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/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_shift | [392, 1] | [394, 36] | simp [eval_rew, Function.comp] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
ε : ℕ → M
a : M
p : SyntacticSemiformula L n
⊢ (Eval s e (a :>ₙ ε)) ((Rew.hom ⇑) p) ↔ (Eval s e ε) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | unfold fvar? at h | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
p : Semiformula L μ n
h : Function.funEqOn p.fvar? ε ε'
⊢ (Eval s e ε) p ↔ (Eval s e ε') p | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
p : Semiformula L μ n
h : Function.funEqOn (fun x => x ∈ p.fvarList) ε ε'
⊢ (Eval s e ε) p ↔ (Eval s e ε') p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | induction p using Semiformula.rec' | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
p : Semiformula L μ n
h : Function.funEqOn (fun x => x ∈ p.fvarList) ε ε'
⊢ (Eval s e ε) p ↔ (Eval s e ε') p | case hverum
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ ⊤.fvarList) ε ε'
⊢ (Eval s e ε) ⊤ ↔ (Eval s e ε') ⊤
case hfalsum
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hverum => simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ ⊤.fvarList) ε ε'
⊢ (Eval s e ε) ⊤ ↔ (Eval s e ε') ⊤ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hfalsum => simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ ⊥.fvarList) ε ε'
⊢ (Eval s e ε) ⊥ ↔ (Eval s e ε') ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hrel k r v =>
simp [eval_rel]; apply iff_of_eq; congr
funext i
exact Semiterm.val_eq_of_funEqOn (v i) (fun x hx ↦ h x (by simp [fvarList]; exact ⟨i, hx⟩)) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Eval s e ε) (rel r v) ↔ (Eval s e ε') (rel r v) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hnrel k r v =>
simp [eval_nrel]; apply iff_of_eq; congr
funext i
exact Semiterm.val_eq_of_funEqOn (v i) (fun x hx ↦ h x (by simp [fvarList]; exact ⟨i, hx⟩)) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (Eval s e ε) (nrel r v) ↔ (Eval s e ε') (nrel r v) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hall p ih =>
simp; apply forall_congr'; intro x
exact ih (fun x hx ↦ h _ $ by simp [fvar?]; exact hx) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | case hex p ih =>
simp; apply exists_congr; intro x
exact ih (fun x hx ↦ h _ $ by simp [fvar?]; exact hx) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ ⊤.fvarList) ε ε'
⊢ (Eval s e ε) ⊤ ↔ (Eval s e ε') ⊤ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ ⊥.fvarList) ε ε'
⊢ (Eval s e ε) ⊥ ↔ (Eval s e ε') ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [eval_rel] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Eval s e ε) (rel r v) ↔ (Eval s e ε') (rel r v) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Structure.rel r fun i => Semiterm.val s e ε (v i)) ↔... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply iff_of_eq | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Structure.rel r fun i => Semiterm.val s e ε (v i)) ↔... | case a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Structure.rel r fun i => Semiterm.val s e ε (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | congr | case a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (Structure.rel r fun i => Semiterm.val s e ε (... | case a.e_a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (fun i => Semiterm.val s e ε (v i)) = fun ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | funext i | case a.e_a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
⊢ (fun i => Semiterm.val s e ε (v i)) = fun ... | case a.e_a.h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
i : Fin k
⊢ Semiterm.val s e ε (v i) = Sem... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact Semiterm.val_eq_of_funEqOn (v i) (fun x hx ↦ h x (by simp [fvarList]; exact ⟨i, hx⟩)) | case a.e_a.h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
i : Fin k
⊢ Semiterm.val s e ε (v i) = Sem... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ (fun x => x ∈ (rel... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ ∃ a, x ∈ (v a).fva... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ⟨i, hx⟩ | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (rel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ ∃ a, x ∈ (v a).fva... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [eval_nrel] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (Eval s e ε) (nrel r v) ↔ (Eval s e ε') (nrel r v) | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (¬Structure.rel r fun i => Semiterm.val s e ε (v i))... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply iff_of_eq | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (¬Structure.rel r fun i => Semiterm.val s e ε (v i))... | case a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (¬Structure.rel r fun i => Semiterm.val s e ε... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | congr | case a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (¬Structure.rel r fun i => Semiterm.val s e ε... | case a.e_a.e_a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (fun i => Semiterm.val s e ε (v i)) =... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | funext i | case a.e_a.e_a
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
⊢ (fun i => Semiterm.val s e ε (v i)) =... | case a.e_a.e_a.h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
i : Fin k
⊢ Semiterm.val s e ε (v i) ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact Semiterm.val_eq_of_funEqOn (v i) (fun x hx ↦ h x (by simp [fvarList]; exact ⟨i, hx⟩)) | case a.e_a.e_a.h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
i : Fin k
⊢ Semiterm.val s e ε (v i) ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ (fun x => x ∈ (nr... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ ∃ a, x ∈ (v a).fv... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ⟨i, hx⟩ | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ k : ℕ
r : L.Rel k
v : Fin k → Semiterm L μ n✝
e : Fin n✝ → M
h : Function.funEqOn (fun x => x ∈ (nrel r v).fvarList) ε ε'
i : Fin k
x : μ
hx : (v i).fvar? x
⊢ ∃ a, x ∈ (v a).fv... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply and_congr | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | case h₁
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ihp (fun x (hx : x ∈ p.fvarList) ↦ h x $ by simp [hx, fvarList]) | case h₁
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [hx, fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ihq (fun x (hx : x ∈ q.fvarList) ↦ h x $ by simp [hx, fvarList]) | case h₂
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [hx, fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply or_congr | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | case h₁
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ihp (fun x (hx : x ∈ p.fvarList) ↦ h x $ by simp [hx, fvarList]) | case h₁
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [hx, fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ihq (fun x (hx : x ∈ q.fvarList) ↦ h x $ by simp [hx, fvarList]) | case h₂
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Fu... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [hx, fvarList] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p q : Semiformula L μ n✝
ihp : ∀ {e : Fin n✝ → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
ihq : ∀ {e : Fin n✝ → M}, Function.f... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply forall_congr' | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | intro x | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ih (fun x hx ↦ h _ $ by simp [fvar?]; exact hx) | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [fvar?] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact hx | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | apply exists_congr | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | intro x | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact ih (fun x hx ↦ h _ $ by simp [fvar?]; exact hx) | case h
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : F... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | simp [fvar?] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_iff_of_funEqOn | [398, 1] | [425, 58] | exact hx | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
ε' : μ → M
n✝ : ℕ
p : Semiformula L μ (n✝ + 1)
ih : ∀ {e : Fin (n✝ + 1) → M}, Function.funEqOn (fun x => x ∈ p.fvarList) ε ε' → ((Eval s e ε) p ↔ (Eval s e ε') p)
e : Fin n✝ → M
h : Function... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | simp [Formula.fvUnivClosure, eval_rewriteMap] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) ↔ ∀ (ε : μ → M), (Evalf s ε) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | haveI : Inhabited μ := ⟨Classical.ofNonempty⟩ | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) ↔ ∀ (ε : μ → M)... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | constructor | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) ↔ ∀ (ε : μ → M)... | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) → ∀ (ε ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | intro h ε | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
⊢ (∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p) → ∀ (ε ... | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : μ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | have := h (fun i ↦ ε $ Semiformula.fvListingInv _ i) | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : μ... | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this✝ : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | exact (eval_iff_of_funEqOn p (by intro x hx; simp [fvListingInv_fvListing hx])).mp this | case mp
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this✝ : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | intro x hx | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this✝ : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : μ → M
th... | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this✝ : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : μ → M
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | simp [fvListingInv_fvListing hx] | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε✝ : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this✝ : Inhabited μ
h : ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x => e (fvListing p x)) p
ε : μ → M
th... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | intro h e | case mpr
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
⊢ (∀ (ε : μ → M), (Evalf s ε) p) → ∀ (e : Fin ((fvarList p).length + 1) → M), (Eval s ![] fun x... | case mpr
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e✝ : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
h : ∀ (ε : μ → M), (Evalf s ε) p
e : Fin ((fvarList p).length + 1) → M
⊢ (Eval s ![] fun x =>... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure_inhabited | [427, 1] | [436, 52] | exact h (fun x ↦ e $ Semiformula.fvListing p x) | case mpr
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e✝ : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
h✝ : Nonempty μ
inst✝ : DecidableEq μ
p : Formula L μ
this : Inhabited μ
h : ∀ (ε : μ → M), (Evalf s ε) p
e : Fin ((fvarList p).length + 1) → M
⊢ (Eval s ![] fun x =>... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure | [438, 1] | [444, 27] | by_cases hμ : Nonempty μ | n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | case pos
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ : Nonempty μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p
case neg
n₂ : ℕ
L : Language
μ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure | [438, 1] | [444, 27] | exact val_fvUnivClosure_inhabited | case pos
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ : Nonempty μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure | [438, 1] | [444, 27] | haveI hμ : IsEmpty μ := not_nonempty_iff.mp hμ | case neg
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ : ¬Nonempty μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | case neg
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ✝ : ¬Nonempty μ
hμ : IsEmpty μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure | [438, 1] | [444, 27] | simp [Formula.fvUnivClosure_sentence p, IsEmpty.eq_elim] | case neg
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ✝ : ¬Nonempty μ
hμ : IsEmpty μ
⊢ (Evalf s Empty.elim) (∀ᶠ* p) ↔ ∀ (ε : μ → M), (Evalf s ε) p | case neg
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ✝ : ¬Nonempty μ
hμ : IsEmpty μ
⊢ (Eval s Fin.isEmpty'.elim' hμ.elim') p ↔ (Evalf s hμ.elim') p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.val_fvUnivClosure | [438, 1] | [444, 27] | simp [Matrix.empty_eq] | case neg
n₂ : ℕ
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type w
s : Structure L M
n : ℕ
e : Fin n → M
e₂ : Fin n₂ → M
ε : μ → M
ε₂ : μ₂ → M
inst✝¹ : Nonempty M
inst✝ : DecidableEq μ
p : Formula L μ
hμ✝ : ¬Nonempty μ
hμ : IsEmpty μ
⊢ (Eval s Fin.isEmpty'.elim' hμ.elim') p ↔ (Evalf s hμ.elim') p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
⊢ (Eval (ofEquiv φ) e ε) ⊤ ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) ⊤ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
⊢ (Eval (ofEquiv φ) e ε) ⊥ ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp [Function.comp, eval_rel, ofEquiv_rel φ, Structure.ofEquiv_val φ] | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L μ x✝
⊢ (Eval (ofEquiv φ) e ε) (Semiformula.rel r v) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (Semiformula.rel r v) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp [Function.comp, eval_nrel, ofEquiv_rel φ, Structure.ofEquiv_val φ] | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L μ x✝
⊢ (Eval (ofEquiv φ) e ε) (nrel r v) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (nrel r v) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp [eval_ofEquiv_iff (p := p), eval_ofEquiv_iff (p := q)] | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p q : Semiformula L μ x✝
⊢ (Eval (ofEquiv φ) e ε) (p ⋏ q) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (p ⋏ q) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp [eval_ofEquiv_iff (p := p), eval_ofEquiv_iff (p := q)] | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p q : Semiformula L μ x✝
⊢ (Eval (ofEquiv φ) e ε) (p ⋎ q) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (p ⋎ q) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (Eval (ofEquiv φ) e ε) (∀' p) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (∀' p) | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∀ (x : N), (Eval (ofEquiv φ) (x :> e) ε) p) ↔ ∀ (x : M), (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | exact
⟨fun h x => by simpa[Matrix.comp_vecCons''] using eval_ofEquiv_iff.mp (h (φ x)),
fun h x => eval_ofEquiv_iff.mpr (by simpa[Matrix.comp_vecCons''] using h (φ.symm x))⟩ | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∀ (x : N), (Eval (ofEquiv φ) (x :> e) ε) p) ↔ ∀ (x : M), (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simpa[Matrix.comp_vecCons''] using eval_ofEquiv_iff.mp (h (φ x)) | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
h : ∀ (x : N), (Eval (ofEquiv φ) (x :> e) ε) p
x : M
⊢ (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simpa[Matrix.comp_vecCons''] using h (φ.symm x) | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
h : ∀ (x : M), (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
x : N
⊢ (Eval s (⇑φ.symm ∘ (x :> e)) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simp | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (Eval (ofEquiv φ) e ε) (∃' p) ↔ (Eval s (⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) (∃' p) | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∃ x, (Eval (ofEquiv φ) (x :> e) ε) p) ↔ ∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | exact
⟨by rintro ⟨x, h⟩; exists φ.symm x; simpa[Matrix.comp_vecCons''] using eval_ofEquiv_iff.mp h,
by rintro ⟨x, h⟩; exists φ x; apply eval_ofEquiv_iff.mpr; simpa[Matrix.comp_vecCons''] using h⟩ | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∃ x, (Eval (ofEquiv φ) (x :> e) ε) p) ↔ ∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | rintro ⟨x, h⟩ | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∃ x, (Eval (ofEquiv φ) (x :> e) ε) p) → ∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : N
h : (Eval (ofEquiv φ) (x :> e) ε) p
⊢ ∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | exists φ.symm x | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : N
h : (Eval (ofEquiv φ) (x :> e) ε) p
⊢ ∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : N
h : (Eval (ofEquiv φ) (x :> e) ε) p
⊢ (Eval s (φ.symm x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simpa[Matrix.comp_vecCons''] using eval_ofEquiv_iff.mp h | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : N
h : (Eval (ofEquiv φ) (x :> e) ε) p
⊢ (Eval s (φ.symm x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | rintro ⟨x, h⟩ | M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
⊢ (∃ x, (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p) → ∃ x, (Eval (ofEquiv φ) (x :> e) ε) p | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ ∃ x, (Eval (ofEquiv φ) (x :> e) ε) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | exists φ x | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ ∃ x, (Eval (ofEquiv φ) (x :> e) ε) p | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ (Eval (ofEquiv φ) (φ x :> e) ε) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | apply eval_ofEquiv_iff.mpr | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ (Eval (ofEquiv φ) (φ x :> e) ε) p | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ (Eval s (⇑φ.symm ∘ (φ x :> e)) (⇑φ.symm ∘ ε)) p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.eval_ofEquiv_iff | [458, 1] | [473, 101] | simpa[Matrix.comp_vecCons''] using h | case intro
M : Type u_2
N : Type u_1
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
s : Structure L M
φ : M ≃ N
x✝ : ℕ
e : Fin x✝ → N
ε : μ → N
p : Semiformula L μ (x✝ + 1)
x : M
h : (Eval s (x :> ⇑φ.symm ∘ e) (⇑φ.symm ∘ ε)) p
⊢ (Eval s (⇑φ.symm ∘ (φ x :> e)) (⇑φ.symm ∘ ε)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.models_iff | [519, 1] | [519, 102] | simp [models_def] | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝ : Nonempty M
s : Structure L M
T U : Theory L
σ : Sentence L
⊢ M ⊧ₘ σ ↔ (Semiformula.Evalf s Empty.elim) σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.satisfiable_iff | [541, 1] | [543, 110] | rintro ⟨s, hs⟩ | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝ : Nonempty M
s : Structure L M
T U : Theory L
⊢ Semantics.Satisfiable (Struc L) T → ∃ M, ∃ (x : Nonempty M), ∃ x_1, M ⊧ₘ* T | case intro
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝ : Nonempty M
s✝ : Structure L M
T U : Theory L
s : Struc L
hs : s ⊧* T
⊢ ∃ M, ∃ (x : Nonempty M), ∃ x_1, M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.satisfiable_iff | [541, 1] | [543, 110] | exact ⟨s.Dom, s.nonempty, s.struc, hs⟩ | case intro
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝ : Nonempty M
s✝ : Structure L M
T U : Theory L
s : Struc L
hs : s ⊧* T
⊢ ∃ M, ∃ (x : Nonempty M), ∃ x_1, M ⊧ₘ* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.satisfiable_iff | [541, 1] | [543, 110] | rintro ⟨M, i, s, hT⟩ | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝ : Nonempty M
s : Structure L M
T U : Theory L
⊢ (∃ M, ∃ (x : Nonempty M), ∃ x_1, M ⊧ₘ* T) → Semantics.Satisfiable (Struc L) T | case intro.intro.intro
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M✝ : Type u_1
inst✝ : Nonempty M✝
s✝ : Structure L M✝
T U : Theory L
M : Type v
i : Nonempty M
s : Structure L M
hT : M ⊧ₘ* T
⊢ Semantics.Satisfiable (Struc L) T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.satisfiable_iff | [541, 1] | [543, 110] | exact ⟨s.toStruc, hT⟩ | case intro.intro.intro
L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M✝ : Type u_1
inst✝ : Nonempty M✝
s✝ : Structure L M✝
T U : Theory L
M : Type v
i : Nonempty M
s : Structure L M
hT : M ⊧ₘ* T
⊢ Semantics.Satisfiable (Struc L) T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.eval_lMap | [574, 1] | [577, 74] | induction p using rec' <;>
simp [*, Semiterm.val_lMap, lMap_rel, lMap_nrel, eval_rel, eval_nrel] | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ : Language
L₂ : Language
Φ : L₁ →ᵥ L₂
M : Type u
inst✝ : Nonempty M
s₂ : Structure L₂ M
n : ℕ
e : Fin n → M
ε : μ → M
p : Semiformula L₁ μ n
⊢ (Eval s₂ e ε) ((lMap Φ) p) ↔ (Eval (Structure.lMap Φ s₂) e ε) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Semiformula.models_lMap | [579, 1] | [581, 48] | simp [Semantics.Realize, Evalf, eval_lMap] | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ : Language
L₂ : Language
Φ : L₁ →ᵥ L₂
M : Type u
inst✝ : Nonempty M
s₂ : Structure L₂ M
n : ℕ
e : Fin n → M
ε : μ → M
σ : Sentence L₁
⊢ s₂.toStruc ⊧ (lMap Φ) σ ↔ (Structure.lMap Φ s₂).toStruc ⊧ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.lMap_models_lMap | [587, 1] | [592, 41] | intro s hM | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
T : Theory L₁
σ : Sentence L₁
h : T ⊨[Struc L₁] σ
⊢ Theory.lMap Φ T ⊨[Struc L₂] (Semiformula.lMap Φ) σ | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
T : Theory L₁
σ : Sentence L₁
h : T ⊨[Struc L₁] σ
s : Struc L₂
hM : s ∈ Semantics.models (Struc L₂) (Theory.lMap Φ T)
⊢ s ⊧ (Semiformula.lMap Φ) σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.lMap_models_lMap | [587, 1] | [592, 41] | have : (s.struc.lMap Φ).toStruc ⊧ σ :=
h ⟨fun _ hq => Semiformula.models_lMap.mp <| hM.realize (Set.mem_image_of_mem _ hq)⟩ | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
T : Theory L₁
σ : Sentence L₁
h : T ⊨[Struc L₁] σ
s : Struc L₂
hM : s ∈ Semantics.models (Struc L₂) (Theory.lMap Φ T)
⊢ s ⊧ (Semiformula.lMap Φ) σ | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
T : Theory L₁
σ : Sentence L₁
h : T ⊨[Struc L₁] σ
s : Struc L₂
hM : s ∈ Semantics.models (Struc L₂) (Theory.lMap Φ T)
this : (Structure.lMap Φ s.struc).toStruc ⊧ σ
⊢ s ⊧ (Semiformula.lMap Φ) σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.lMap_models_lMap | [587, 1] | [592, 41] | exact Semiformula.models_lMap.mpr this | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
T : Theory L₁
σ : Sentence L₁
h : T ⊨[Struc L₁] σ
s : Struc L₂
hM : s ∈ Semantics.models (Struc L₂) (Theory.lMap Φ T)
this : (Structure.lMap Φ s.struc).toStruc ⊧ σ
⊢ s ⊧ (Semiformula.lMap Φ) σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.ModelsTheory.add_iff | [605, 1] | [606, 64] | simp [Theory.add_def] | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u_1
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U : Theory L
⊢ M ⊧ₘ* T + U ↔ M ⊧ₘ* T ∧ M ⊧ₘ* U | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Theory.modelsTheory_onTheory₁ | [619, 1] | [621, 96] | simp [Semiformula.models_lMap, Theory.lMap, modelsTheory_iff, @modelsTheory_iff (T := T₁)] | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
L₁ L₂ : Language
Φ : L₁ →ᵥ L₂
M : Type u
inst✝ : Nonempty M
s₂ : Structure L₂ M
T₁ : Theory L₁
⊢ M ⊧ₘ* lMap Φ T₁ ↔ M ⊧ₘ* T₁ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Semantics/Semantics.lean | LO.FirstOrder.Structure.mem_theory_iff | [633, 1] | [633, 69] | rfl | L : Language
μ : Type v
μ₁ : Type v₁
μ₂ : Type v₂
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
σ : Sentence L
⊢ σ ∈ theory L M ↔ M ⊧ₘ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | intro d v | α : Type u_1
Δ : Sequent α
⊢ ⊢¹ Δ → Semantics.Valid (Valuation α) (List.disj Δ) | α : Type u_1
Δ : Sequent α
d : ⊢¹ Δ
v : Valuation α
⊢ v ⊧ List.disj Δ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | induction d | α : Type u_1
Δ : Sequent α
d : ⊢¹ Δ
v : Valuation α
⊢ v ⊧ List.disj Δ | case axL
α : Type u_1
Δ : Sequent α
v : Valuation α
Δ✝ : List (Formula α)
a✝ : α
⊢ v ⊧ (Formula.atom a✝ :: Formula.natom a✝ :: Δ✝).disj
case verum
α : Type u_1
Δ : Sequent α
v : Valuation α
Δ✝ : List (Formula α)
⊢ v ⊧ (⊤ :: Δ✝).disj
case or
α : Type u_1
Δ : Sequent α
v : Valuation α
Δ✝ : List (Formula α)
p✝ q✝ : Form... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | case axL Δ a =>
simp[List.map_disj]
by_cases v a <;> simp[*] | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
a : α
⊢ v ⊧ (Formula.atom a :: Formula.natom a :: Δ).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | case verum => simp[List.map_disj] | α : Type u_1
Δ : Sequent α
v : Valuation α
Δ✝ : List (Formula α)
⊢ v ⊧ (⊤ :: Δ✝).disj | no goals |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.