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.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | by_contra hC | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
⊢ p ∈ t.tableau.1 ∨ q ∈ t.tableau.1 | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : ¬(p ∈ t.tableau.1 ∨ q ∈ t.tableau.1)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | simp [not_or] at hC | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : ¬(p ∈ t.tableau.1 ∨ q ∈ t.tableau.1)
⊢ False | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | have : p ∈ t.tableau.2 := t.not_mem₁_iff_mem₂.mp hC.1 | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
⊢ False | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this : p ∈ t.tableau.2
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | have : q ∈ t.tableau.2 := t.not_mem₁_iff_mem₂.mp hC.2 | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this : p ∈ t.tableau.2
⊢ False | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝ : p ∈ t.tableau.2
this : q ∈ t.tableau.2
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | have : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].disj' := by simp; | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝ : p ∈ t.tableau.2
this : q ∈ t.tableau.2
⊢ False | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝¹ : p ∈ t.tableau.2
this✝ : q ∈ t.tableau.2
this : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].di... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | have : 𝓓 ⊬! [p ⋎ q].conj' ⟶ [p, q].disj' := t.consistent (by simp_all) (by simp_all) | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝¹ : p ∈ t.tableau.2
this✝ : q ∈ t.tableau.2
this : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].di... | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝² : p ∈ t.tableau.2
this✝¹ : q ∈ t.tableau.2
this✝ : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | contradiction | case mp
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝² : p ∈ t.tableau.2
this✝¹ : q ∈ t.tableau.2
this✝ : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | simp | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝ : p ∈ t.tableau.2
this : q ∈ t.tableau.2
⊢ 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | simp_all | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝¹ : p ∈ t.tableau.2
this✝ : q ∈ t.tableau.2
this : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].disj'
⊢ ∀ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | simp_all | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ⋎ q ∈ t.tableau.1
hC : p ∉ t.tableau.1 ∧ q ∉ t.tableau.1
this✝¹ : p ∈ t.tableau.2
this✝ : q ∈ t.tableau.2
this : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].disj'
⊢ ∀ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | intro h | case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
⊢ p ∈ t.tableau.1 ∨ q ∈ t.tableau.1 → p ⋎ q ∈ t.tableau.1 | case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ∈ t.tableau.1 ∨ q ∈ t.tableau.1
⊢ p ⋎ q ∈ t.tableau.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | cases h with
| inl h => exact mdp₁ h disj₁!
| inr h => exact mdp₁ h disj₂! | case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ∈ t.tableau.1 ∨ q ∈ t.tableau.1
⊢ p ⋎ q ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | exact mdp₁ h disj₁! | case mpr.inl
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : p ∈ t.tableau.1
⊢ p ⋎ q ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_or | [375, 1] | [388, 35] | exact mdp₁ h disj₂! | case mpr.inr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p q : Formula α
h : q ∈ t.tableau.1
⊢ p ⋎ q ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_of_provable | [390, 1] | [392, 35] | intro h | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p : Formula α
⊢ 𝓓 ⊢! p → p ∈ t.tableau.1 | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
⊢ p ∈ t.tableau.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_of_provable | [390, 1] | [392, 35] | exact mdp₁ mem₁_verum $ dhyp! h | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t✝ t t₁ t₂ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
⊢ p ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.CanonicalModel.frame_def | [422, 1] | [422, 111] | rfl | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t₁ t₂ : SCT 𝓓
⊢ (CanonicalModel 𝓓).Rel t₁ t₂ ↔ t₁.tableau.1 ⊆ t₂.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.CanonicalModel.valuation_def | [423, 1] | [423, 118] | rfl | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
a : α
⊢ (CanonicalModel 𝓓).Valuation t a ↔ atom a ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | induction p using rec' generalizing t with
| himp p q ihp ihq =>
constructor;
. contrapose;
simp_all;
intro h;
replace h := t.not_mem₁_iff_mem₂.mp h;
have : (𝓓)-Consistent (insert p t.tableau.1, {q}) := by
simp only [Tableau.ParametricConsistent];
intro Γ Δ hΓ hΔ;
replace hΓ : ∀ r... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
t : (CanonicalModel 𝓓).World
⊢ t ⊧ p ↔ p ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | constructor | case himp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.table... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | . contrapose;
simp_all;
intro h;
replace h := t.not_mem₁_iff_mem₂.mp h;
have : (𝓓)-Consistent (insert p t.tableau.1, {q}) := by
simp only [Tableau.ParametricConsistent];
intro Γ Δ hΓ hΔ;
replace hΓ : ∀ r, r ∈ Γ.remove p → r ∈ t.tableau.1 := by
intro r hr;
have ⟨hr₁, hr₂⟩ := List.mem_rem... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.table... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | . simp [Satisfies.imp_def];
intro h t' htt' hp;
replace hp := ihp.mp hp;
have hpq := htt' h;
apply ihq.mpr;
apply t'.not_mem₂_iff_mem₁.mp;
exact not_mem₂
(by simp_all)
(show 𝓓 ⊢! [p, p ⟶ q].conj' ⟶ q by
simp;
apply andImplyIffImplyImply'!.mpr;
apply deduct'!;
apply deduct!;
... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | contrapose | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.table... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.table... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp_all | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.table... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | intro h | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | replace h := t.not_mem₁_iff_mem₂.mp h | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have : (𝓓)-Consistent (insert p t.tableau.1, {q}) := by
simp only [Tableau.ParametricConsistent];
intro Γ Δ hΓ hΔ;
replace hΓ : ∀ r, r ∈ Γ.remove p → r ∈ t.tableau.1 := by
intro r hr;
have ⟨hr₁, hr₂⟩ := List.mem_remove_iff.mp hr;
have := by simpa using hΓ r hr₁;
simp_all;
by_contra hC;
have :... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | obtain ⟨t', ⟨⟨_, _⟩, _⟩⟩ := by simpa [Set.insert_subset_iff] using lindenbaum this; | case himp.mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ih... | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | existsi t' | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp_all | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply t'.not_mem₁_iff_mem₂.mpr | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | assumption | case himp.mp.intro.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp only [Tableau.ParametricConsistent] | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | intro Γ Δ hΓ hΔ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | replace hΓ : ∀ r, r ∈ Γ.remove p → r ∈ t.tableau.1 := by
intro r hr;
have ⟨hr₁, hr₂⟩ := List.mem_remove_iff.mp hr;
have := by simpa using hΓ r hr₁;
simp_all; | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | by_contra hC | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have : 𝓓 ⊢! (Γ.remove p).conj' ⟶ (p ⟶ q) := imp_trans! (andImplyIffImplyImply'!.mp $ implyLeftRemoveConj' hC) (by
apply deduct'!;
apply deduct!;
have : [p, p ⟶ Δ.disj'] ⊢[𝓓]! p := by_axm! (by simp);
have : [p, p ⟶ Δ.disj'] ⊢[𝓓]! Δ.disj' := (by_axm! (by simp)) ⨀ this;
exact disj_allsame'! (by simpa using hΔ... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have : 𝓓 ⊬! (Γ.remove p).conj' ⟶ (p ⟶ q) := by simpa only [List.disj'_singleton] using (t.consistent hΓ (show ∀ r ∈ [p ⟶ q], r ∈ t.tableau.2 by simp_all)); | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | contradiction | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | intro r hr | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have ⟨hr₁, hr₂⟩ := List.mem_remove_iff.mp hr | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have := by simpa using hΓ r hr₁; | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp_all | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simpa using hΓ r hr₁ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply deduct'! | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply deduct! | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ ... | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have : [p, p ⟶ Δ.disj'] ⊢[𝓓]! p := by_axm! (by simp) | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have : [p, p ⟶ Δ.disj'] ⊢[𝓓]! Δ.disj' := (by_axm! (by simp)) ⨀ this | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | exact disj_allsame'! (by simpa using hΔ) this | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simpa using hΔ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simpa only [List.disj'_singleton] using (t.consistent hΓ (show ∀ r ∈ [p ⟶ q], r ∈ t.tableau.2 by simp_all)) | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp_all | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simpa [Set.insert_subset_iff] using lindenbaum this | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
t : (CanonicalModel 𝓓).World
ihp : ∀ {t : (CanonicalModel 𝓓).World}, Satisfies (CanonicalModel 𝓓) t p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (C... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp [Satisfies.imp_def] | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | intro h t' htt' hp | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | replace hp := ihp.mp hp | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | have hpq := htt' h | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply ihq.mpr | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply t'.not_mem₂_iff_mem₁.mp | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | exact not_mem₂
(by simp_all)
(show 𝓓 ⊢! [p, p ⟶ q].conj' ⟶ q by
simp;
apply andImplyIffImplyImply'!.mpr;
apply deduct'!;
apply deduct!;
exact by_axm! ⨀ (by_axm! (p := p));
) | case himp.mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tabl... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp_all | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply andImplyIffImplyImply'!.mpr | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply deduct'! | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t : (Can... | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | apply deduct! | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1
t... | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | exact by_axm! ⨀ (by_axm! (p := p)) | case h.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p q : Formula α
ihp : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p ↔ p ∈ t.tableau.1
ihq : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q ↔ q ∈ t.tableau.1... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.truthlemma | [433, 1] | [479, 52] | simp [Satisfies.iff_models, Satisfies, *] | case hor
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p✝ q✝ : Formula α
a✝¹ : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ p✝ ↔ p✝ ∈ t.tableau.1
a✝ : ∀ {t : (CanonicalModel 𝓓).World}, t ⊧ q✝ ↔ q✝ ∈ t.tabl... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | constructor | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ CanonicalModel 𝓓 ⊧ p ↔ 𝓓 ⊢! p | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ CanonicalModel 𝓓 ⊧ p → 𝓓 ⊢! p
case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝²... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | . contrapose;
intro h;
have : (𝓓)-Consistent (∅, {p}) := by
simp only [Tableau.ParametricConsistent, Collection.not_mem_empty, imp_false, Set.mem_singleton_iff];
rintro Γ Δ hΓ hΔ;
by_contra hC;
replace hΓ : Γ = [] := List.nil_iff.mpr hΓ;
subst hΓ;
have : 𝓓 ⊢! p := disj_allsame'! hΔ (hC ⨀ v... | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ CanonicalModel 𝓓 ⊧ p → 𝓓 ⊢! p
case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝²... | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ 𝓓 ⊢! p → CanonicalModel 𝓓 ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | . intro h t;
suffices p ∈ t.tableau.1 by exact truthlemma.mpr this;
exact mem₁_of_provable h; | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ 𝓓 ⊢! p → CanonicalModel 𝓓 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | contrapose | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ CanonicalModel 𝓓 ⊧ p → 𝓓 ⊢! p | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ ¬𝓓 ⊢! p → ¬CanonicalModel 𝓓 ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | intro h | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ ¬𝓓 ⊢! p → ¬CanonicalModel 𝓓 ⊧ p | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
⊢ ¬CanonicalModel 𝓓 ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | have : (𝓓)-Consistent (∅, {p}) := by
simp only [Tableau.ParametricConsistent, Collection.not_mem_empty, imp_false, Set.mem_singleton_iff];
rintro Γ Δ hΓ hΔ;
by_contra hC;
replace hΓ : Γ = [] := List.nil_iff.mpr hΓ;
subst hΓ;
have : 𝓓 ⊢! p := disj_allsame'! hΔ (hC ⨀ verum!);
contradiction; | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
⊢ ¬CanonicalModel 𝓓 ⊧ p | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
⊢ ¬CanonicalModel 𝓓 ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | obtain ⟨t', ht'⟩ := lindenbaum this | case mp
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
⊢ ¬CanonicalModel 𝓓 ⊧ p | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ¬CanonicalModel �... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | simp [ValidOnModel.iff_models, ValidOnModel] | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ¬CanonicalModel �... | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ∃ x, ¬Satisfies (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | existsi t' | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ∃ x, ¬Satisfies (... | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ¬Satisfies (Canon... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | apply truthlemma.not.mpr | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ ¬Satisfies (Canon... | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ p ∉ t'.tableau.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | apply t'.not_mem₁_iff_mem₂.mpr | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ p ∉ t'.tableau.1 | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ p ∈ t'.tableau.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | simp_all | case mp.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
this : (𝓓)-Consistent (∅, {p})
t' : SaturatedConsistentTableau 𝓓
ht' : (∅, {p}) ⊆ t'.tableau
⊢ p ∈ t'.tableau.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | simp only [Tableau.ParametricConsistent, Collection.not_mem_empty, imp_false, Set.mem_singleton_iff] | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
⊢ (𝓓)-Consistent (∅, {p}) | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
⊢ ∀ {Γ Δ : List (Formula α)}, (∀ (p : Formula α), p ∉ Γ) → (∀ p_1 ∈ Δ, p_1 = p) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | rintro Γ Δ hΓ hΔ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
⊢ ∀ {Γ Δ : List (Formula α)}, (∀ (p : Formula α), p ∉ Γ) → (∀ p_1 ∈ Δ, p_1 = p) → 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΓ : ∀ (p : Formula α), p ∉ Γ
hΔ : ∀ p_1 ∈ Δ, p_1 = p
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | by_contra hC | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΓ : ∀ (p : Formula α), p ∉ Γ
hΔ : ∀ p_1 ∈ Δ, p_1 = p
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΓ : ∀ (p : Formula α), p ∉ Γ
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | replace hΓ : Γ = [] := List.nil_iff.mpr hΓ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΓ : ∀ (p : Formula α), p ∉ Γ
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
⊢ False | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
hΓ : Γ = []
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | subst hΓ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Γ Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
hΓ : Γ = []
⊢ False | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! [].conj' ⟶ Δ.disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | have : 𝓓 ⊢! p := disj_allsame'! hΔ (hC ⨀ verum!) | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! [].conj' ⟶ Δ.disj'
⊢ False | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! [].conj' ⟶ Δ.disj'
this : 𝓓 ⊢! p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | contradiction | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
h : ¬𝓓 ⊢! p
Δ : List (Formula α)
hΔ : ∀ p_1 ∈ Δ, p_1 = p
hC : 𝓓 ⊢! [].conj' ⟶ Δ.disj'
this : 𝓓 ⊢! p
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | intro h t | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t : SCT 𝓓
p : Formula α
⊢ 𝓓 ⊢! p → CanonicalModel 𝓓 ⊧ p | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
t : (CanonicalModel 𝓓).World
⊢ t ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | suffices p ∈ t.tableau.1 by exact truthlemma.mpr this; | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
t : (CanonicalModel 𝓓).World
⊢ t ⊧ p | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
t : (CanonicalModel 𝓓).World
⊢ p ∈ t.tableau.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | exact mem₁_of_provable h | case mpr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
t : (CanonicalModel 𝓓).World
⊢ p ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.deducible_of_validOnCanonicelModel | [481, 1] | [501, 30] | exact truthlemma.mpr this | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Encodable α
inst✝ : Inhabited (SCT 𝓓)
t✝ : SCT 𝓓
p : Formula α
h : 𝓓 ⊢! p
t : (CanonicalModel 𝓓).World
this : p ∈ t.tableau.1
⊢ t ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.complete! | [508, 1] | [511, 33] | intro h | α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : Inhabited α
𝓓 : DeductionParameter α
inst✝³ : 𝓓.IncludeEFQ
inst✝² : Encodable α
p : Formula α
inst✝¹ : Consistent 𝓓
inst✝ : Canonical 𝓓
⊢ 𝔽(Ax(𝓓)) ⊧ p → 𝓓 ⊢! p | α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : Inhabited α
𝓓 : DeductionParameter α
inst✝³ : 𝓓.IncludeEFQ
inst✝² : Encodable α
p : Formula α
inst✝¹ : Consistent 𝓓
inst✝ : Canonical 𝓓
h : 𝔽(Ax(𝓓)) ⊧ p
⊢ 𝓓 ⊢! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.complete! | [508, 1] | [511, 33] | apply deducible_of_validOnCanonicelModel.mp | α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : Inhabited α
𝓓 : DeductionParameter α
inst✝³ : 𝓓.IncludeEFQ
inst✝² : Encodable α
p : Formula α
inst✝¹ : Consistent 𝓓
inst✝ : Canonical 𝓓
h : 𝔽(Ax(𝓓)) ⊧ p
⊢ 𝓓 ⊢! p | α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : Inhabited α
𝓓 : DeductionParameter α
inst✝³ : 𝓓.IncludeEFQ
inst✝² : Encodable α
p : Formula α
inst✝¹ : Consistent 𝓓
inst✝ : Canonical 𝓓
h : 𝔽(Ax(𝓓)) ⊧ p
⊢ CanonicalModel 𝓓 ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Kripke.complete! | [508, 1] | [511, 33] | exact h Canonical.realize _ _ | α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : Inhabited α
𝓓 : DeductionParameter α
inst✝³ : 𝓓.IncludeEFQ
inst✝² : Encodable α
p : Formula α
inst✝¹ : Consistent 𝓓
inst✝ : Canonical 𝓓
h : 𝔽(Ax(𝓓)) ⊧ p
⊢ CanonicalModel 𝓓 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | simp [-ValidOnFrame.models_iff] | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
d : 𝓓 ⊢ p
⊢ 𝔽(Ax(𝓓)) ⊧ p | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
d : 𝓓 ⊢ p
⊢ ∀ {F : Frame' α}, F ∈ 𝔽(Ax(𝓓)) → F ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | intro F hF | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
d : 𝓓 ⊢ p
⊢ ∀ {F : Frame' α}, F ∈ 𝔽(Ax(𝓓)) → F ⊧ p | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
d : 𝓓 ⊢ p
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
⊢ F ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | induction d with
| eaxm h => exact validOnAxiomSetFrameClass_axiom h hF;
| mdp _ _ ihpq ihp => exact ValidOnFrame.mdp ihpq ihp;
| _ =>
intros;
first
| apply ValidOnFrame.verum
| apply ValidOnFrame.imply₁
| apply ValidOnFrame.imply₂
| apply ValidOnFrame.disj₁
| apply ValidOnFrame.disj₂
| apply ValidOnFra... | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
d : 𝓓 ⊢ p
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
⊢ F ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | exact validOnAxiomSetFrameClass_axiom h hF | case eaxm
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ : Formula α
h : p✝ ∈ Ax(𝓓)
⊢ F ⊧ p✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | exact ValidOnFrame.mdp ihpq ihp | case mdp
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
a✝¹ : Deduction 𝓓 (p✝ ⟶ q✝)
a✝ : Deduction 𝓓 p✝
ihpq : F ⊧ p✝ ⟶ q✝
ihp : F ⊧ p✝
⊢ F ⊧ q✝ | no goals |
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