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/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | simp at hf | case mk.intro.refl
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
f : F
hf : ¬∅ ⊧ f
A : ¬∅.Nonempty
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | rintro ⟨𝓜, h𝓜⟩ | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
⊢ s.Nonempty → Meaningful s | case intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
⊢ Meaningful s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | rcases Meaningful.exists_unrealize (𝓜 := 𝓜) with ⟨f, hf⟩ | case intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
⊢ Meaningful s | case intro.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
f : F
hf : ¬𝓜 ⊧ f
⊢ Meaningful s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | exact ⟨f, by simp [set_models_iff]; exact ⟨𝓜, h𝓜, hf⟩⟩ | case intro.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
f : F
hf : ¬𝓜 ⊧ f
⊢ Meaningful s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | simp [set_models_iff] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
f : F
hf : ¬𝓜 ⊧ f
⊢ ¬s ⊧ f | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
f : F
hf : ¬𝓜 ⊧ f
⊢ ∃ x ∈ s, ¬x ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | exact ⟨𝓜, h𝓜, hf⟩ | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
𝓜 : M
h𝓜 : 𝓜 ∈ s
f : F
hf : ¬𝓜 ⊧ f
⊢ ∃ x ∈ s, ¬x ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.meaningful_iff_satisfiableSet | [188, 1] | [189, 73] | simp [set_meaningful_iff_nonempty, satisfiableSet_iff_models_nonempty] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
⊢ Satisfiable M T ↔ Meaningful (models M T) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | simp [consequence_iff, Satisfiable] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ T ⊨[M] f ↔ ¬Satisfiable M (insert (~f) T) | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f) ↔ ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | constructor | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f) ↔ ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f) → ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T
case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ (x : M), ¬x ⊧ f → ¬x... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | intro h 𝓜 hf hT | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f) → ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f
𝓜 : M
hf : ¬𝓜 ⊧ f
hT : 𝓜 ⊧* T
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | have : 𝓜 ⊧ f := h hT | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f
𝓜 : M
hf : ¬𝓜 ⊧ f
hT : 𝓜 ⊧* T
⊢ False | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f
𝓜 : M
hf : ¬𝓜 ⊧ f
hT : 𝓜 ⊧* T
this : 𝓜 ⊧ f
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | contradiction | case mp
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f
𝓜 : M
hf : ¬𝓜 ⊧ f
hT : 𝓜 ⊧* T
this : 𝓜 ⊧ f
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | intro h 𝓜 | case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
⊢ (∀ (x : M), ¬x ⊧ f → ¬x ⊧* T) → ∀ {𝓜 : M}, 𝓜 ⊧* T → 𝓜 ⊧ f | case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T
𝓜 : M
⊢ 𝓜 ⊧* T → 𝓜 ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | contrapose | case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T
𝓜 : M
⊢ 𝓜 ⊧* T → 𝓜 ⊧ f | case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T
𝓜 : M
⊢ ¬𝓜 ⊧ f → ¬𝓜 ⊧* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.consequence_iff_not_satisfiable | [196, 1] | [200, 37] | exact h 𝓜 | case mpr
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
inst✝ : Tarski M
f : F
h : ∀ (x : M), ¬x ⊧ f → ¬x ⊧* T
𝓜 : M
⊢ ¬𝓜 ⊧ f → ¬𝓜 ⊧* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | suffices ∀ s d, T s ⊆ T (s + d) by
simpa[Nat.add_sub_of_le h] using this s₁ (s₂ - s₁) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
⊢ T s₁ ⊆ T s₂ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
⊢ ∀ (s d : ℕ), T s ⊆ T (s + d) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | intro s d | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
⊢ ∀ (s d : ℕ), T s ⊆ T (s + d) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s d : ℕ
⊢ T s ⊆ T (s + d) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | induction' d with d ih | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s d : ℕ
⊢ T s ⊆ T (s + d) | case zero
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s : ℕ
⊢ T s ⊆ T (s + 0)
case succ
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s d : ℕ
ih : T s ⊆ T (s... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | simpa[Nat.add_sub_of_le h] using this s₁ (s₂ - s₁) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
this : ∀ (s d : ℕ), T s ⊆ T (s + d)
⊢ T s₁ ⊆ T s₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | simp | case zero
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s : ℕ
⊢ T s ⊆ T (s + 0) | case zero
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s : ℕ
⊢ T s ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | rfl | case zero
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s : ℕ
⊢ T s ⊆ T s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.subset_of_le | [213, 1] | [220, 74] | simpa only [Nat.add_succ, add_zero] using subset_trans ih (H (s + d)) | case succ
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
s₁ s₂ : ℕ
h : s₁ ≤ s₂
s d : ℕ
ih : T s ⊆ T (s + d)
⊢ T s ⊆ T (s + (d + 1)) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | haveI := Classical.decEq | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
⊢ ∃ s, ↑u ⊆ T s | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
this : (α : Sort ?u.87136) → DecidableEq α
⊢ ∃ s, ↑u ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | induction u using Finset.induction | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
this : (α : Sort ?u.87136) → DecidableEq α
⊢ ∃ s, ↑u ⊆ T s | case empty
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
hu : ↑∅ ⊆ ⋃ s, T s
⊢ ∃ s, ↑∅ ⊆ T s
case insert
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Typ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | case empty => exact ⟨0, by simp⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
hu : ↑∅ ⊆ ⋃ s, T s
⊢ ∃ s, ↑∅ ⊆ T s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | case insert f u _ ih =>
simp at hu
have : ∃ s, ↑u ⊆ T s := ih (subset_trans (Set.subset_insert _ _) hu)
rcases this with ⟨s, hs⟩
have : ∃ s', f ∈ T s' := by simpa using (Set.insert_subset_iff.mp hu).1
rcases this with ⟨s', hs'⟩
exact ⟨max s s', by
simp; exact Set.insert_subset
(subset_of_le H (Nat... | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : ↑(insert f u) ⊆ ⋃ s, T s
⊢ ∃ s, ↑(insert f u) ⊆ T s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | exact ⟨0, by simp⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
hu : ↑∅ ⊆ ⋃ s, T s
⊢ ∃ s, ↑∅ ⊆ T s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | simp | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
hu : ↑∅ ⊆ ⋃ s, T s
⊢ ↑∅ ⊆ T 0 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | simp at hu | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : ↑(insert f u) ⊆ ⋃ s, T s
⊢ ∃ s, ↑(insert f u) ⊆ T s | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
⊢ ∃ s, ↑(insert f u) ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | have : ∃ s, ↑u ⊆ T s := ih (subset_trans (Set.subset_insert _ _) hu) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
⊢ ∃ s, ↑(insert f u) ⊆ T s | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this✝ : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
this : ∃ s, ↑u ⊆ T s
⊢ ∃ s, ↑(insert f u) ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | rcases this with ⟨s, hs⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this✝ : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
this : ∃ s, ↑u ⊆ T s
⊢ ∃ s, ↑(insert f u) ⊆ T s | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
⊢ ∃ s, ↑(insert f u) ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | have : ∃ s', f ∈ T s' := by simpa using (Set.insert_subset_iff.mp hu).1 | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
⊢ ∃ s, ↑(insert f u) ⊆ T s | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this✝ : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
this : ∃ s', f ∈ T s'
⊢ ∃ s, ↑(insert f u) ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | rcases this with ⟨s', hs'⟩ | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this✝ : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
this : ∃ s', f ∈ T s'
⊢ ∃ s, ↑(insert f u) ⊆ T s | case intro.intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
s' : ℕ
hs' : f ∈ T s'
⊢ ∃ s, ↑(insert f u) ⊆ T s |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | exact ⟨max s s', by
simp; exact Set.insert_subset
(subset_of_le H (Nat.le_max_right s s') hs')
(subset_trans hs (subset_of_le H $ Nat.le_max_left s s'))⟩ | case intro.intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
s' : ℕ
hs' : f ∈ T s'
⊢ ∃ s, ↑(insert f u) ⊆ T s | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | simpa using (Set.insert_subset_iff.mp hu).1 | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
⊢ ∃ s', f ∈ T s' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | simp | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
s' : ℕ
hs' : f ∈ T s'
⊢ ↑(insert f u) ⊆ T (max s s') | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
s' : ℕ
hs' : f ∈ T s'
⊢ insert f ↑u ⊆ T (max s s') |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Cumulative.finset_mem | [222, 1] | [236, 67] | exact Set.insert_subset
(subset_of_le H (Nat.le_max_right s s') hs')
(subset_trans hs (subset_of_le H $ Nat.le_max_left s s')) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : ℕ → Set F
H : Cumulative T
this : (α : Type u_2) → DecidableEq α
f : F
u : Finset F
a✝ : f ∉ u
ih : ↑u ⊆ ⋃ s, T s → ∃ s, ↑u ⊆ T s
hu : insert f ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
s' : ℕ
hs' : f ∈ T s'
⊢ insert f ↑u ⊆ T (max s s') | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simp [Semantics.consequence_iff_not_satisfiable, compact (T := insert (~f) T)] | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ T ⊨[M] f ↔ ∃ u, ↑u ⊆ T ∧ ↑u ⊨[M] f | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ (∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x) ↔ ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | constructor | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ (∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x) ↔ ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u) | case mp
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ (∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x) → ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u)
case mpr
M : Type u_1
F : Typ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | intro ⟨u, ss, hu⟩ | case mp
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ (∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x) → ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u) | case mp
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | exact ⟨Finset.erase u (~f), by simp [ss],
by simp; intro h; exact hu (Semantics.Satisfiable.of_subset h (by simp))⟩ | case mp
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simp [ss] | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ↑(u.erase (~f)) ⊆ T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simp | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ¬Semantics.Satisfiable M (insert (~f) ↑(u.erase (~f))) | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ¬Semantics.Satisfiable M (insert (~f) ↑u) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | intro h | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
⊢ ¬Semantics.Satisfiable M (insert (~f) ↑u) | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
h : Semantics.Satisfiable M (insert (~f) ↑u)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | exact hu (Semantics.Satisfiable.of_subset h (by simp)) | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
h : Semantics.Satisfiable M (insert (~f) ↑u)
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simp | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ insert (~f) T
hu : ¬Semantics.Satisfiable M ↑u
h : Semantics.Satisfiable M (insert (~f) ↑u)
⊢ ↑u ⊆ insert (~f) ↑u | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | intro ⟨u, ss, hu⟩ | case mpr
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
⊢ (∃ u, ↑u ⊆ T ∧ ¬Semantics.Satisfiable M (insert (~f) ↑u)) → ∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x | case mpr
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ T
hu : ¬Semantics.Satisfiable M (insert (~f) ↑u)
⊢ ∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | exact ⟨insert (~f) u,
by simpa using Set.insert_subset_insert ss, by simpa using hu⟩ | case mpr
M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ T
hu : ¬Semantics.Satisfiable M (insert (~f) ↑u)
⊢ ∃ x, ↑x ⊆ insert (~f) T ∧ ¬Semantics.Satisfiable M ↑x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simpa using Set.insert_subset_insert ss | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ T
hu : ¬Semantics.Satisfiable M (insert (~f) ↑u)
⊢ ↑(insert (~f) u) ⊆ insert (~f) T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.conseq_compact | [254, 1] | [263, 69] | simpa using hu | M : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
𝓢 : Semantics F M
inst✝² : Compact M
𝓜 : M
T : Set F
inst✝¹ : Semantics.Tarski M
inst✝ : DecidableEq F
f : F
u : Finset F
ss : ↑u ⊆ T
hu : ¬Semantics.Satisfiable M (insert (~f) ↑u)
⊢ ¬Semantics.Satisfiable M ↑(insert (~f) u) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | intro H s | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
⊢ Semantics.Satisfiable M (⋃ s, T s) → ∀ (s : ℕ), Semantics.Satisfiable M (T s) | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : Semantics.Satisfiable M (⋃ s, T s)
s : ℕ
⊢ Semantics.Satisfiable M (T s) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | exact H.of_subset (Set.subset_iUnion T s) | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : Semantics.Satisfiable M (⋃ s, T s)
s : ℕ
⊢ Semantics.Satisfiable M (T s) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | intro H | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
⊢ (∀ (s : ℕ), Semantics.Satisfiable M (T s)) → Semantics.Satisfiable M (⋃ s, T s) | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
⊢ Semantics.Satisfiable M (⋃ s, T s) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | apply compact.mpr | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
⊢ Semantics.Satisfiable M (⋃ s, T s) | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
⊢ ∀ (u : Finset F), ↑u ⊆ ⋃ s, T s → Semantics.Satisfiable M ↑u |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | intro u hu | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
⊢ ∀ (u : Finset F), ↑u ⊆ ⋃ s, T s → Semantics.Satisfiable M ↑u | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
⊢ Semantics.Satisfiable M ↑u |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | rcases hT.finset_mem hu with ⟨s, hs⟩ | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
⊢ Semantics.Satisfiable M ↑u | case intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
⊢ Semantics.Satisfiable M ↑u |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Compact.compact_cumulative | [265, 1] | [273, 33] | exact (H s).of_subset hs | case intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Compact M
𝓜 : M
T : ℕ → Set F
hT : Cumulative T
H : ∀ (s : ℕ), Semantics.Satisfiable M (T s)
u : Finset F
hu : ↑u ⊆ ⋃ s, T s
s : ℕ
hs : ↑u ⊆ T s
⊢ Semantics.Satisfiable M ↑u | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.unprovable_iff_isEmpty | [62, 1] | [63, 63] | simp [Provable, Unprovable] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
f : F
⊢ 𝓢 ⊬! f ↔ IsEmpty (𝓢 ⊢ f) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.provableSet_iff | [68, 1] | [70, 79] | simp [ProvableSet, PrfSet, Provable, Classical.nonempty_pi, ←imp_iff_not_or] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
s : Set F
⊢ 𝓢 ⊢!* s ↔ Nonempty (𝓢 ⊢* s) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.not_reducible_iff | [98, 1] | [98, 99] | simp [reducible_iff, Unprovable] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ ¬𝓢 ≤ₛ 𝓣 ↔ ∃ f, 𝓢 ⊢! f ∧ 𝓣 ⊬! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.strictReducible_iff | [100, 1] | [102, 50] | simp [StrictReducible, reducible_iff] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ 𝓢 <ₛ 𝓣 ↔ (∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f) ∧ ∃ f, 𝓢 ⊬! f ∧ 𝓣 ⊢! f | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ (∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f) → ((∃ x, 𝓣 ⊢! x ∧ ¬𝓢 ⊢! x) ↔ ∃ f, 𝓢 ⊬! f ∧ 𝓣 ⊢! f) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.strictReducible_iff | [100, 1] | [102, 50] | intro _ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ (∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f) → ((∃ x, 𝓣 ⊢! x ∧ ¬𝓢 ⊢! x) ↔ ∃ f, 𝓢 ⊬! f ∧ 𝓣 ⊢! f) | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
a✝ : ∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f
⊢ (∃ x, 𝓣 ⊢! x ∧ ¬𝓢 ⊢! x) ↔ ∃ f, 𝓢 ⊬! f ∧ 𝓣 ⊢! f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.strictReducible_iff | [100, 1] | [102, 50] | exact exists_congr (fun _ ↦ by simp [and_comm]) | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
a✝ : ∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f
⊢ (∃ x, 𝓣 ⊢! x ∧ ¬𝓢 ⊢! x) ↔ ∃ f, 𝓢 ⊬! f ∧ 𝓣 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.strictReducible_iff | [100, 1] | [102, 50] | simp [and_comm] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
a✝ : ∀ {f : F}, 𝓢 ⊢! f → 𝓣 ⊢! f
x✝ : F
⊢ 𝓣 ⊢! x✝ ∧ ¬𝓢 ⊢! x✝ ↔ 𝓢 ⊬! x✝ ∧ 𝓣 ⊢! x✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Equiv.iff | [106, 1] | [106, 90] | simp [Equiv, Set.ext_iff, theory] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ 𝓢 =ₛ 𝓣 ↔ ∀ (f : F), 𝓢 ⊢! f ↔ 𝓣 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Equiv.le | [118, 1] | [118, 80] | simp [Equiv, Reducible] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ 𝓢 =ₛ 𝓣 → 𝓢 ≤ₛ 𝓣 | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ theory 𝓢 = theory 𝓣 → theory 𝓢 ⊆ theory 𝓣 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Equiv.le | [118, 1] | [118, 80] | intro e | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
⊢ theory 𝓢 = theory 𝓣 → theory 𝓢 ⊆ theory 𝓣 | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
e : theory 𝓢 = theory 𝓣
⊢ theory 𝓢 ⊆ theory 𝓣 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Equiv.le | [118, 1] | [118, 80] | rw [e] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
𝓣 : T
𝓤 : U
e : theory 𝓢 = theory 𝓣
⊢ theory 𝓢 ⊆ theory 𝓣 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_def | [166, 1] | [167, 59] | simp [Inconsistent] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ Inconsistent 𝓢 ↔ ∀ (f : F), 𝓢 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_iff_theory_eq | [169, 1] | [171, 43] | simp [Inconsistent, Set.ext_iff, theory] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ Inconsistent 𝓢 ↔ theory 𝓢 = Set.univ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.not_inconsistent_iff_consistent | [173, 1] | [175, 40] | rintro ⟨h⟩ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ Consistent 𝓢 → ¬Inconsistent 𝓢 | case mk
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
h : ¬Inconsistent 𝓢
⊢ ¬Inconsistent 𝓢 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.not_inconsistent_iff_consistent | [173, 1] | [175, 40] | exact h | case mk
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
h : ¬Inconsistent 𝓢
⊢ ¬Inconsistent 𝓢 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.not_consistent_iff_inconsistent | [179, 1] | [180, 81] | simp [←not_inconsistent_iff_consistent] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ ¬Consistent 𝓢 ↔ Inconsistent 𝓢 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.consistent_iff_exists_unprovable | [184, 1] | [186, 60] | simp [←not_inconsistent_iff_consistent, inconsistent_def] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ Consistent 𝓢 ↔ ∃ f, 𝓢 ⊬! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_iff_theory_eq_univ | [193, 1] | [194, 92] | simp [inconsistent_def, theory, Set.ext_iff] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 : S
⊢ Inconsistent 𝓢 ↔ theory 𝓢 = Set.univ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Inconsistent.equiv | [198, 1] | [199, 54] | simp [h.theory_eq, h'.theory_eq] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝² : System F S
inst✝¹ : System F T
inst✝ : System F U
𝓢 𝓣 : S
h : Inconsistent 𝓢
h' : Inconsistent 𝓣
f : F
⊢ f ∈ theory 𝓢 ↔ f ∈ theory 𝓣 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.incomplete_iff_exists_undecidable | [354, 1] | [355, 89] | simp [Complete, Undecidable, not_or] | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝³ : System F S
inst✝² : System F T
inst✝¹ : System F U
inst✝ : LogicalConnective F
𝓢 : S
⊢ ¬Complete 𝓢 ↔ ∃ f, Undecidable 𝓢 f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Axiomatized.le_of_subset | [381, 1] | [381, 82] | rintro f ⟨b⟩ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
⊢ 𝓢 ≤ₛ 𝓣 | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
f : F
b : 𝓢 ⊢ f
⊢ f ∈ theory 𝓣 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Axiomatized.le_of_subset | [381, 1] | [381, 82] | exact ⟨weakening h b⟩ | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
f : F
b : 𝓢 ⊢ f
⊢ f ∈ theory 𝓣 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Axiomatized.weakening! | [383, 1] | [383, 91] | rintro ⟨b⟩ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
f : F
⊢ 𝓢 ⊢! f → 𝓣 ⊢! f | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
f : F
b : 𝓢 ⊢ f
⊢ 𝓣 ⊢! f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Axiomatized.weakening! | [383, 1] | [383, 91] | exact ⟨weakening h b⟩ | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁴ : System F S
inst✝³ : System F T
inst✝² : System F U
inst✝¹ : Collection F S
inst✝ : Axiomatized S
𝓢 𝓣 : S
h : 𝓢 ⊆ 𝓣
f : F
b : 𝓢 ⊢ f
⊢ 𝓣 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.StrongCut.cut! | [410, 1] | [411, 52] | rcases hp with ⟨b⟩ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : StrongCut S T
𝓢 : S
𝓣 : T
p : F
H : 𝓢 ⊢!* Collection.set 𝓣
hp : 𝓣 ⊢! p
⊢ 𝓢 ⊢! p | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : StrongCut S T
𝓢 : S
𝓣 : T
p : F
H : 𝓢 ⊢!* Collection.set 𝓣
b : 𝓣 ⊢ p
⊢ 𝓢 ⊢! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.StrongCut.cut! | [410, 1] | [411, 52] | exact ⟨StrongCut.cut H.get b⟩ | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : StrongCut S T
𝓢 : S
𝓣 : T
p : F
H : 𝓢 ⊢!* Collection.set 𝓣
b : 𝓣 ⊢ p
⊢ 𝓢 ⊢! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Compact.finite_provable | [435, 1] | [437, 48] | rcases h with ⟨b⟩ | F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : Compact S
f : F
𝓢 : S
h : 𝓢 ⊢! f
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ 𝓕 ⊢! f | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : Compact S
f : F
𝓢 : S
b : 𝓢 ⊢ f
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ 𝓕 ⊢! f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Compact.finite_provable | [435, 1] | [437, 48] | exact ⟨φ b, φ_subset b, φ_finite b, ⟨φPrf b⟩⟩ | case intro
F : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
inst✝⁶ : System F S
inst✝⁵ : System F T
inst✝⁴ : System F U
inst✝³ : Collection F S
inst✝² : Collection F T
inst✝¹ : Axiomatized S
inst✝ : Compact S
f : F
𝓢 : S
b : 𝓢 ⊢ f
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ 𝓕 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.consistent_iff_unprovable_bot | [468, 1] | [470, 73] | simp [inconsistent_iff_provable_bot, ←not_inconsistent_iff_consistent] | S : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
inst✝¹ : System F S
inst✝ : DeductiveExplosion S
𝓢 : S
⊢ Consistent 𝓢 ↔ 𝓢 ⊬! ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_compact | [476, 1] | [479, 48] | rcases Compact.finite_provable (H ⊥) with ⟨𝓕, h𝓕, fin, h⟩ | S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 : S
H : Inconsistent 𝓢
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ Inconsistent 𝓕 | case intro.intro.intro
S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 : S
H : Inconsistent 𝓢
𝓕 : S
h𝓕 : 𝓕 ⊆ 𝓢
fin : Collection.Finite 𝓕
h : 𝓕 ⊢! ⊥
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ Inco... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_compact | [476, 1] | [479, 48] | exact ⟨𝓕, h𝓕, fin, inconsistent_of_provable h⟩ | case intro.intro.intro
S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 : S
H : Inconsistent 𝓢
𝓕 : S
h𝓕 : 𝓕 ⊆ 𝓢
fin : Collection.Finite 𝓕
h : 𝓕 ⊢! ⊥
⊢ ∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ Inco... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_compact | [476, 1] | [479, 48] | rintro ⟨𝓕, h𝓕, _, H⟩ | S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 : S
⊢ (∃ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 ∧ Inconsistent 𝓕) → Inconsistent 𝓢 | case intro.intro.intro
S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 𝓕 : S
h𝓕 : 𝓕 ⊆ 𝓢
left✝ : Collection.Finite 𝓕
H : Inconsistent 𝓕
⊢ Inconsistent 𝓢 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.inconsistent_compact | [476, 1] | [479, 48] | exact H.of_supset h𝓕 | case intro.intro.intro
S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 𝓕 : S
h𝓕 : 𝓕 ⊆ 𝓢
left✝ : Collection.Finite 𝓕
H : Inconsistent 𝓕
⊢ Inconsistent 𝓢 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.consistent_compact | [481, 1] | [483, 73] | simp [←not_inconsistent_iff_consistent, inconsistent_compact (𝓢 := 𝓢)] | S : Type u_1
F : Type u_2
inst✝⁵ : LogicalConnective F
inst✝⁴ : System F S
inst✝³ : DeductiveExplosion S
inst✝² : Collection F S
inst✝¹ : Axiomatized S
inst✝ : Compact S
𝓢 : S
⊢ Consistent 𝓢 ↔ ∀ 𝓕 ⊆ 𝓢, Collection.Finite 𝓕 → Consistent 𝓕 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Deduction.of_insert! | [501, 1] | [502, 50] | rcases h with ⟨b⟩ | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
h : cons p 𝓢 ⊢! q
⊢ 𝓢 ⊢! p ⟶ q | case intro
S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
b : cons p 𝓢 ⊢ q
⊢ 𝓢 ⊢! p ⟶ q |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Deduction.of_insert! | [501, 1] | [502, 50] | exact ⟨Deduction.ofInsert b⟩ | case intro
S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
b : cons p 𝓢 ⊢ q
⊢ 𝓢 ⊢! p ⟶ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Deduction.inv! | [506, 1] | [507, 45] | rcases h with ⟨b⟩ | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
h : 𝓢 ⊢! p ⟶ q
⊢ cons p 𝓢 ⊢! q | case intro
S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
b : 𝓢 ⊢ p ⟶ q
⊢ cons p 𝓢 ⊢! q |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.System.Deduction.inv! | [506, 1] | [507, 45] | exact ⟨Deduction.inv b⟩ | case intro
S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
inst✝¹ : Cons F S
inst✝ : Deduction S
p q : F
𝓢 : S
b : 𝓢 ⊢ p ⟶ q
⊢ cons p 𝓢 ⊢! q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Sound.consistent_of_meaningful | [536, 1] | [537, 75] | rcases H with ⟨f, hf⟩ | S : Type u_1
F : Type u_2
inst✝⁴ : LogicalConnective F
inst✝³ : System F S
M : Type u_3
inst✝² : Semantics F M
𝓢 𝓣 : S
𝓜 𝓝 : M
inst✝¹ : Sound 𝓢 𝓜
inst✝ : Sound 𝓣 𝓝
H : Semantics.Meaningful 𝓜
h : System.Inconsistent 𝓢
⊢ False | case mk.intro
S : Type u_1
F : Type u_2
inst✝⁴ : LogicalConnective F
inst✝³ : System F S
M : Type u_3
inst✝² : Semantics F M
𝓢 𝓣 : S
𝓜 𝓝 : M
inst✝¹ : Sound 𝓢 𝓜
inst✝ : Sound 𝓣 𝓝
h : System.Inconsistent 𝓢
f : F
hf : ¬𝓜 ⊧ f
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Sound.consistent_of_meaningful | [536, 1] | [537, 75] | exact hf (Sound.sound (h f)) | case mk.intro
S : Type u_1
F : Type u_2
inst✝⁴ : LogicalConnective F
inst✝³ : System F S
M : Type u_3
inst✝² : Semantics F M
𝓢 𝓣 : S
𝓜 𝓝 : M
inst✝¹ : Sound 𝓢 𝓜
inst✝ : Sound 𝓣 𝓝
h : System.Inconsistent 𝓢
f : F
hf : ¬𝓜 ⊧ f
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | contrapose | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
⊢ System.Consistent 𝓢 → Semantics.Meaningful 𝓜 | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
⊢ ¬Semantics.Meaningful 𝓜 → ¬System.Consistent 𝓢 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | intro h | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
⊢ ¬Semantics.Meaningful 𝓜 → ¬System.Consistent 𝓢 | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
h : ¬Semantics.Meaningful 𝓜
⊢ ¬System.Consistent 𝓢 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | simp [Semantics.not_meaningful_iff, System.not_consistent_iff_inconsistent] at h ⊢ | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
h : ¬Semantics.Meaningful 𝓜
⊢ ¬System.Consistent 𝓢 | S : Type u_1
F : Type u_2
inst✝³ : LogicalConnective F
inst✝² : System F S
M : Type u_3
inst✝¹ : Semantics F M
𝓢 : S
𝓜 : M
inst✝ : Complete 𝓢 𝓜
h : ∀ (f : F), 𝓜 ⊧ f
⊢ System.Inconsistent 𝓢 |
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