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.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | case subset => constructor <;> apply Set.subset_iUnion_of_subset 0 (by simp); | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ t ⊆ t∞ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | case saturated =>
intro p;
cases mem_lindenbaum_next_indexed t p with
| inl h => left; simp [lindenbaum_maximal]; use (encode p + 1);
| inr h => right; simp [lindenbaum_maximal]; use (encode p + 1); | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ t∞.Saturated | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | case consistent =>
intro Γ Δ hΓ hΔ;
simp_all [lindenbaum_maximal];
obtain ⟨m, hΓ⟩ : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1 := by
induction Γ with
| nil => simp;
| cons p Γ ih =>
simp_all;
obtain ⟨i, hi⟩ := hΓ.1;
obtain ⟨m, hm⟩ := ih;
use (i + m);
constructor;
. exact lindenbaum_n... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ (𝓓)-Consistent t∞ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | constructor <;> apply Set.subset_iUnion_of_subset 0 (by simp) | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ t ⊆ t∞ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ t.2 ⊆ t[0].2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | intro p | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ t∞.Saturated | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
⊢ p ∈ t∞.1 ∨ p ∈ t∞.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | cases mem_lindenbaum_next_indexed t p with
| inl h => left; simp [lindenbaum_maximal]; use (encode p + 1);
| inr h => right; simp [lindenbaum_maximal]; use (encode p + 1); | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
⊢ p ∈ t∞.1 ∨ p ∈ t∞.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | left | case inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ (lindenbaum_next_indexed ?m.372754 t (encode p + 1)).1
⊢ p ∈ t∞.1 ∨ p ∈ t∞.2 | case inl.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ (lindenbaum_next_indexed ?m.372754 t (encode p + 1)).1
⊢ p ∈ t∞.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp [lindenbaum_maximal] | case inl.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ (lindenbaum_next_indexed ?m.372754 t (encode p + 1)).1
⊢ p ∈ t∞.1 | case inl.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ (lindenbaum_next_indexed ?m.372754 t (encode p + 1)).1
⊢ ∃ i, p ∈ t[i].1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | use (encode p + 1) | case inl.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ (lindenbaum_next_indexed ?m.372754 t (encode p + 1)).1
⊢ ∃ i, p ∈ t[i].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | right | case inr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ t[encode p + 1].2
⊢ p ∈ t∞.1 ∨ p ∈ t∞.2 | case inr.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ t[encode p + 1].2
⊢ p ∈ t∞.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp [lindenbaum_maximal] | case inr.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ t[encode p + 1].2
⊢ p ∈ t∞.2 | case inr.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ t[encode p + 1].2
⊢ ∃ i, p ∈ t[i].2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | use (encode p + 1) | case inr.h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
p : Formula α
h : p ∈ t[encode p + 1].2
⊢ ∃ i, p ∈ t[i].2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | 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 : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
⊢ (𝓓)-Consistent t∞ | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t∞.1
hΔ : ∀ p ∈ Δ, p ∈ t∞.2
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp_all [lindenbaum_maximal] | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ t∞.1
hΔ : ∀ p ∈ Δ, p ∈ t∞.2
⊢ 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 : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨m, hΓ⟩ : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1 := by
induction Γ with
| nil => simp;
| cons p Γ ih =>
simp_all;
obtain ⟨i, hi⟩ := hΓ.1;
obtain ⟨m, hm⟩ := ih;
use (i + m);
constructor;
. exact lindenbaum_next_indexed_subset₁_of_lt (by simp) hi;
. intro q hq;
exact lindenbaum_next_index... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | case intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
⊢ 𝓓 ⊬! ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨n, hΔ⟩ : ∃ n, ∀ p ∈ Δ, p ∈ t[n].2 := by
induction Δ with
| nil => simp;
| cons p Δ ih =>
simp_all;
obtain ⟨i, hi⟩ := hΔ.1;
obtain ⟨n, hn⟩ := ih;
use (i + n);
constructor;
. exact lindenbaum_next_indexed_subset₂_of_lt (by simp) hi;
. intro q hq;
exact lindenbaum_next_index... | case intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
⊢ 𝓓 ⊬! ... | case intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | rcases (lt_trichotomy m n) with hm | hmn | hn | case intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n... | case intro.intro.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . exact lindenbaum_next_indexed_parametricConsistent hCon n (by intro p hp; exact lindenbaum_next_indexed_subset₁_of_lt hm.le $ hΓ p hp) hΔ; | case intro.intro.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m]... | case intro.intro.inr.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . subst hmn;
exact lindenbaum_next_indexed_parametricConsistent hCon m hΓ hΔ; | case intro.intro.inr.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... | case intro.intro.inr.inr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . exact lindenbaum_next_indexed_parametricConsistent hCon m hΓ (by intro p hp; exact lindenbaum_next_indexed_subset₂_of_lt hn.le $ hΔ p hp); | case intro.intro.inr.inr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | induction Γ with
| nil => simp;
| cons p Γ ih =>
simp_all;
obtain ⟨i, hi⟩ := hΓ.1;
obtain ⟨m, hm⟩ := ih;
use (i + m);
constructor;
. exact lindenbaum_next_indexed_subset₁_of_lt (by simp) hi;
. intro q hq;
exact lindenbaum_next_indexed_subset₁_of_lt (by simp) $ hm q hq; | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
⊢ ∃ m, ∀ p ∈ Γ, p ∈ t[m].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | case nil
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
hΓ : ∀ p ∈ [], ∃ i, p ∈ t[i].1
⊢ ∃ m, ∀ p ∈ [], p ∈ t[m].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp_all | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
ih : (∀ p ∈ Γ, ∃ i, p ∈ t[i].1) → ∃ m,... | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
ih : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1
hΓ : (∃ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨i, hi⟩ := hΓ.1 | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
ih : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1
hΓ : (∃ ... | case cons.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
ih : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1
hΓ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨m, hm⟩ := ih | case cons.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
ih : ∃ m, ∀ p ∈ Γ, p ∈ t[m].1
hΓ... | case cons.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | use (i + m) | case cons.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀... | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ i, a ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | constructor | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ i, a ... | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . exact lindenbaum_next_indexed_subset₁_of_lt (by simp) hi; | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ ... | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . intro q hq;
exact lindenbaum_next_indexed_subset₁_of_lt (by simp) $ hm q hq; | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₁_of_lt (by simp) hi | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ i, a ∈ t[i].... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | intro q hq | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃... | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₁_of_lt (by simp) $ hm q hq | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Δ : List (Formula α)
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
p : Formula α
Γ : List (Formula α)
hΓ : (∃ i, p ∈ t[i].1) ∧ ∀ a ∈ Γ, ∃ i, a ∈ t[i].... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | induction Δ with
| nil => simp;
| cons p Δ ih =>
simp_all;
obtain ⟨i, hi⟩ := hΔ.1;
obtain ⟨n, hn⟩ := ih;
use (i + n);
constructor;
. exact lindenbaum_next_indexed_subset₂_of_lt (by simp) hi;
. intro q hq;
exact lindenbaum_next_indexed_subset₂_of_lt (by simp) $ hn q hq; | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
⊢ ∃ n, ∀ p ∈ Δ, p ∈... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | case nil
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
hΔ : ∀ p ∈ [], ∃ i, p ∈ t[i].2
⊢ ∃ n, ∀ p ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp_all | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
ih : (... | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
ih : ∃... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨i, hi⟩ := hΔ.1 | case cons
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
ih : ∃... | case cons.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | obtain ⟨n, hn⟩ := ih | case cons.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
... | case cons.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formu... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | use (i + n) | case cons.intro.intro
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formu... | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ : (∃ i... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | constructor | case h
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ : (∃ i... | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . exact lindenbaum_next_indexed_subset₂_of_lt (by simp) hi; | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ :... | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | . intro q hq;
exact lindenbaum_next_indexed_subset₂_of_lt (by simp) $ hn q hq; | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₂_of_lt (by simp) hi | case h.left
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ :... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ : (∃ i, p ∈ t... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | intro q hq | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ ... | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₂_of_lt (by simp) $ hn q hq | case h.right
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | simp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
p : Formula α
Δ : List (Formula α)
hΔ : (∃ i, p ∈ t... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_parametricConsistent hCon n (by intro p hp; exact lindenbaum_next_indexed_subset₁_of_lt hm.le $ hΓ p hp) hΔ | case intro.intro.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m]... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | intro p hp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₁_of_lt hm.le $ hΓ p hp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | subst hmn | case intro.intro.inr.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... | case intro.intro.inr.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_parametricConsistent hCon m hΓ hΔ | case intro.intro.inr.inl
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_parametricConsistent hCon m hΓ (by intro p hp; exact lindenbaum_next_indexed_subset₂_of_lt hn.le $ hΔ p hp) | case intro.intro.inr.inr
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | intro p hp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.exists_parametricConsistent_saturated_tableau | [257, 1] | [297, 145] | exact lindenbaum_next_indexed_subset₂_of_lt hn.le $ hΔ p hp | α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.IncludeEFQ
inst✝¹ : Inhabited α
inst✝ : Encodable α
t : Tableau α
hCon : (𝓓)-Consistent t
Γ Δ : List (Formula α)
hΓ✝ : ∀ p ∈ Γ, ∃ i, p ∈ t[i].1
hΔ✝ : ∀ p ∈ Δ, ∃ i, p ∈ t[i].2
m : ℕ
hΓ : ∀ p ∈ Γ, p ∈ t[m].1
n : ℕ
hΔ : ∀ p ∈ Δ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.lindenbaum | [318, 1] | [320, 31] | obtain ⟨t, ht, hCon, hMax⟩ := Tableau.lindenbaum hCon | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t₀ : Tableau α
hCon : (𝓓)-Consistent t₀
⊢ ∃ t, t₀ ⊆ t.tableau | case intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t₀ : Tableau α
hCon✝ : (𝓓)-Consistent t₀
t : Tableau α
ht : t₀ ⊆ t
hCon : (𝓓)-Consistent t
hMax : t.Saturated
⊢ ∃ t, t₀ ⊆ t.tableau |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.lindenbaum | [318, 1] | [320, 31] | exact ⟨⟨t, hMax, hCon⟩, ht⟩ | case intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Encodable α
t₀ : Tableau α
hCon✝ : (𝓓)-Consistent t₀
t : Tableau α
ht : t₀ ⊆ t
hCon : (𝓓)-Consistent t
hMax : t.Saturated
⊢ ∃ t, t₀ ⊆ t.tableau | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.equality_of₁ | [336, 1] | [341, 62] | have e := Tableau.equality_def.mpr ⟨e₁, (saturated_duality.mp e₁)⟩ | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
⊢ t₁ = t₂ | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
e : t₁.tableau = t₂.tableau
⊢ t₁ = t₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.equality_of₁ | [336, 1] | [341, 62] | calc
t₁ = ⟨t₁.tableau, t₁.saturated, t₁.consistent⟩ := by rfl;
_ = ⟨t₂.tableau, t₂.saturated, t₂.consistent⟩ := by simp [e];
_ = t₂ := by rfl; | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
e : t₁.tableau = t₂.tableau
⊢ t₁ = t₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.equality_of₁ | [336, 1] | [341, 62] | rfl | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
e : t₁.tableau = t₂.tableau
⊢ t₁ = { tableau := t₁.tableau, saturated := ⋯, consistent := ⋯ } | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.equality_of₁ | [336, 1] | [341, 62] | simp [e] | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
e : t₁.tableau = t₂.tableau
⊢ { tableau := t₁.tableau, saturated := ⋯, consistent := ⋯ } =
{ tableau := t₂.tableau, saturated := ⋯, consi... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.equality_of₁ | [336, 1] | [341, 62] | rfl | 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
e₁ : t₁.tableau.1 = t₂.tableau.1
e : t₁.tableau = t₂.tableau
⊢ { tableau := t₂.tableau, saturated := ⋯, consistent := ⋯ } = t₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mdp₁ | [347, 1] | [348, 94] | exact t.not_mem₂_iff_mem₁.mp $ not_mem₂ (by simpa) (show 𝓓 ⊢! List.conj' [p] ⟶ q by simpa;) | 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
hp : p ∈ t.tableau.1
h : 𝓓 ⊢! p ⟶ q
⊢ q ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mdp₁ | [347, 1] | [348, 94] | simpa | 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
hp : p ∈ t.tableau.1
h : 𝓓 ⊢! p ⟶ q
⊢ ∀ p_1 ∈ [p], p_1 ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mdp₁ | [347, 1] | [348, 94] | simpa | 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
hp : p ∈ t.tableau.1
h : 𝓓 ⊢! p ⟶ q
⊢ 𝓓 ⊢! [p].conj' ⟶ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | apply t.not_mem₂_iff_mem₁.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
⊢ ⊤ ∈ 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
⊢ ⊤ ∉ t.tableau.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | by_contra hC | 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
⊢ ⊤ ∉ t.tableau.2 | 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
hC : ⊤ ∈ t.tableau.2
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | have : 𝓓 ⊬! [].conj' ⟶ [⊤].disj' := t.consistent (by simp) (by simpa) | 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
hC : ⊤ ∈ t.tableau.2
⊢ False | 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
hC : ⊤ ∈ t.tableau.2
this : 𝓓 ⊬! [].conj' ⟶ [⊤].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | have : 𝓓 ⊢! [].conj' ⟶ [⊤].disj' := by 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
hC : ⊤ ∈ t.tableau.2
this : 𝓓 ⊬! [].conj' ⟶ [⊤].disj'
⊢ False | 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
hC : ⊤ ∈ t.tableau.2
this✝ : 𝓓 ⊬! [].conj' ⟶ [⊤].disj'
this : 𝓓 ⊢! [].conj' ⟶ [⊤].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | contradiction | 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
hC : ⊤ ∈ t.tableau.2
this✝ : 𝓓 ⊬! [].conj' ⟶ [⊤].disj'
this : 𝓓 ⊢! [].conj' ⟶ [⊤].disj'
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | 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
hC : ⊤ ∈ t.tableau.2
⊢ ∀ p ∈ [], p ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | simpa | 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
hC : ⊤ ∈ t.tableau.2
⊢ ∀ p ∈ [⊤], p ∈ t.tableau.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.mem₁_verum | [350, 1] | [356, 17] | 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
hC : ⊤ ∈ t.tableau.2
this : 𝓓 ⊬! [].conj' ⟶ [⊤].disj'
⊢ 𝓓 ⊢! [].conj' ⟶ [⊤].disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | by_contra hC | 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
⊢ ⊥ ∉ 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
hC : ⊥ ∈ t.tableau.1
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | have : 𝓓 ⊬! [⊥].conj' ⟶ [].disj' := t.consistent (by simpa) (by 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
hC : ⊥ ∈ t.tableau.1
⊢ False | 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
hC : ⊥ ∈ t.tableau.1
this : 𝓓 ⊬! [⊥].conj' ⟶ [].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | have : 𝓓 ⊢! [⊥].conj' ⟶ [].disj' := by 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
hC : ⊥ ∈ t.tableau.1
this : 𝓓 ⊬! [⊥].conj' ⟶ [].disj'
⊢ False | 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
hC : ⊥ ∈ t.tableau.1
this✝ : 𝓓 ⊬! [⊥].conj' ⟶ [].disj'
this : 𝓓 ⊢! [⊥].conj' ⟶ [].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | contradiction | 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
hC : ⊥ ∈ t.tableau.1
this✝ : 𝓓 ⊬! [⊥].conj' ⟶ [].disj'
this : 𝓓 ⊢! [⊥].conj' ⟶ [].disj'
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | simpa | 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
hC : ⊥ ∈ t.tableau.1
⊢ ∀ p ∈ [⊥], p ∈ t.tableau.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | 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
hC : ⊥ ∈ t.tableau.1
⊢ ∀ p ∈ [], p ∈ t.tableau.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.not_mem₁_falsum | [358, 1] | [363, 17] | 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
hC : ⊥ ∈ t.tableau.1
this : 𝓓 ⊬! [⊥].conj' ⟶ [].disj'
⊢ 𝓓 ⊢! [⊥].conj' ⟶ [].disj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | constructor | 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 ⋏ 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
⊢ p ⋏ q ∈ t.tableau.1 → p ∈ t.tableau.1 ∧ q ∈ t.tableau.1
case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | . intro h; constructor <;> exact mdp₁ h (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
⊢ p ⋏ q ∈ t.tableau.1 → p ∈ t.tableau.1 ∧ q ∈ t.tableau.1
case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α... | 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 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | . rintro ⟨hp, hq⟩;
by_contra hC;
have : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj' := by simp;
have : 𝓓 ⊬! [p, q].conj' ⟶ [p ⋏ q].disj' := t.consistent (by aesop) (by simpa using t.not_mem₁_iff_mem₂.mp hC);
contradiction; | 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 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | intro h | 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
⊢ 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
⊢ p ∈ t.tableau.1 ∧ q ∈ t.tableau.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | constructor <;> exact mdp₁ h (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
⊢ p ∈ t.tableau.1 ∧ 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_and | [365, 1] | [373, 19] | 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
⊢ 𝓓 ⊢! p ⋏ q ⟶ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | rintro ⟨hp, hq⟩ | 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.intro
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
hp : p ∈ t.tableau.1
hq : 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_and | [365, 1] | [373, 19] | by_contra hC | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
⊢ p ⋏ q ∈ t.tableau.1 | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ 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_and | [365, 1] | [373, 19] | have : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj' := by simp; | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
⊢ False | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | have : 𝓓 ⊬! [p, q].conj' ⟶ [p ⋏ q].disj' := t.consistent (by aesop) (by simpa using t.not_mem₁_iff_mem₂.mp hC) | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
⊢ False | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this✝ : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
this : 𝓓 ⊬! [p, q].conj' ⟶ [p ⋏ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | contradiction | case mpr.intro
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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this✝ : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
this : 𝓓 ⊬! [p, q].conj' ⟶ [p ⋏ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.SaturatedConsistentTableau.iff_mem₁_and | [365, 1] | [373, 19] | 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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
⊢ 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_and | [365, 1] | [373, 19] | aesop | 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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
⊢ ∀ p_1 ∈ [p, q], p_1 ∈ 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_and | [365, 1] | [373, 19] | simpa using t.not_mem₁_iff_mem₂.mp hC | 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
hp : p ∈ t.tableau.1
hq : q ∈ t.tableau.1
hC : p ⋏ q ∉ t.tableau.1
this : 𝓓 ⊢! [p, q].conj' ⟶ [p ⋏ q].disj'
⊢ ∀ p_1 ∈ [p ⋏ q], p_1 ∈ t.tableau.2 | 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] | constructor | 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 ⋎ 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
⊢ p ⋎ q ∈ t.tableau.1 → p ∈ t.tableau.1 ∨ q ∈ t.tableau.1
case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α... |
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;
by_contra hC; simp [not_or] at hC;
have : p ∈ t.tableau.2 := t.not_mem₁_iff_mem₂.mp hC.1;
have : q ∈ t.tableau.2 := t.not_mem₁_iff_mem₂.mp hC.2;
have : 𝓓 ⊢! [p ⋎ q].conj' ⟶ [p, q].disj' := by simp;
have : 𝓓 ⊬! [p ⋎ q].conj' ⟶ [p, q].disj' := t.consistent (by simp_all) (by simp_all);
contradicti... | 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
⊢ p ⋎ q ∈ t.tableau.1 → p ∈ t.tableau.1 ∨ q ∈ t.tableau.1
case mpr
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α... | 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 |
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;
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
⊢ 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] | intro h | 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
⊢ 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
⊢ p ∈ t.tableau.1 ∨ q ∈ t.tableau.1 |
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