url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | first
| apply ValidOnFrame.verum
| apply ValidOnFrame.imply₁
| apply ValidOnFrame.imply₂
| apply ValidOnFrame.disj₁
| apply ValidOnFrame.disj₂
| apply ValidOnFrame.disj₃
| apply ValidOnFrame.conj₁
| apply ValidOnFrame.conj₂
| apply ValidOnFrame.conj₃ | case disj₃
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ r✝ : Formula α
⊢ F ⊧ (p✝ ⟶ r✝) ⟶ (q✝ ⟶ r✝) ⟶ p✝ ⋎ q✝ ⟶ r✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.verum | case verum
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
⊢ F ⊧ ⊤ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.imply₁ | case imply₁
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ p✝ ⟶ q✝ ⟶ p✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.imply₂ | case imply₂
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ r✝ : Formula α
⊢ F ⊧ (p✝ ⟶ q✝ ⟶ r✝) ⟶ (p✝ ⟶ q✝) ⟶ p✝ ⟶ r✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.disj₁ | case disj₁
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ p✝ ⟶ p✝ ⋎ q✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.disj₂ | case disj₂
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ q✝ ⟶ p✝ ⋎ q✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.disj₃ | case disj₃
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ r✝ : Formula α
⊢ F ⊧ (p✝ ⟶ r✝) ⟶ (q✝ ⟶ r✝) ⟶ p✝ ⋎ q✝ ⟶ r✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.conj₁ | case conj₁
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ p✝ ⋏ q✝ ⟶ p✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.conj₂ | case conj₂
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ p✝ ⋏ q✝ ⟶ q✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.sound | [14, 1] | [31, 31] | apply ValidOnFrame.conj₃ | case conj₃
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
p : Formula α
F : Frame' α
hF : F ∈ 𝔽(Ax(𝓓))
p✝ q✝ : Formula α
⊢ F ⊧ p✝ ⟶ q✝ ⟶ p✝ ⋏ q✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.unprovable_bot | [37, 1] | [40, 27] | intro h | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
ne : IsNonempty 𝔽(Ax(𝓓))
⊢ 𝓓 ⊬! ⊥ | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
ne : IsNonempty 𝔽(Ax(𝓓))
h : 𝓓 ⊢! ⊥
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.unprovable_bot | [37, 1] | [40, 27] | obtain ⟨F, hF⟩ := ne | α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
ne : IsNonempty 𝔽(Ax(𝓓))
h : 𝓓 ⊢! ⊥
⊢ False | case mk.intro
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
h : 𝓓 ⊢! ⊥
F : Frame
hF : F ∈ 𝔽(Ax(𝓓))
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Soundness.lean | LO.Propositional.Superintuitionistic.Kripke.unprovable_bot | [37, 1] | [40, 27] | simpa using sound! h hF | case mk.intro
α : Type u_1
inst✝ : Inhabited α
𝓓 : DeductionParameter α
h : 𝓓 ⊢! ⊥
F : Frame
hF : F ∈ 𝔽(Ax(𝓓))
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.completeness_of_encodable | [29, 1] | [33, 23] | have : T ⊢'' [Rew.emb.hom σ] :=
Disjconseq.completeness_of_encodable (T := T) (Γ := {σ}) (fun M i s hM ↦ ⟨σ, List.mem_of_mem_head? rfl, h hM⟩) | L : Language
T : Theory L
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
σ : Sentence L
h : T ⊨ σ
⊢ T ⊢! σ | L : Language
T : Theory L
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
σ : Sentence L
h : T ⊨ σ
this : T ⊢'' [Rew.emb.hom σ]
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.completeness_of_encodable | [29, 1] | [33, 23] | exact ⟨toProof this⟩ | L : Language
T : Theory L
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
σ : Sentence L
h : T ⊨ σ
this : T ⊢'' [Rew.emb.hom σ]
⊢ T ⊢! σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | letI := Classical.typeDecidableEq | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
⊢ T ⊢! σ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : ∃ u : Finset (Sentence L), ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable (u : Theory L) := by
simpa using (compact (T := insert (~σ) T)).not.mp (Semantics.consequence_iff_not_satisfiable.mp h) | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
⊢ T ⊢! σ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝ : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
this : ∃ u, ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable ↑u
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | choose u hu using this | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝ : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
this : ∃ u, ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable ↑u
⊢ T ⊢! σ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
hu : ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable ↑u
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | rcases hu with ⟨ssu, hu⟩ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
hu : ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable ↑u
⊢ T ⊢! σ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | haveI : ∀ k, Encodable ((languageFinset u).Func k) := fun _ ↦ Fintype.toEncodable _ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
⊢ T ⊢! σ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this : (k : ℕ) → Encodable ((languageFinset u).Func k)
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | haveI : ∀ k, Encodable ((languageFinset u).Rel k) := fun _ ↦ Fintype.toEncodable _ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this : (k : ℕ) → Encodable ((languageFinset u).Func k)
⊢ T ⊢! σ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
⊢ T ⊢! σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | let u' : Finset (Sentence (languageFinset u)) := Finset.imageOfFinset u (fun _ hσ ↦ toSubLanguageFinsetSelf hσ) | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
⊢ T ⊢! σ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have image_u' : u'.image (Semiformula.lMap L.ofSubLanguage) = u := by
ext τ; simp [u', Finset.mem_imageOfFinset_iff]
exact ⟨by rintro ⟨a, ⟨τ, hτ, rfl⟩, rfl⟩; simp[hτ],
by intro hτ; exact ⟨toSubLanguageFinsetSelf hτ, ⟨τ, hτ, rfl⟩, Semiformula.lMap_toSubLanguageFinsetSelf hτ⟩⟩ | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset... | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : ¬Satisfiable (u' : Theory (languageFinset u)) := by
intro h
have : Satisfiable (u : Theory L) := by
rw[←image_u']
simpa using (satisfiable_lMap L.ofSubLanguage (fun k ↦ Subtype.val_injective) (fun _ ↦ Subtype.val_injective) h)
contradiction | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset... | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝² : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fins... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : System.Inconsistent (u' : Theory (languageFinset u)) := Complete.inconsistent_of_unsatisfiable this | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝² : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fins... | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝³ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝² : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : System.Inconsistent (u : Theory L) := by rw[←image_u']; simpa using System.inconsistent_lMap L.ofSubLanguage this | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝³ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝² : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝⁴ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝³ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝² : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : System.Inconsistent (insert (~σ) T) := this.of_supset ssu | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝⁴ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝³ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝² : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝⁵ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝⁴ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝³ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | exact Gentzen.provable_iff_inconsistent.mpr this | case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝⁵ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝⁴ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝³ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Fin... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | simpa using (compact (T := insert (~σ) T)).not.mp (Semantics.consequence_iff_not_satisfiable.mp h) | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this : (α : Sort ?u.10884) → DecidableEq α := Classical.typeDecidableEq
⊢ ∃ u, ↑u ⊆ insert (~σ) T ∧ ¬Satisfiable ↑u | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | ext τ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... | case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Se... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | simp [u', Finset.mem_imageOfFinset_iff] | case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Se... | case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Se... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | exact ⟨by rintro ⟨a, ⟨τ, hτ, rfl⟩, rfl⟩; simp[hτ],
by intro hτ; exact ⟨toSubLanguageFinsetSelf hτ, ⟨τ, hτ, rfl⟩, Semiformula.lMap_toSubLanguageFinsetSelf hτ⟩⟩ | case a
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Se... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | rintro ⟨a, ⟨τ, hτ, rfl⟩, rfl⟩ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... | case intro.intro.intro.intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | simp[hτ] | case intro.intro.intro.intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | intro hτ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | exact ⟨toSubLanguageFinsetSelf hτ, ⟨τ, hτ, rfl⟩, Semiformula.lMap_toSubLanguageFinsetSelf hτ⟩ | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | intro h | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence ... | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | have : Satisfiable (u : Theory L) := by
rw[←image_u']
simpa using (satisfiable_lMap L.ofSubLanguage (fun k ↦ Subtype.val_injective) (fun _ ↦ Subtype.val_injective) h) | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence... | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝² : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Senten... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | contradiction | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝² : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Senten... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | rw[←image_u'] | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence... | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | simpa using (satisfiable_lMap L.ofSubLanguage (fun k ↦ Subtype.val_injective) (fun _ ↦ Subtype.val_injective) h) | L : Language
T : Theory L
σ : Sentence L
h✝ : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | rw[←image_u'] | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝³ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝² : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Senten... | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝³ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝² : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Senten... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.complete | [39, 1] | [61, 51] | simpa using System.inconsistent_lMap L.ofSubLanguage this | L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝³ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝² : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Senten... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.satisfiable_of_consistent | [71, 1] | [73, 122] | let ⟨M, _, _, h⟩ := satisfiable_iff.mp (satisfiable_of_consistent' h) | L : Language
T : Theory L
h : System.Consistent T
⊢ Semantics.Satisfiable (Struc L) T | L : Language
T : Theory L
h✝ : System.Consistent T
M : Type u
w✝¹ : Nonempty M
w✝ : Structure L M
h : M ⊧ₘ* T
⊢ Semantics.Satisfiable (Struc L) T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.satisfiable_of_consistent | [71, 1] | [73, 122] | exact satisfiable_iff.mpr ⟨ULift.{w} M, inferInstance, inferInstance, ((uLift_elementaryEquiv L M).modelsTheory).mpr h⟩ | L : Language
T : Theory L
h✝ : System.Consistent T
M : Type u
w✝¹ : Nonempty M
w✝ : Structure L M
h : M ⊧ₘ* T
⊢ Semantics.Satisfiable (Struc L) T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.satidfiable_iff_satisfiable | [80, 1] | [81, 72] | simp [satisfiable_iff_consistent'.{u, w}, satisfiable_iff_consistent] | L : Language
T : Theory L
⊢ Semantics.Satisfiable (Struc L) T ↔ Satisfiable T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Completeness.lean | LO.FirstOrder.consequence_iff_consequence | [83, 1] | [84, 87] | simp [Semantics.consequence_iff_not_satisfiable, satidfiable_iff_satisfiable.{u, w}] | L : Language
T : Theory L
σ : Sentence L
⊢ T ⊨[Struc L] σ ↔ T ⊨ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₀ | [92, 1] | [93, 79] | simp [val_func, Matrix.empty_eq] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 0
v : Fin 0 → Semiterm L μ n
⊢ val s e ε (func f v) = Structure.func f ![] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₁ | [95, 1] | [97, 80] | simp [val_func] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
⊢ val s e ε (func f ![t]) = Structure.func f ![val s e ε t] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
⊢ (Structure.func f fun i => val s e ε t) = Structure.func f ![val s e ε t] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₁ | [95, 1] | [97, 80] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
⊢ (Structure.func f fun i => val s e ε t) = Structure.func f ![val s e ε t] | case e_a
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
⊢ (fun i => val s e ε t) = ![val s e ε t] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₁ | [95, 1] | [97, 80] | funext i | case e_a
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
⊢ (fun i => val s e ε t) = ![val s e ε t] | case e_a.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
i : Fin 1
⊢ val s e ε t = ![val s e ε t] i |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₁ | [95, 1] | [97, 80] | cases' i using Fin.cases with i <;> simp | case e_a.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 1
t : Semiterm L μ n
i : Fin 1
⊢ val s e ε t = ![val s e ε t] i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₂ | [99, 1] | [101, 80] | simp [val_func] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
⊢ val s e ε (func f ![t, u]) = Structure.func f ![val s e ε t, val s e ε u] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
⊢ (Structure.func f fun i => val s e ε (![t, u] i)) = Structure.func f ![val s e ε t, val s e ε u] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₂ | [99, 1] | [101, 80] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
⊢ (Structure.func f fun i => val s e ε (![t, u] i)) = Structure.func f ![val s e ε t, val s e ε u] | case e_a
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
⊢ (fun i => val s e ε (![t, u] i)) = ![val s e ε t, val s e ε u] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₂ | [99, 1] | [101, 80] | funext i | case e_a
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
⊢ (fun i => val s e ε (![t, u] i)) = ![val s e ε t, val s e ε u] | case e_a.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
i : Fin 2
⊢ val s e ε (![t, u] i) = ![val s e ε t, val s e ε u] i |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_func₂ | [99, 1] | [101, 80] | cases' i using Fin.cases with i <;> simp | case e_a.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : L.Func 2
t u : Semiterm L μ n
i : Fin 2
⊢ val s e ε (![t, u] i) = ![val s e ε t, val s e ε u] i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_rew | [103, 1] | [105, 50] | induction t <;> simp [*, Rew.func, val_func] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ω : Rew L μ₁ n₁ μ₂ n₂
t : Semiterm L μ₁ n₁
⊢ val s e₂ ε₂ (ω t) = val s (val s e₂ ε₂ ∘ ⇑ω ∘ bvar) (val s e₂ ε₂ ∘ ⇑ω ∘ fvar) t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_rewrite | [107, 1] | [109, 27] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → Semiterm L μ₂ n
t : Semiterm L μ₁ n
⊢ val s e ε₂ ((Rew.rewrite f) t) = val s e (fun x => val s e ε₂ (f x)) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → Semiterm L μ₂ n
t : Semiterm L μ₁ n
⊢ val s (val s e ε₂ ∘ ⇑(Rew.rewrite f) ∘ bvar) (val s e ε₂ ∘ ⇑(Rew.rewrite f) ∘ fvar) t =
val s e (f... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_rewrite | [107, 1] | [109, 27] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → Semiterm L μ₂ n
t : Semiterm L μ₁ n
⊢ val s (val s e ε₂ ∘ ⇑(Rew.rewrite f) ∘ bvar) (val s e ε₂ ∘ ⇑(Rew.rewrite f) ∘ fvar) t =
val s e (f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_rewriteMap | [111, 1] | [113, 27] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → μ₂
t : Semiterm L μ₁ n
⊢ val s e ε₂ ((Rew.rewriteMap f) t) = val s e (fun x => ε₂ (f x)) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → μ₂
t : Semiterm L μ₁ n
⊢ val s (val s e ε₂ ∘ ⇑(Rew.rewriteMap f) ∘ bvar) (val s e ε₂ ∘ ⇑(Rew.rewriteMap f) ∘ fvar) t =
val s e (fun x =>... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_rewriteMap | [111, 1] | [113, 27] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
f : μ₁ → μ₂
t : Semiterm L μ₁ n
⊢ val s (val s e ε₂ ∘ ⇑(Rew.rewriteMap f) ∘ bvar) (val s e ε₂ ∘ ⇑(Rew.rewriteMap f) ∘ fvar) t =
val s e (fun x =>... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_substs | [115, 1] | [117, 27] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
w : Fin n₁ → Semiterm L μ n₂
t : Semiterm L μ n₁
⊢ val s e₂ ε ((Rew.substs w) t) = val s (fun x => val s e₂ ε (w x)) ε t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
w : Fin n₁ → Semiterm L μ n₂
t : Semiterm L μ n₁
⊢ val s (val s e₂ ε ∘ ⇑(Rew.substs w) ∘ bvar) (val s e₂ ε ∘ ⇑(Rew.substs w) ∘ fvar) t =
val s (f... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_substs | [115, 1] | [117, 27] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
w : Fin n₁ → Semiterm L μ n₂
t : Semiterm L μ n₁
⊢ val s (val s e₂ ε ∘ ⇑(Rew.substs w) ∘ bvar) (val s e₂ ε ∘ ⇑(Rew.substs w) ∘ fvar) t =
val s (f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_bShift | [119, 1] | [120, 86] | simp [val_rew, Function.comp] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
a : M
t : Semiterm L μ n
⊢ val s (a :> e) ε (Rew.bShift t) = val s e ε t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_bShift' | [122, 1] | [123, 88] | simp [val_rew, Function.comp] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin (n + 1) → M
t : Semiterm L μ n
⊢ val s e ε (Rew.bShift t) = val s (fun x => e x.succ) ε t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_emb | [125, 1] | [127, 55] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s e ε (Rew.emb t) = val s e (fun a => i.elim a) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s (val s e ε ∘ ⇑Rew.emb ∘ bvar) (val s e ε ∘ ⇑Rew.emb ∘ fvar) t = val s e (fun a => i.elim a) t |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_emb | [125, 1] | [127, 55] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s (val s e ε ∘ ⇑Rew.emb ∘ bvar) (val s e ε ∘ ⇑Rew.emb ∘ fvar) t = val s e (fun a => i.elim a) t | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s e ε ∘ ⇑Rew.emb ∘ fvar = fun a => i.elim a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_emb | [125, 1] | [127, 55] | { funext x; exact i.elim' x } | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s e ε ∘ ⇑Rew.emb ∘ fvar = fun a => i.elim a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_emb | [125, 1] | [127, 55] | funext x | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
⊢ val s e ε ∘ ⇑Rew.emb ∘ fvar = fun a => i.elim a | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
x : o
⊢ (val s e ε ∘ ⇑Rew.emb ∘ fvar) x = i.elim x |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_emb | [125, 1] | [127, 55] | exact i.elim' x | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
o : Type v'
i : IsEmpty o
t : Semiterm L o n
x : o
⊢ (val s e ε ∘ ⇑Rew.emb ∘ fvar) x = i.elim x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_castLE | [129, 1] | [131, 24] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
h : n₁ ≤ n₂
t : Semiterm L μ n₁
⊢ val s e₂ ε ((Rew.castLE h) t) = val s (fun x => e₂ (Fin.castLE h x)) ε t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
h : n₁ ≤ n₂
t : Semiterm L μ n₁
⊢ val s (val s e₂ ε ∘ ⇑(Rew.castLE h) ∘ bvar) (val s e₂ ε ∘ ⇑(Rew.castLE h) ∘ fvar) t =
val s (fun x => e₂ (Fin.c... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_castLE | [129, 1] | [131, 24] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
h : n₁ ≤ n₂
t : Semiterm L μ n₁
⊢ val s (val s e₂ ε ∘ ⇑(Rew.castLE h) ∘ bvar) (val s e₂ ε ∘ ⇑(Rew.castLE h) ∘ fvar) t =
val s (fun x => e₂ (Fin.c... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_embSubsts | [133, 1] | [135, 39] | simp [val_rew, Empty.eq_elim] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
k : ℕ
w : Fin k → Semiterm L μ n
t : Semiterm L Empty k
⊢ val s e ε ((Rew.embSubsts w) t) = valb s (fun x => val s e ε (w x)) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
k : ℕ
w : Fin k → Semiterm L μ n
t : Semiterm L Empty k
⊢ val s (val s e ε ∘ ⇑(Rew.embSubsts w) ∘ bvar) Empty.elim t = valb s (fun x => val s e ε (w ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_embSubsts | [133, 1] | [135, 39] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
k : ℕ
w : Fin k → Semiterm L μ n
t : Semiterm L Empty k
⊢ val s (val s e ε ∘ ⇑(Rew.embSubsts w) ∘ bvar) Empty.elim t = valb s (fun x => val s e ε (w ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toS | [137, 1] | [139, 41] | simp [val_rew, Matrix.empty_eq] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L (Fin n) 0
⊢ valb s e (Rew.toS t) = val s ![] e t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L (Fin n) 0
⊢ val s ![] (val s e Empty.elim ∘ ⇑Rew.toS ∘ fvar) t = val s ![] e t |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toS | [137, 1] | [139, 41] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L (Fin n) 0
⊢ val s ![] (val s e Empty.elim ∘ ⇑Rew.toS ∘ fvar) t = val s ![] e t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toF | [141, 1] | [144, 32] | simp [val_rew, Matrix.empty_eq] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
⊢ val s ![] e (Rew.toF t) = valb s e t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
⊢ val s (val s ![] e ∘ ⇑Rew.toF ∘ bvar) (val s ![] e ∘ ⇑Rew.toF ∘ fvar) t = valb s e t |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toF | [141, 1] | [144, 32] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
⊢ val s (val s ![] e ∘ ⇑Rew.toF ∘ bvar) (val s ![] e ∘ ⇑Rew.toF ∘ fvar) t = valb s e t | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
⊢ val s ![] e ∘ ⇑Rew.toF ∘ fvar = Empty.elim |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toF | [141, 1] | [144, 32] | funext i | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
⊢ val s ![] e ∘ ⇑Rew.toF ∘ fvar = Empty.elim | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
i : Empty
⊢ (val s ![] e ∘ ⇑Rew.toF ∘ fvar) i = i.elim |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toF | [141, 1] | [144, 32] | simp | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
i : Empty
⊢ (val s ![] e ∘ ⇑Rew.toF ∘ fvar) i = i.elim | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
i : Empty
⊢ val s ![] e (Rew.toF &i) = i.elim |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_toF | [141, 1] | [144, 32] | contradiction | case e_ε.h
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
e : Fin n → M
t : Semiterm L Empty n
i : Empty
⊢ val s ![] e (Rew.toF &i) = i.elim | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_lMap | [150, 1] | [152, 81] | induction t <;> simp [*, valm, Function.comp, val_func, Semiterm.lMap_func] | n n₁ n₂ : ℕ
L₁ : Language
L₂ : Language
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e✝¹ : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝¹ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
φ✝ : L₁ →ᵥ L₂
e✝ : Fin n → M
ε✝ : μ → M
φ : L₁ →ᵥ L₂
s₂ : Structure L₂ M
e : Fin n → M
ε : μ → M
t : Semiterm L₁ μ n
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_shift | [160, 1] | [161, 83] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
t : SyntacticSemiterm L n
⊢ val s e ε (⇑ t) = val s e (ε ∘ Nat.succ) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
t : SyntacticSemiterm L n
⊢ val s (val s e ε ∘ ⇑⇑ ∘ bvar) (val s e ε ∘ ⇑⇑ ∘ fvar) t = val s e (ε ∘ Nat.succ) t |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_shift | [160, 1] | [161, 83] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
t : SyntacticSemiterm L n
⊢ val s (val s e ε ∘ ⇑⇑ ∘ bvar) (val s e ε ∘ ⇑⇑ ∘ fvar) t = val s e (ε ∘ Nat.succ) t | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_free | [163, 1] | [165, 77] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ val s e (a :>ₙ ε) (Rew.free t) = val s (e <: a) ε t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ val s (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar) (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ fvar) t = val s (e <... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_free | [163, 1] | [165, 77] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ val s (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar) (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ fvar) t = val s (e <... | case e_e
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar = e <: a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_free | [163, 1] | [165, 77] | exact funext $ Fin.lastCases (by simp) (by simp) | case e_e
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar = e <: a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_free | [163, 1] | [165, 77] | simp | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar) (Fin.last n) = (e <: a) (Fin.last n) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_free | [163, 1] | [165, 77] | simp | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L (n + 1)
⊢ ∀ (i : Fin n), (val s e (a :>ₙ ε) ∘ ⇑Rew.free ∘ bvar) i.castSucc = (e <: a) i.castSucc | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_fix | [167, 1] | [169, 98] | simp [val_rew] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ val s (e <: a) ε (Rew.fix t) = val s e (a :>ₙ ε) t | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ val s (val s (e <: a) ε ∘ ⇑Rew.fix ∘ bvar) (val s (e <: a) ε ∘ ⇑Rew.fix ∘ fvar) t = val s e (a :>ₙ ε) t |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_fix | [167, 1] | [169, 98] | congr <;> simp [Function.comp] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ val s (val s (e <: a) ε ∘ ⇑Rew.fix ∘ bvar) (val s (e <: a) ε ∘ ⇑Rew.fix ∘ fvar) t = val s e (a :>ₙ ε) t | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ (fun x => val s (e <: a) ε (Rew.fix &x)) = a :>ₙ ε |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_fix | [167, 1] | [169, 98] | exact funext (Nat.cases (by simp) (by simp)) | case e_ε
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ (fun x => val s (e <: a) ε (Rew.fix &x)) = a :>ₙ ε | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_fix | [167, 1] | [169, 98] | simp | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ val s (e <: a) ε (Rew.fix &0) = (a :>ₙ ε) 0 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_fix | [167, 1] | [169, 98] | simp | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
t : SyntacticSemiterm L n
⊢ ∀ (n_1 : ℕ), val s (e <: a) ε (Rew.fix &(n_1 + 1)) = (a :>ₙ ε) (n_1 + 1) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | induction t <;> simp [val_func] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
t : Semiterm L μ n
h : Function.funEqOn t.fvar? ε ε'
⊢ val s e ε t = val s e ε' t | case fvar
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
a✝ : μ
h : Function.funEqOn (&a✝).fvar? ε ε'
⊢ ε a✝ = ε' a✝
case func
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Typ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | case fvar x => exact h x (by simp [fvar?]) | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
x : μ
h : Function.funEqOn (&x).fvar? ε ε'
⊢ ε x = ε' x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | case func k f v ih =>
congr; funext i; exact ih i (by intro x hx; exact h x (by simp; exact ⟨i, hx⟩)) | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
k : ℕ
f : L.Func k
v : Fin k → Semiterm L μ n
ih : ∀ (a : Fin k), Function.funEqOn (v a).fvar? ε ε' → val s e ε (v a) = val s e ε' (v a)
h... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | exact h x (by simp [fvar?]) | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
x : μ
h : Function.funEqOn (&x).fvar? ε ε'
⊢ ε x = ε' x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | simp [fvar?] | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
x : μ
h : Function.funEqOn (&x).fvar? ε ε'
⊢ (&x).fvar? x | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiterm.val_eq_of_funEqOn | [173, 1] | [178, 84] | congr | n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
k : ℕ
f : L.Func k
v : Fin k → Semiterm L μ n
ih : ∀ (a : Fin k), Function.funEqOn (v a).fvar? ε ε' → val s e ε (v a) = val s e ε' (v a)
h... | case e_a
n n₁ n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
e : Fin n → M
e₁ : Fin n₁ → M
e₂ : Fin n₂ → M
ε : μ → M
ε₁ : μ₁ → M
ε₂ : μ₂ → M
ε' : μ → M
k : ℕ
f : L.Func k
v : Fin k → Semiterm L μ n
ih : ∀ (a : Fin k), Function.funEqOn (v a).fvar? ε ε' → val s e ε (v a) = val s e ε... |
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