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/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | intro f | 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 | 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
f : F
⊢ 𝓢 ⊢! f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | exact Complete.complete (h f) | 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
f : F
⊢ 𝓢 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.inconsistent_of_unsatisfiable | [582, 1] | [583, 94] | contrapose | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬Semantics.Satisfiable M T → System.Inconsistent 𝓢 | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬System.Inconsistent 𝓢 → ¬¬Semantics.Satisfiable M T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.inconsistent_of_unsatisfiable | [582, 1] | [583, 94] | simpa [←System.not_consistent_iff_inconsistent] using satisfiable_of_consistent | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬System.Inconsistent 𝓢 → ¬¬Semantics.Satisfiable M T | 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 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | intro f | 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 | 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
f : F
⊢ 𝓢 ⊢! f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.meaningful_of_consistent | [566, 1] | [569, 41] | exact Complete.complete (h f) | 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
f : F
⊢ 𝓢 ⊢! f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.inconsistent_of_unsatisfiable | [582, 1] | [583, 94] | contrapose | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬Semantics.Satisfiable M T → System.Inconsistent 𝓢 | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬System.Inconsistent 𝓢 → ¬¬Semantics.Satisfiable M T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/System.lean | LO.Complete.inconsistent_of_unsatisfiable | [582, 1] | [583, 94] | simpa [←System.not_consistent_iff_inconsistent] using satisfiable_of_consistent | 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
inst✝¹ : ∀ (𝓜 : M), Semantics.Meaningful 𝓜
𝓢 : S
T : Set F
inst✝ : Complete 𝓢 (Semantics.models M T)
⊢ ¬System.Inconsistent 𝓢 → ¬¬Semantics.Satisfiable M T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.FiniteContext.not_mem_empty | [39, 1] | [39, 112] | simp [EmptyCollection.emptyCollection] | F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
p : F
⊢ p ∉ ∅ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.FiniteContext.by_axm! | [94, 1] | [94, 94] | simpa | F : Type u_1
inst✝³ : LogicalConnective F
inst✝² : DecidableEq F
S : Type u_2
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
Γ Δ E : List F
p : F
h : autoParam (p ∈ Γ) _auto✝
⊢ p ∈ Collection.set { ctx := Γ } | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.FiniteContext.weakening! | [98, 1] | [98, 106] | simpa | F : Type u_1
inst✝³ : LogicalConnective F
inst✝² : DecidableEq F
S : Type u_2
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
Γ Δ E : List F
h✝ : Γ ⊆ Δ
p : F
h : Γ ⊢[𝓢]! p
⊢ { ctx := Γ } ⊆ { ctx := Δ } | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.FiniteContext.of'! | [107, 1] | [107, 89] | simp | F : Type u_1
inst✝³ : LogicalConnective F
inst✝² : DecidableEq F
S : Type u_2
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
Γ Δ E : List F
p : F
h : 𝓢 ⊢! p
⊢ [] ⊆ Γ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.not_mem_empty | [213, 1] | [213, 119] | simp [EmptyCollection.emptyCollection, Set.mem_def] | F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
p : F
⊢ p ∉ ∅ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.provable_iff | [252, 1] | [253, 87] | rintro ⟨⟨Δ, h, b⟩⟩ | F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
⊢ Γ *⊢[𝓢]! p → ∃ Δ, (∀ q ∈ Δ, q ∈ Γ) ∧ Δ ⊢[𝓢]! p | case intro.mk
F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
Δ : List F
h : ∀ q ∈ Δ, q ∈ { ctx := Γ }
b : Δ ⊢[𝓢] p
⊢ ∃ Δ, (∀ q ∈ Δ, q ∈ Γ) ∧ Δ ⊢[𝓢]! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.provable_iff | [252, 1] | [253, 87] | exact ⟨Δ, h, ⟨b⟩⟩ | case intro.mk
F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
Δ : List F
h : ∀ q ∈ Δ, q ∈ { ctx := Γ }
b : Δ ⊢[𝓢] p
⊢ ∃ Δ, (∀ q ∈ Δ, q ∈ Γ) ∧ Δ ⊢[𝓢]! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.provable_iff | [252, 1] | [253, 87] | rintro ⟨Δ, h, ⟨d⟩⟩ | F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
⊢ (∃ Δ, (∀ q ∈ Δ, q ∈ Γ) ∧ Δ ⊢[𝓢]! p) → Γ *⊢[𝓢]! p | case intro.intro.intro
F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
Δ : List F
h : ∀ q ∈ Δ, q ∈ Γ
d : { ctx := Δ } ⊢ p
⊢ Γ *⊢[𝓢]! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.provable_iff | [252, 1] | [253, 87] | exact ⟨⟨Δ, h, d⟩⟩ | case intro.intro.intro
F : Type u_1
inst✝² : LogicalConnective F
inst✝¹ : DecidableEq F
S : Type u_2
inst✝ : System F S
𝓢 : S
Γ : Set F
p : F
Δ : List F
h : ∀ q ∈ Δ, q ∈ Γ
d : { ctx := Δ } ⊢ p
⊢ Γ *⊢[𝓢]! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/HilbertStyle/Context.lean | LO.System.Context.by_axm! | [302, 1] | [302, 69] | simpa | F : Type u_1
inst✝³ : LogicalConnective F
inst✝² : DecidableEq F
S : Type u_2
inst✝¹ : System F S
𝓢 : S
inst✝ : Minimal 𝓢
p : F
Γ : Set F
h : p ∈ Γ
⊢ p ∈ Collection.set { ctx := Γ } | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | serial_of_refl | [46, 1] | [49, 17] | rintro w | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Serial rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w : α
⊢ ∃ w₂, rel w w₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | serial_of_refl | [46, 1] | [49, 17] | existsi w | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w : α
⊢ ∃ w₂, rel w w₂ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w : α
⊢ rel w w |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | serial_of_refl | [46, 1] | [49, 17] | exact hRefl w | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w : α
⊢ rel w w | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | eucl_of_symm_trans | [52, 1] | [55, 32] | intro w₁ w₂ w₃ r₁₂ r₁₃ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Euclidean rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₁₃ : rel w₁ w₃
⊢ rel w₃ w₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | eucl_of_symm_trans | [52, 1] | [55, 32] | have r₂₁ := hSymm r₁₂ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₁₃ : rel w₁ w₃
⊢ rel w₃ w₂ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₁₃ : rel w₁ w₃
r₂₁ : rel w₂ w₁
⊢ rel w₃ w₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | eucl_of_symm_trans | [52, 1] | [55, 32] | exact hSymm $ hTrans r₂₁ r₁₃ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₁₃ : rel w₁ w₃
r₂₁ : rel w₂ w₁
⊢ rel w₃ w₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | trans_of_symm_eucl | [58, 1] | [60, 39] | rintro w₁ w₂ w₃ r₁₂ r₂₃ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Transitive rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₂₃ : rel w₂ w₃
⊢ rel w₁ w₃ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | trans_of_symm_eucl | [58, 1] | [60, 39] | exact hSymm $ hEucl (hSymm r₁₂) r₂₃ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ w₃ : α
r₁₂ : rel w₁ w₂
r₂₃ : rel w₂ w₃
⊢ rel w₁ w₃ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | symm_of_refl_eucl | [63, 1] | [65, 30] | intro w₁ w₂ r₁₂ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Symmetric rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
⊢ rel w₂ w₁ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | symm_of_refl_eucl | [63, 1] | [65, 30] | exact hEucl (hRefl w₁) r₁₂ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
⊢ rel w₂ w₁ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | trans_of_refl_eucl | [68, 1] | [70, 40] | have hSymm := symm_of_refl_eucl hRefl hEucl | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Transitive rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm✝ : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
hSymm : Symmetric rel
⊢ Transitive rel |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | trans_of_refl_eucl | [68, 1] | [70, 40] | exact trans_of_symm_eucl hSymm hEucl | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm✝ : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
hSymm : Symmetric rel
⊢ Transitive rel | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | refl_of_symm_serial_eucl | [73, 1] | [77, 48] | rintro w₁ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
⊢ Reflexive rel | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ : α
⊢ rel w₁ w₁ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | refl_of_symm_serial_eucl | [73, 1] | [77, 48] | obtain ⟨w₂, r₁₂⟩ := hSerial w₁ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ : α
⊢ rel w₁ w₁ | case intro
α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
⊢ rel w₁ w₁ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | refl_of_symm_serial_eucl | [73, 1] | [77, 48] | have r₂₁ := hSymm r₁₂ | case intro
α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
⊢ rel w₁ w₁ | case intro
α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
r₂₁ : rel w₂ w₁
⊢ rel w₁ w₁ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | refl_of_symm_serial_eucl | [73, 1] | [77, 48] | exact trans_of_symm_eucl hSymm hEucl r₁₂ r₂₁ | case intro
α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
w₁ w₂ : α
r₁₂ : rel w₁ w₂
r₂₁ : rel w₂ w₁
⊢ rel w₁ w₁ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | ConverseWellFounded.iff_has_max | [82, 1] | [83, 72] | simp [ConverseWellFounded, WellFounded.wellFounded_iff_has_min, flip] | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
r : α → α → Prop
⊢ ConverseWellFounded r ↔ ∀ (s : Set α), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬r m x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | Finite.converseWellFounded_of_trans_irrefl | [87, 1] | [90, 38] | apply @Finite.wellFounded_of_trans_of_irrefl _ _ _
⟨by intro a b c rba rcb; exact IsTrans.trans c b a rcb rba⟩
⟨by simp [flip, IsIrrefl.irrefl]⟩ | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
inst✝² : Finite α
inst✝¹ : IsTrans α rel
inst✝ : IsIrrefl α rel
⊢ ConverseWellFounded rel | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | Finite.converseWellFounded_of_trans_irrefl | [87, 1] | [90, 38] | intro a b c rba rcb | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
inst✝² : Finite α
inst✝¹ : IsTrans α rel
inst✝ : IsIrrefl α rel
⊢ ∀ (a b c : α),
flip (fun x x_1 => rel x x_1) a b → flip (fun x x_1 => rel x x_1) b c → flip (fun x x_1 => rel... | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
inst✝² : Finite α
inst✝¹ : IsTrans α rel
inst✝ : IsIrrefl α rel
a b c : α
rba : flip (fun x x_1 => rel x x_1) a b
rcb : flip (fun x x_1 => rel x x_1) b c
⊢ flip (fun x x_1 => rel ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | Finite.converseWellFounded_of_trans_irrefl | [87, 1] | [90, 38] | exact IsTrans.trans c b a rcb rba | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
inst✝² : Finite α
inst✝¹ : IsTrans α rel
inst✝ : IsIrrefl α rel
a b c : α
rba : flip (fun x x_1 => rel x x_1) a b
rcb : flip (fun x x_1 => rel x x_1) b c
⊢ flip (fun x x_1 => rel ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | Finite.converseWellFounded_of_trans_irrefl | [87, 1] | [90, 38] | simp [flip, IsIrrefl.irrefl] | α : Type u
rel : α → α → Prop
hRefl : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl : Euclidean rel
inst✝² : Finite α
inst✝¹ : IsTrans α rel
inst✝ : IsIrrefl α rel
⊢ ∀ (a : α), ¬flip (fun x x_1 => rel x x_1) a a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | extensive_of_reflex_antisymm_eucl | [95, 1] | [98, 39] | intro x y rxy | α : Type u
rel : α → α → Prop
hRefl✝ : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl✝ : Euclidean rel
hRefl : Reflexive rel
hAntisymm : Antisymmetric rel
hEucl : Euclidean rel
⊢ Extensive rel | α : Type u
rel : α → α → Prop
hRefl✝ : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl✝ : Euclidean rel
hRefl : Reflexive rel
hAntisymm : Antisymmetric rel
hEucl : Euclidean rel
x y : α
rxy : rel x y
⊢ x = y |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | extensive_of_reflex_antisymm_eucl | [95, 1] | [98, 39] | have rxx := hRefl x | α : Type u
rel : α → α → Prop
hRefl✝ : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl✝ : Euclidean rel
hRefl : Reflexive rel
hAntisymm : Antisymmetric rel
hEucl : Euclidean rel
x y : α
rxy : rel x y
⊢ x = y | α : Type u
rel : α → α → Prop
hRefl✝ : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl✝ : Euclidean rel
hRefl : Reflexive rel
hAntisymm : Antisymmetric rel
hEucl : Euclidean rel
x y : α
rxy : rel x y
rxx : rel x x
⊢ x = y |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Vorspiel/BinaryRelations.lean | extensive_of_reflex_antisymm_eucl | [95, 1] | [98, 39] | exact hAntisymm rxy (hEucl rxx rxy) | α : Type u
rel : α → α → Prop
hRefl✝ : Reflexive rel
hSymm : Symmetric rel
hSerial : Serial rel
hTrans : Transitive rel
hEucl✝ : Euclidean rel
hRefl : Reflexive rel
hAntisymm : Antisymmetric rel
hEucl : Euclidean rel
x y : α
rxy : rel x y
rxx : rel x x
⊢ x = y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | contrapose | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ F ⊧* 𝗟 → Transitive F.Rel | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ ¬Transitive F.Rel → ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | intro hT | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ ¬Transitive F.Rel → ¬F ⊧* 𝗟 | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hT : ¬Transitive F.Rel
⊢ ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | simp [Transitive] at hT | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hT : ¬Transitive F.Rel
⊢ ¬F ⊧* 𝗟 | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hT : ∃ x x_1, F.Rel x x_1 ∧ ∃ x_2, F.Rel x_1 x_2 ∧ ¬F.Rel x x_2
⊢ ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | obtain ⟨w₁, w₂, r₁₂, w₃, r₂₃, nr₁₃⟩ := hT | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hT : ∃ x x_1, F.Rel x x_1 ∧ ∃ x_2, F.Rel x_1 x_2 ∧ ¬F.Rel x x_2
⊢ ¬F ⊧* 𝗟 | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | simp only [Semantics.RealizeSet.setOf_iff, ValidOnFrame.models_iff, ValidOnFrame,
ValidOnModel.iff_models, ValidOnModel, Satisfies.iff_models, forall_exists_index,
forall_apply_eq_imp_iff, Satisfies.imp_def, Satisfies.box_def, not_forall, exists_prop] | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ¬F ⊧* 𝗟 | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∃ x x_1 x_2,
(∀ (w' : F.World),
x_2 ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | existsi (atom default), (λ w' _ => w' ≠ w₂ ∧ w' ≠ w₃), w₁ | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∃ x x_1 x_2,
(∀ (w' : F.World),
x_2 ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation... | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ (∀ (w' : F.World),
w₁ ≺ w' →
(∀ (w'_1 : F.World),
w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w'... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | constructor | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ (∀ (w' : F.World),
w₁ ≺ w' →
(∀ (w'_1 : F.World),
w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w'... | case intro.intro.intro.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∀ (w' : F.World),
w₁ ≺ w' →
(∀ (w'_1 : F.World),
w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w' x... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | . intro x hx h;
by_cases hx₂ : x = w₂;
. subst hx₂; simpa [Satisfies] using h w₃ r₂₃;
. by_cases hx₃ : x = w₃ <;> simp_all [Satisfies, hx₃]; | case intro.intro.intro.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∀ (w' : F.World),
w₁ ≺ w' →
(∀ (w'_1 : F.World),
w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w' x... | case intro.intro.intro.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∃ x, w₁ ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } x (atom default) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | . existsi w₂; simpa [Satisfies]; | case intro.intro.intro.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∃ x, w₁ ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } x (atom default) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | intro x hx h | case intro.intro.intro.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∀ (w' : F.World),
w₁ ≺ w' →
(∀ (w'_1 : F.World),
w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w' x... | case intro.intro.intro.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | by_cases hx₂ : x = w₂ | case intro.intro.intro.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (... | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : x = w₂
⊢ Sa... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | . subst hx₂; simpa [Satisfies] using h w₃ r₂₃; | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : x = w₂
⊢ Sa... | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : ¬x = w₂
⊢ S... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | . by_cases hx₃ : x = w₃ <;> simp_all [Satisfies, hx₃]; | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : ¬x = w₂
⊢ S... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | subst hx₂ | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : x = w₂
⊢ Sa... | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₃ : F.World
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
r₁₂ : F.Rel w₁ x
r₂₃ : F.Rel x w₃
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x_1 => w' ≠ x ∧ w' ≠ w₃ } w' (atom default)
⊢ Satisfies { Frame := F, Valua... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | simpa [Satisfies] using h w₃ r₂₃ | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₃ : F.World
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
r₁₂ : F.Rel w₁ x
r₂₃ : F.Rel x w₃
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x_1 => w' ≠ x ∧ w' ≠ w₃ } w' (atom default)
⊢ Satisfies { Frame := F, Valua... | no goals |
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