insert p U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : In... | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') → ¬(𝓓)-Consistent (T, insert p U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | . contrapose; push_neg; apply iff_ParametricConsistent_insert₂.mp; | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') → ¬(𝓓)-Consistent (T, insert p U) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | contrapose | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬(𝓓)-Consistent (T, insert p U) → ∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') → ¬¬(𝓓)-Consistent (T, insert p U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | push_neg | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') → ¬¬(𝓓)-Consistent (T, insert p U) | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') →
(𝓓)-Consistent (T, insert p U) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | apply iff_ParametricConsistent_insert₂.mpr | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') →
(𝓓)-Consistent (T, insert p U) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | contrapose | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj') → ¬(𝓓)-Consistent (T, insert p U) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬¬(𝓓)-Consistent (T, insert p U) → ¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | push_neg | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ ¬¬(𝓓)-Consistent (T, insert p U) → ¬∃ Γ Δ, (∀ p ∈ Γ, p ∈ T) ∧ (∀ p ∈ Δ, p ∈ U) ∧ 𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) →
∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_not_ParametricConsistent_insert₂ | [76, 1] | [79, 69] | apply iff_ParametricConsistent_insert₂.mp | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
T : Theory α
p : Formula α
U : Theory α
⊢ (𝓓)-Consistent (T, insert p U) →
∀ (Γ Δ : List (Formula α)), (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → ¬𝓓 ⊢! Γ.conj' ⟶ p ⋎ Δ.disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | by_contra hC | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
⊢ (𝓓)-Consistent (insert p t.1, t.2) ∨ (𝓓)-Consistent (t.1, insert p t.2) | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬((𝓓)-Consistent (insert p t.1, t.2) ∨ (𝓓)-Consistent (t.1, insert p t.2))
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | push_neg at hC | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬((𝓓)-Consistent (insert p t.1, t.2) ∨ (𝓓)-Consistent (t.1, insert p t.2))
⊢ False | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | have ⟨hC₁, hC₂⟩ := hC | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
⊢ False | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | obtain ⟨Γ₁, Δ₁, hΓ₁, hΔ₁, h₁⟩ := iff_not_ParametricConsistent_insert₁.mp hC₁ | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... | case intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(�... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | replace h₁ := imply_left_and_comm'! h₁ | case intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(�... | case intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(�... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | obtain ⟨Γ₂, Δ₂, hΓ₂, hΔ₂, h₂⟩ := iff_not_ParametricConsistent_insert₂.mp hC₂ | case intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(�... | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | have : 𝓓 ⊢! (Γ₁ ++ Γ₂).conj' ⟶ (Δ₁ ++ Δ₂).disj' := imp_trans! (conj₁'! iff_concat_conj!) $ imp_trans! (cut! h₁ h₂) (conj₂'! iff_concact_disj!) | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | have : 𝓓 ⊬! (Γ₁ ++ Γ₂).conj' ⟶ (Δ₁ ++ Δ₂).disj' := hCon (by simp; rintro q (hq₁ | hq₂); exact hΓ₁ q hq₁; exact hΓ₂ q hq₂) (by simp; rintro q (hq₁ | hq₂); exact hΔ₁ q hq₁; exact hΔ₂ q hq₂) | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | contradiction | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (inse... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | simp | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | rintro q (hq₁ | hq₂) | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | exact hΓ₁ q hq₁ | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | exact hΓ₂ q hq₂ | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | simp | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | rintro q (hq₁ | hq₂) | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, insert p ... | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | exact hΔ₁ q hq₁ | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.consistent_either | [85, 1] | [97, 17] | exact hΔ₂ q hq₂ | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
p : Formula α
hC : ¬(𝓓)-Consistent (insert p t.1, t.2) ∧ ¬(𝓓)-Consistent (t.1, insert p t.2)
hC₁ : ¬(𝓓)-Consistent (insert p t.1, t.2)
hC₂ : ¬(𝓓)-Consistent (t.1, ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | by_contra h | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
⊢ Disjoint t.1 t.2 | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | obtain ⟨T, hp₁, hp₂, hp⟩ := by simpa [Disjoint] using h; | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
⊢ False | case intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp : ¬T ⊆ ∅
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | obtain ⟨p, hp, _⟩ := Set.not_subset.mp hp | case intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp : ¬T ⊆ ∅
⊢ False | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T
right✝ : p ∉ ∅
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | simp [ParametricConsistent] at hCon | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T
right✝ : p ∉ ∅
⊢ False | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | have : 𝓓 ⊬! [p].conj' ⟶ [p].disj' := hCon
(by simp_all; apply hp₁; assumption)
(by simp_all; apply hp₂; assumption) | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | have : 𝓓 ⊢! [p].conj' ⟶ [p].disj' := by simp; | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | contradiction | case intro.intro.intro.intro.intro
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | simpa [Disjoint] using h | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
h : ¬Disjoint t.1 t.2
⊢ ?m.113495 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | simp_all | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | apply hp₁ | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... | case a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | assumption | case a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | simp_all | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | apply hp₂ | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... | case a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.disjoint_of_consistent | [99, 1] | [108, 17] | simp | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : ∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ t.1) → (∀ p ∈ Δ, p ∈ t.2) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj'
h : ¬Disjoint t.1 t.2
T : Theory α
hp₁ : T ⊆ t.1
hp₂ : T ⊆ t.2
hp✝ : ¬T ⊆ ∅
p : Formula α
hp : p ∈ T... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | by_contra hC | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
⊢ q ∉ t.2 | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | have : 𝓓 ⊢! Γ.conj' ⟶ [q].disj' := by simpa; | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
⊢ False | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | have : 𝓓 ⊬! Γ.conj' ⟶ [q].disj' := hCon (by aesop) (by aesop) | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
⊢ False | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this✝ : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
this : 𝓓 ⊬! Γ.conj' ⟶ [q].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | contradiction | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this✝ : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
this : 𝓓 ⊬! Γ.conj' ⟶ [q].disj'
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | simpa | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
⊢ 𝓓 ⊢! Γ.conj' ⟶ [q].disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | aesop | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
⊢ ∀ p ∈ Γ, p ∈ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂ | [110, 1] | [114, 17] | aesop | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : (𝓓)-Consistent t
q : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t.1
h : 𝓓 ⊢! Γ.conj' ⟶ q
hC : q ∈ t.2
this : 𝓓 ⊢! Γ.conj' ⟶ [q].disj'
⊢ ∀ p ∈ [q], p ∈ t.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₂_of_not_mem₁ | [125, 1] | [129, 25] | intro h | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.1 → p ∈ t.2 | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.1
⊢ p ∈ t.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₂_of_not_mem₁ | [125, 1] | [129, 25] | cases (hMat p) with
| inl h' => exact absurd h' h;
| inr _ => assumption; | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.1
⊢ p ∈ t.2 | no goals |
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