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.mem₂_of_not_mem₁ | [125, 1] | [129, 25] | exact absurd h' h | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.1
h' : 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.mem₂_of_not_mem₁ | [125, 1] | [129, 25] | assumption | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.1
h✝ : p ∈ t.2
⊢ p ∈ t.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₁_of_not_mem₂ | [131, 1] | [135, 33] | intro h | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.2 → p ∈ t.1 | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.2
⊢ p ∈ t.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₁_of_not_mem₂ | [131, 1] | [135, 33] | cases (hMat p) with
| inl _ => assumption;
| inr h' => exact absurd h' h; | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.2
⊢ p ∈ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₁_of_not_mem₂ | [131, 1] | [135, 33] | assumption | case inl
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.2
h✝ : p ∈ t.1
⊢ p ∈ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem₁_of_not_mem₂ | [131, 1] | [135, 33] | exact absurd h' h | case inr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
h : p ∉ t.2
h' : p ∈ t.2
⊢ p ∈ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₁_iff_mem₂ | [137, 1] | [140, 63] | constructor | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.1 ↔ p ∈ t.2 | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.1 → p ∈ t.2
case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₁_iff_mem₂ | [137, 1] | [140, 63] | . apply mem₂_of_not_mem₁ hMat; | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.1 → p ∈ t.2
case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited ... | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.2 → p ∉ t.1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₁_iff_mem₂ | [137, 1] | [140, 63] | . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon; | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.2 → p ∉ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₁_iff_mem₂ | [137, 1] | [140, 63] | apply mem₂_of_not_mem₁ hMat | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.1 → p ∈ t.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₁_iff_mem₂ | [137, 1] | [140, 63] | apply Set.disjoint_right.mp $ disjoint_of_consistent hCon | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.2 → p ∉ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂_iff_mem₁ | [142, 1] | [145, 62] | constructor | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.2 ↔ p ∈ t.1 | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.2 → p ∈ t.1
case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂_iff_mem₁ | [142, 1] | [145, 62] | . apply mem₁_of_not_mem₂ hMat; | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.2 → p ∈ t.1
case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited ... | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.1 → p ∉ t.2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂_iff_mem₁ | [142, 1] | [145, 62] | . apply Set.disjoint_left.mp $ disjoint_of_consistent hCon; | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.1 → p ∉ t.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂_iff_mem₁ | [142, 1] | [145, 62] | apply mem₁_of_not_mem₂ hMat | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∉ t.2 → p ∈ t.1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.not_mem₂_iff_mem₁ | [142, 1] | [145, 62] | apply Set.disjoint_left.mp $ disjoint_of_consistent hCon | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
p : Formula α
⊢ p ∈ t.1 → p ∉ t.2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | constructor | α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.1 = t₂.1 ↔... | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.1 ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro h;
apply Set.eq_of_subset_of_subset;
. intro p hp;
apply not_mem₁_iff_mem₂ ct₂ |>.mp; rw [←h];
apply not_mem₁_iff_mem₂ ct₁ |>.mpr hp;
. intro p hp;
apply not_mem₁_iff_mem₂ ct₁ |>.mp; rw [h];
apply not_mem₁_iff_mem₂ ct₂ |>.mpr hp; | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.1 ... | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.2... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro h;
apply Set.eq_of_subset_of_subset;
. intro p hp;
apply not_mem₂_iff_mem₁ ct₂ |>.mp; rw [←h];
apply not_mem₂_iff_mem₁ ct₁ |>.mpr hp;
. intro p hp;
apply not_mem₂_iff_mem₁ ct₁ |>.mp; rw [h];
apply not_mem₂_iff_mem₁ ct₂ |>.mpr hp; | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.2... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro h | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.1 ... | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t₁.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply Set.eq_of_subset_of_subset | case mp
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t₁.... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro p hp;
apply not_mem₁_iff_mem₂ ct₂ |>.mp; rw [←h];
apply not_mem₁_iff_mem₂ ct₁ |>.mpr hp; | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro p hp;
apply not_mem₁_iff_mem₂ ct₁ |>.mp; rw [h];
apply not_mem₁_iff_mem₂ ct₂ |>.mpr hp; | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro p hp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₁_iff_mem₂ ct₂ |>.mp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | rw [←h] | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₁_iff_mem₂ ct₁ |>.mpr hp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro p hp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₁_iff_mem₂ ct₁ |>.mp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | rw [h] | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₁_iff_mem₂ ct₂ |>.mpr hp | case mp.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro h | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
⊢ t₁.2... | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t₁... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply Set.eq_of_subset_of_subset | case mpr
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : t₁... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro p hp;
apply not_mem₂_iff_mem₁ ct₂ |>.mp; rw [←h];
apply not_mem₂_iff_mem₁ ct₁ |>.mpr hp; | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | . intro p hp;
apply not_mem₂_iff_mem₁ ct₁ |>.mp; rw [h];
apply not_mem₂_iff_mem₁ ct₂ |>.mpr hp; | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro p hp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₂_iff_mem₁ ct₂ |>.mp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | rw [←h] | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₂_iff_mem₁ ct₁ |>.mpr hp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | intro p hp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₂_iff_mem₁ ct₁ |>.mp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | rw [h] | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.saturated_duality | [147, 1] | [168, 45] | apply not_mem₂_iff_mem₁ ct₂ |>.mpr hp | case mpr.a
α : Type u_1
t : Tableau α
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
hCon : autoParam ((𝓓)-Consistent t) _auto✝
hMat : autoParam t.Saturated _auto✝
t₁ t₂ : Tableau α
ct₁ : (𝓓)-Consistent t₁
ct₂ : (𝓓)-Consistent t₂
st₁ : t₁.Saturated
st₂ : t₂.Saturated
h : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | intro Γ Δ hΓ hΔ | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
⊢ (𝓓)-Consistent (Ax(𝓓), ∅) | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : ∀ p ∈ Δ, p ∈ (Ax(𝓓), ∅).2
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | replace hΔ : Δ = [] := List.nil_iff.mpr hΔ | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : ∀ p ∈ Δ, p ∈ (Ax(𝓓), ∅).2
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | obtain ⟨q, hq⟩ := h.exists_unprovable | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | by_contra hC | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
⊢ 𝓓 ⊬! Γ.conj' ⟶ Δ.disj' | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | have : 𝓓 ⊢! q := by
subst hΔ;
simp [List.disj'_nil] at hC;
exact imp_trans! hC efq! ⨀ (by
apply iff_provable_list_conj.mpr;
exact λ _ hp => ⟨Deduction.eaxm $ hΓ _ hp⟩;
); | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
⊢ False | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
this : 𝓓 ⊢! q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | contradiction | case intro
α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
this : 𝓓 ⊢! q
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | subst hΔ | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
hΔ : Δ = []
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ Δ.disj'
⊢ 𝓓 ⊢! q | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ [].disj'
⊢ 𝓓 ⊢! q |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | simp [List.disj'_nil] at hC | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ [].disj'
⊢ 𝓓 ⊢! q | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ ⊥
⊢ 𝓓 ⊢! q |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | exact imp_trans! hC efq! ⨀ (by
apply iff_provable_list_conj.mpr;
exact λ _ hp => ⟨Deduction.eaxm $ hΓ _ hp⟩;
) | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ ⊥
⊢ 𝓓 ⊢! q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | apply iff_provable_list_conj.mpr | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ ⊥
⊢ 𝓓 ⊢! Γ.conj' | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ ⊥
⊢ ∀ p ∈ Γ, 𝓓 ⊢! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.self_ParametricConsistent | [174, 1] | [186, 17] | exact λ _ hp => ⟨Deduction.eaxm $ hΓ _ hp⟩ | α : Type u_1
inst✝³ : DecidableEq α
inst✝² : Inhabited α
𝓓 : DeductionParameter α
inst✝¹ : 𝓓.IncludeEFQ
inst✝ : Inhabited α
h : Consistent 𝓓
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ (Ax(𝓓), ∅).1
q : Formula α
hq : 𝓓 ⊬! q
hC : 𝓓 ⊢! Γ.conj' ⟶ ⊥
⊢ ∀ p ∈ Γ, 𝓓 ⊢! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | simp [lindenbaum_next] | 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
consistent : (𝓓)-Consistent t
p : Formula α
⊢ (𝓓)-Consistent (lindenbaum_next 𝓓 p 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
consistent : (𝓓)-Consistent t
p : Formula α
⊢ (𝓓)-Consistent (if (𝓓)-Consistent (insert p t.1, t.2) then (insert p t.1, t.2) else (t.1, insert p t.2)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | split | 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
consistent : (𝓓)-Consistent t
p : Formula α
⊢ (𝓓)-Consistent (if (𝓓)-Consistent (insert p t.1, t.2) then (insert p t.1, t.2) else (t.1, insert p t.2)) | case isTrue
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : (𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (insert p t.1, t.2)
case isFalse
α : Type ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | . simpa; | case isTrue
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : (𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (insert p t.1, t.2)
case isFalse
α : Type ... | case isFalse
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (t.1, insert p t.2) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | . have := consistent_either consistent p;
simp_all only [false_or]; | case isFalse
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (t.1, insert p t.2) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | simpa | case isTrue
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : (𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (insert p t.1, t.2) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | have := consistent_either consistent p | case isFalse
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t.1, t.2)
⊢ (𝓓)-Consistent (t.1, insert p t.2) | case isFalse
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t.1, t.2)
this : (𝓓)-Consistent (insert p t.1, t.2) ∨ (𝓓)-Consisten... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.next_parametericConsistent | [212, 1] | [217, 30] | simp_all only [false_or] | case isFalse
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
consistent : (𝓓)-Consistent t
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t.1, t.2)
this : (𝓓)-Consistent (insert p t.1, t.2) ∨ (𝓓)-Consisten... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_zero | [219, 1] | [220, 126] | simp [lindenbaum_next_indexed] | 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✝ t : Tableau α
⊢ t[0] = t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | simp [lindenbaum_next_indexed] | 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
i : ℕ
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent t[i + 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
i : ℕ
⊢ (𝓓)-Consistent t[i] →
(𝓓)-Consistent
(match decode i with
| some p => lindenbaum_next 𝓓 p t[i]
| none => t[i]) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | split | 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
i : ℕ
⊢ (𝓓)-Consistent t[i] →
(𝓓)-Consistent
(match decode i with
| some p => lindenbaum_next 𝓓 p t[i]
| none => t[i]) | case h_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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent (lindenbaum_next 𝓓 p✝ t[i])
case h_2... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | . intro h; apply next_parametericConsistent; assumption; | case h_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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent (lindenbaum_next 𝓓 p✝ t[i])
case h_2... | case h_2
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
i : ℕ
x✝ : Option (Formula α)
heq✝ : decode i = none
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent t[i] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | . tauto; | case h_2
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
i : ℕ
x✝ : Option (Formula α)
heq✝ : decode i = none
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent t[i] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | intro h | case h_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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent (lindenbaum_next 𝓓 p✝ t[i]) | case h_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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
h : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent (lindenbaum_next 𝓓 p✝ t[i]) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | apply next_parametericConsistent | case h_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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
h : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent (lindenbaum_next 𝓓 p✝ t[i]) | case h_1.consistent
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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
h : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent t[i] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | assumption | case h_1.consistent
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
i : ℕ
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode i = some p✝
h : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent t[i] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent_succ | [222, 1] | [226, 11] | tauto | case h_2
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
i : ℕ
x✝ : Option (Formula α)
heq✝ : decode i = none
⊢ (𝓓)-Consistent t[i] → (𝓓)-Consistent t[i] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | simp [lindenbaum_next_indexed, lindenbaum_next] | 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✝ t : Tableau α
p : Formula α
⊢ p ∈ t[encode p + 1].1 ∨ p ∈ t[encode p + 1].2 | 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✝ t : Tableau α
p : Formula α
⊢ p ∈
(if (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2) then (insert p t[encode p].1, t[encode p].2)
else (t[encode p]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | split | 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✝ t : Tableau α
p : Formula α
⊢ p ∈
(if (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2) then (insert p t[encode p].1, t[encode p].2)
else (t[encode p]... | case isTrue
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✝ t : Tableau α
p : Formula α
h✝ : (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (insert p t[encode p].1, t[encode p].2).1 ∨ p ∈ (insert p t[enc... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | . left; tauto; | case isTrue
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✝ t : Tableau α
p : Formula α
h✝ : (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (insert p t[encode p].1, t[encode p].2).1 ∨ p ∈ (insert p t[enc... | case isFalse
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✝ t : Tableau α
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (t[encode p].1, insert p t[encode p].2).1 ∨ p ∈ (t[encode p].... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | . right; tauto; | case isFalse
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✝ t : Tableau α
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (t[encode p].1, insert p t[encode p].2).1 ∨ p ∈ (t[encode p].... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | left | case isTrue
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✝ t : Tableau α
p : Formula α
h✝ : (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (insert p t[encode p].1, t[encode p].2).1 ∨ p ∈ (insert p t[enc... | case isTrue.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✝ t : Tableau α
p : Formula α
h✝ : (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (insert p t[encode p].1, t[encode p].2).1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | tauto | case isTrue.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✝ t : Tableau α
p : Formula α
h✝ : (𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (insert p t[encode p].1, t[encode p].2).1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | right | case isFalse
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✝ t : Tableau α
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (t[encode p].1, insert p t[encode p].2).1 ∨ p ∈ (t[encode p].... | case isFalse.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✝ t : Tableau α
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (t[encode p].1, insert p t[encode p].2).2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.mem_lindenbaum_next_indexed | [228, 1] | [232, 18] | tauto | case isFalse.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✝ t : Tableau α
p : Formula α
h✝ : ¬(𝓓)-Consistent (insert p t[encode p].1, t[encode p].2)
⊢ p ∈ (t[encode p].1, insert p t[encode p].2).2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent | [234, 1] | [237, 86] | induction i with
| zero => simpa;
| succ i ih => apply lindenbaum_next_indexed_parametricConsistent_succ; assumption; | 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
consistent : (𝓓)-Consistent t
i : ℕ
⊢ (𝓓)-Consistent t[i] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent | [234, 1] | [237, 86] | simpa | case zero
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
consistent : (𝓓)-Consistent t
⊢ (𝓓)-Consistent t[0] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent | [234, 1] | [237, 86] | apply lindenbaum_next_indexed_parametricConsistent_succ | case succ
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
consistent : (𝓓)-Consistent t
i : ℕ
ih : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent t[i + 1] | case succ.a
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
consistent : (𝓓)-Consistent t
i : ℕ
ih : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent t[i] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_parametricConsistent | [234, 1] | [237, 86] | assumption | case succ.a
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
consistent : (𝓓)-Consistent t
i : ℕ
ih : (𝓓)-Consistent t[i]
⊢ (𝓓)-Consistent t[i] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | induction h with
| refl => simp;
| step h ih =>
simp [lindenbaum_next_indexed, lindenbaum_next];
split;
. split <;> tauto;
. tauto; | 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
m n : ℕ
h : m ≤ n
⊢ t[m].1 ⊆ t[n].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | simp | case refl
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
m n : ℕ
⊢ t[m].1 ⊆ t[m].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | simp [lindenbaum_next_indexed, lindenbaum_next] | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
⊢ t[m].1 ⊆ t[m✝.succ].1 | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
⊢ t[m].1 ⊆
(match decode m✝ with
| some p =>
if (𝓓)-Consistent (insert p t[m✝].1, t[m... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | split | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
⊢ t[m].1 ⊆
(match decode m✝ with
| some p =>
if (𝓓)-Consistent (insert p t[m✝].1, t[m... | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].1 ⊆
(if (𝓓)-Consistent... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | . split <;> tauto; | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].1 ⊆
(if (𝓓)-Consistent... | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].1 ⊆ t[m✝].1 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | . tauto; | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].1 ⊆ t[m✝].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | split <;> tauto | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].1 ⊆
(if (𝓓)-Consistent... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₁_of_lt | [239, 1] | [246, 13] | tauto | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].1 ⊆ t[m✝].1
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].1 ⊆ t[m✝].1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | induction h with
| refl => simp;
| step h ih =>
simp [lindenbaum_next_indexed, lindenbaum_next];
split;
. split <;> tauto;
. tauto; | 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
m n : ℕ
h : m ≤ n
⊢ t[m].2 ⊆ t[n].2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | simp | case refl
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
m n : ℕ
⊢ t[m].2 ⊆ t[m].2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | simp [lindenbaum_next_indexed, lindenbaum_next] | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
⊢ t[m].2 ⊆ t[m✝.succ].2 | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
⊢ t[m].2 ⊆
(match decode m✝ with
| some p =>
if (𝓓)-Consistent (insert p t[m✝].1, t[m... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | split | case step
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
⊢ t[m].2 ⊆
(match decode m✝ with
| some p =>
if (𝓓)-Consistent (insert p t[m✝].1, t[m... | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].2 ⊆
(if (𝓓)-Consistent... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | . split <;> tauto; | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].2 ⊆
(if (𝓓)-Consistent... | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].2 ⊆ t[m✝].2 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | . tauto; | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].2 ⊆ t[m✝].2 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | split <;> tauto | case step.h_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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
p✝ : Formula α
heq✝ : decode m✝ = some p✝
⊢ t[m].2 ⊆
(if (𝓓)-Consistent... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.lindenbaum_next_indexed_subset₂_of_lt | [248, 1] | [255, 13] | tauto | case step.h_2
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
m n m✝ : ℕ
h : m.le m✝
ih : t[m].2 ⊆ t[m✝].2
x✝ : Option (Formula α)
heq✝ : decode m✝ = none
⊢ t[m].2 ⊆ t[m✝].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] | use 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
⊢ ∃ u, t ⊆ u ∧ (𝓓)-Consistent u ∧ u.Saturated | 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
⊢ t ⊆ t∞ ∧ (𝓓)-Consistent t∞ ∧ t∞.Saturated |
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] | refine ⟨?subset, ?consistent, ?saturated⟩ | 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
⊢ t ⊆ t∞ ∧ (𝓓)-Consistent t∞ ∧ t∞.Saturated | case subset
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∞
case consistent
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Inhabited α
𝓓 : DeductionParameter α
inst✝² : 𝓓.I... |
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