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/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_refl | [104, 1] | [107, 15] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : M ⊧ₘ (“∀' #0 = #0”)
⊢ eqv L a a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_refl | [104, 1] | [107, 15] | exact this a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | have : M ⊧ₘ “∀ x y, x = y → y = x” := H.realize (Theory.eqAxiom.symm (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ eqv L a b → eqv L b a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : M ⊧ₘ (“∀' ∀' (#1 = #0 → #0 = #1)”)
⊢ eqv L a b → eqv L b a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : M ⊧ₘ (“∀' ∀' (#1 = #0 → #0 = #1)”)
⊢ eqv L a b → eqv L b a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : ∀ (x x_1 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x]
⊢ eqv L a b → eqv L b a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | exact this a b | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : ∀ (x x_1 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x]
⊢ eqv L a b → eqv L b a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | have : M ⊧ₘ “∀ x y z, x = y → y = z → x = z” := H.realize (Theory.eqAxiom.trans (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
⊢ eqv L a b → eqv L b c → eqv L a c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : M ⊧ₘ (“∀' ∀' ∀' (#2 = #1 → (#1 = #0 → #2 = #0))”)
⊢ eqv L a b → eqv L b c → eqv L a c |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : M ⊧ₘ (“∀' ∀' ∀' (#2 = #1 → (#1 = #0 → #2 = #0))”)
⊢ eqv L a b → eqv L b c → eqv L a c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : ∀ (x x_1 x_2 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x_2] → op(=).val ![x, x_2]
⊢ eqv L a b → eqv L b c → eqv L a c |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | exact this a b c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : ∀ (x x_1 x_2 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x_2] → op(=).val ![x, x_2]
⊢ eqv L a b → eqv L b c → eqv L a c | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | have : M ⊧ₘ ∀* (Semiformula.vecEq varSumInL varSumInR ⟶ op(=).operator ![Semiterm.func f varSumInL, Semiterm.func f varSumInR]) :=
H.realize (eqAxiom.funcExt f (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ eqv L (func f v) (func f w) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (“(!(vecEq varSumInL varSumInR) → !!(Semiterm.func f varSumInL) = !!(Semiterm.func f varSumInR))”)
⊢ eqv L (func f v... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simp [varSumInL, varSumInR, models_def, vecEq, Semiterm.val_func] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (“(!(vecEq varSumInL varSumInR) → !!(Semiterm.func f varSumInL) = !!(Semiterm.func f varSumInR))”)
⊢ eqv L (func f v... | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simpa [Matrix.vecAppend_eq_ite] using this (Matrix.vecAppend rfl v w) (fun i => by simpa [Matrix.vecAppend_eq_ite] using h i) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simpa [Matrix.vecAppend_eq_ite] using h i | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt_aux | [126, 1] | [131, 138] | have : M ⊧ₘ ∀* (Semiformula.vecEq varSumInL varSumInR ⟶ Semiformula.rel r varSumInL ⟶ Semiformula.rel r varSumInR) :=
H.realize (eqAxiom.relExt r (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v → rel r w | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (vecEq varSumInL varSumInR ⟶ Semiformula.rel r varSumInL ⟶ Semiformula.rel r varSumInR)
⊢ rel r v → rel r w |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt_aux | [126, 1] | [131, 138] | simp [varSumInL, varSumInR, models_def, vecEq, Semiterm.val_func, eval_rel (r := r)] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (vecEq varSumInL varSumInR ⟶ Semiformula.rel r varSumInL ⟶ Semiformula.rel r varSumInR)
⊢ rel r v → rel r w | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
(rel r fun i => e... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt_aux | [126, 1] | [131, 138] | simpa [eval_rel, Matrix.vecAppend_eq_ite] using this (Matrix.vecAppend rfl v w) (fun i => by simpa [Matrix.vecAppend_eq_ite] using h i) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
(rel r fun i => e... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt_aux | [126, 1] | [131, 138] | simpa [Matrix.vecAppend_eq_ite] using h i | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
(rel r fun i => e... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt | [133, 1] | [137, 57] | simp | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v = rel r w | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v ↔ rel r w |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt | [133, 1] | [137, 57] | constructor | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v ↔ rel r w | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v → rel r w
case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Struct... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt | [133, 1] | [137, 57] | exact eqv_relExt_aux H r h | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v → rel r w | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt | [133, 1] | [137, 57] | exact eqv_relExt_aux H r (fun i => eqv_symm H (h i)) | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r w → rel r v | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.val_mk | [172, 1] | [173, 58] | induction t <;> simp [*, funk_mk, Semiterm.val_func] | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n : ℕ
e : Fin n → M
ε : μ → M
t : Semiterm L μ n
⊢ valm (QuotEq H) (fun i => ⟦e i⟧) (fun i => ⟦ε i⟧) t = ⟦valm M e ε t⟧ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | induction p using Semiformula.rec' <;> simp [*, Semiformula.eval_rel, Semiformula.eval_nrel, val_mk, rel_mk] | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n : ℕ
e : Fin n → M
ε : μ → M
p : Semiformula L μ n
⊢ (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p | case hall
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n : ℕ
ε : μ → M
n✝ : ℕ
p✝ : Semiformula L μ (n✝ + 1)
a✝ : ∀ {e : Fin (n✝ + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p✝ ↔ (Evalm M e ε) p✝
e : Fin n✝ → M
⊢ (∀ (x : QuotEq H), ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | constructor | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∀ (x : QuotEq H), (Eval struc (x :... | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∀ (x : QuotEq H), (Eval st... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | intro h a | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∀ (x : QuotEq H), (Eval st... | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : QuotEq H), (Eval s... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | exact (ih (e := a :> e)).mp (by simpa [Matrix.comp_vecCons] using h ⟦a⟧) | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : QuotEq H), (Eval s... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | simpa [Matrix.comp_vecCons] using h ⟦a⟧ | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : QuotEq H), (Eval struc (x ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | intro h a | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∀ (x : M), (Eval inst✝ (x... | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : M), (Eval inst✝ (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | induction' a using Quotient.ind with a | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : M), (Eval inst✝ (... | case mpr.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : M), (Eval inst✝... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | simpa [Matrix.comp_vecCons] using ih.mpr (h a) | case mpr.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
h : ∀ (x : M), (Eval inst✝... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | constructor | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∃ x, (Eval struc (x :> fun i => ⟦e... | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∃ x, (Eval struc (x :> fun... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | intro ⟨a, h⟩ | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∃ x, (Eval struc (x :> fun... | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : QuotEq H
h : (Eval struc ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | induction' a using Quotient.ind with a | case mp
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : QuotEq H
h : (Eval struc ... | case mp.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval struc (⟦a⟧ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | exact ⟨a, (ih (e := a :> e)).mp (by simpa [Matrix.comp_vecCons] using h)⟩ | case mp.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval struc (⟦a⟧ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | simpa [Matrix.comp_vecCons] using h | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval struc (⟦a⟧ :> fun i =... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | intro ⟨a, h⟩ | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
⊢ (∃ x, (Eval inst✝ (x :> e)... | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval inst✝ (a :> ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | exact ⟨⟦a⟧, by simpa [Matrix.comp_vecCons] using ih.mpr h⟩ | case mpr
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval inst✝ (a :> ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.eval_mk | [175, 1] | [189, 79] | simpa [Matrix.comp_vecCons] using ih.mpr h | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
n✝ : ℕ
ε : μ → M
n : ℕ
p : Semiformula L μ (n + 1)
ih : ∀ {e : Fin (n + 1) → M}, (Evalm (QuotEq H) (fun i => ⟦e i⟧) fun i => ⟦ε i⟧) p ↔ (Evalm M e ε) p
e : Fin n → M
a : M
h : (Eval inst✝ (a :> e) ε) p
⊢... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.models_iff | [191, 1] | [193, 67] | simpa [models_def, Semiformula.Evalf, eq_finZeroElim, Empty.eq_elim] using
eval_mk (H := H) (e := finZeroElim) (ε := Empty.elim) (p := σ) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
σ : Sentence L
⊢ QuotEq H ⊧ₘ σ ↔ M ⊧ₘ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.rel_eq | [201, 1] | [206, 97] | induction' a using Quotient.ind with a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : QuotEq H
⊢ op(=).val ![a, b] ↔ a = b | case a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
b : QuotEq H
a : M
⊢ op(=).val ![⟦a⟧, b] ↔ ⟦a⟧ = b |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.rel_eq | [201, 1] | [206, 97] | induction' b using Quotient.ind with b | case a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
b : QuotEq H
a : M
⊢ op(=).val ![⟦a⟧, b] ↔ ⟦a⟧ = b | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ op(=).val ![⟦a⟧, ⟦b⟧] ↔ ⟦a⟧ = ⟦b⟧ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.rel_eq | [201, 1] | [206, 97] | rw[of_eq_of] | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ op(=).val ![⟦a⟧, ⟦b⟧] ↔ ⟦a⟧ = ⟦b⟧ | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ op(=).val ![⟦a⟧, ⟦b⟧] ↔ eqv L a b |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.rel_eq | [201, 1] | [206, 97] | simp [eqv, Semiformula.Operator.val] | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ op(=).val ![⟦a⟧, ⟦b⟧] ↔ eqv L a b | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ (Eval struc ![⟦a⟧, ⟦b⟧] Empty.elim) op(=).sentence ↔ (Eval inst✝ ![a, b] Empty.elim) op(=).sentence |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.QuotEq.rel_eq | [201, 1] | [206, 97] | simpa [Evalm, Matrix.fun_eq_vec₂, Empty.eq_elim] using
eval_mk (H := H) (e := ![a, b]) (ε := Empty.elim) (p := Semiformula.Operator.Eq.eq.sentence) | case a.a
L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ (Eval struc ![⟦a⟧, ⟦b⟧] Empty.elim) op(=).sentence ↔ (Eval inst✝ ![a, b] Empty.elim) op(=).sentence | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | simp [consequence_iff] | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ T ⊨[Struc L] σ ↔
∀ (M : Type v) [inst : Nonempty M] [inst_1 : Structure L M] [inst_2 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ (∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ) ↔
∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | constructor | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ (∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ) ↔
∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ (∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ) →
∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
case mpr
L : Language
μ : Type u_1
inst... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | intro h M x s _ hM | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ (∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ) →
∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ | case mp
L : Language
μ : Type u_1
inst✝² : Semiformula.Operator.Eq L
T : Theory L
inst✝¹ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
inst✝ : Structure.Eq L M
hM : M ⊧ₘ* T
⊢ M ⊧ₘ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | exact h M x hM | case mp
L : Language
μ : Type u_1
inst✝² : Semiformula.Operator.Eq L
T : Theory L
inst✝¹ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
inst✝ : Structure.Eq L M
hM : M ⊧ₘ* T
⊢ M ⊧ₘ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | intro h M x s hM | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ (∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ) →
∀ (M : Type v) (a : M) [inst : Structure L M], M ⊧ₘ* T → M ⊧ₘ σ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
⊢ M ⊧ₘ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | haveI : Nonempty M := ⟨x⟩ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
⊢ M ⊧ₘ σ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
⊢ M ⊧ₘ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | have H : M ⊧ₘ* (𝐄𝐐 : Theory L) := models_of_subtheory hM | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
⊢ M ⊧ₘ σ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
⊢ M ⊧ₘ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | have e : Structure.Eq.QuotEq H ≡ₑ[L] M := Structure.Eq.QuotEq.elementaryEquiv H | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
⊢ M ⊧ₘ σ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
e : Structure.Eq.QuotEq H ≡ₑ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq | [216, 1] | [224, 78] | exact e.models.mp $ h (Structure.Eq.QuotEq H) ⟦x⟧ (e.modelsTheory.mpr hM) | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
h : ∀ (M : Type v) (a : M) [inst : Structure L M] [inst_1 : Structure.Eq L M], M ⊧ₘ* T → M ⊧ₘ σ
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
e : Structure.Eq.QuotEq H ≡ₑ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_eq' | [226, 1] | [228, 26] | rw [consequence_iff_eq] | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
σ : Sentence L
⊢ T ⊨[Struc L] σ ↔
∀ (M : Type v) [inst : Nonempty M] [inst_1 : Structure L M] [inst_2 : Structure.Eq L M] [inst_3 : M ⊧ₘ* T], M ⊧ₘ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.consequence_iff_add_eq | [230, 1] | [232, 72] | simp | L : Language
μ : Type u_1
inst✝ : Semiformula.Operator.Eq L
T : Theory L
σ : Sentence L
M : Type v
x✝² : Nonempty M
x✝¹ : Structure L M
x✝ : Structure.Eq L M
⊢ M ⊧ₘ* T⁼ → M ⊧ₘ σ ↔ M ⊧ₘ* T → M ⊧ₘ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | simp [satisfiable_iff] | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ Semantics.Satisfiable (Struc L) T ↔ ∃ M, ∃ (x : Nonempty M), ∃ x_1, ∃ (_ : Structure.Eq L M), M ⊧ₘ* T | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, M ⊧ₘ* T) ↔ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | constructor | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, M ⊧ₘ* T) ↔ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, M ⊧ₘ* T) → ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T
case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T) → ∃ M a x, M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | intro ⟨M, x, s, hM⟩ | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, M ⊧ₘ* T) → ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | haveI : Nonempty M := ⟨x⟩ | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | have H : M ⊧ₘ* (𝐄𝐐 : Theory L) := models_of_subtheory hM | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | have e : Structure.Eq.QuotEq H ≡ₑ[L] M := Structure.Eq.QuotEq.elementaryEquiv H | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
e : Structure.Eq.QuotEq H ≡ₑ[L] M
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | exact ⟨Structure.Eq.QuotEq H, ⟦x⟧, inferInstance, inferInstance, e.modelsTheory.mpr hM⟩ | case mp
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
x : M
s : Structure L M
hM : M ⊧ₘ* T
this : Nonempty M
H : M ⊧ₘ* 𝐄𝐐
e : Structure.Eq.QuotEq H ≡ₑ[L] M
⊢ ∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | intro ⟨M, i, s, _, hM⟩ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
⊢ (∃ M a x, Structure.Eq L M ∧ M ⊧ₘ* T) → ∃ M a x, M ⊧ₘ* T | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
i : M
s : Structure L M
left✝ : Structure.Eq L M
hM : M ⊧ₘ* T
⊢ ∃ M a x, M ⊧ₘ* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.satisfiable_iff_eq | [234, 1] | [242, 48] | exact ⟨M, i, s, hM⟩ | case mpr
L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
T : Theory L
inst✝ : 𝐄𝐐 ≼ T
M : Type v
i : M
s : Structure L M
left✝ : Structure.Eq L M
hM : M ⊧ₘ* T
⊢ ∃ M a x, M ⊧ₘ* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Semiformula.eval_existsUnique | [306, 1] | [308, 83] | simp [existsUnique, Semiformula.eval_substs, Matrix.comp_vecCons', ExistsUnique] | L : Language
μ : Type u_1
inst✝¹ : Operator.Eq L
M : Type u_2
s : Structure L M
inst✝ : Structure.Eq L M
n : ℕ
e : Fin n → M
ε : μ → M
p : Semiformula L μ (n + 1)
⊢ (Eval s e ε) (∃'! p) ↔ ∃! x, (Eval s (x :> e) ε) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realize_iff | [64, 1] | [66, 60] | simp [LogicalConnective.iff, iff_iff_implies_and_implies] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
𝓜 : M
p q : F
⊢ 𝓜 ⊧ p ⟷ q ↔ (𝓜 ⊧ p ↔ 𝓜 ⊧ q) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realize_list_conj | [68, 1] | [69, 63] | induction l <;> simp [*] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
𝓜 : M
l : List F
⊢ 𝓜 ⊧ l.conj ↔ ∀ p ∈ l, 𝓜 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realize_finset_conj | [71, 1] | [72, 57] | simp [Finset.conj] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
𝓜 : M
s : Finset F
⊢ 𝓜 ⊧ ⋀s ↔ ∀ p ∈ s, 𝓜 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realize_list_disj | [74, 1] | [75, 63] | induction l <;> simp [*] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
𝓜 : M
l : List F
⊢ 𝓜 ⊧ l.disj ↔ ∃ p ∈ l, 𝓜 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realize_finset_disj | [77, 1] | [78, 57] | simp [Finset.disj] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
𝓜 : M
s : Finset F
⊢ 𝓜 ⊧ ⋁s ↔ ∃ p ∈ s, 𝓜 ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.meaningful_iff | [104, 1] | [105, 40] | rintro ⟨h⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
⊢ Meaningful 𝓜 → ∃ f, ¬𝓜 ⊧ f | case mk
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
h : ∃ f, ¬𝓜 ⊧ f
⊢ ∃ f, ¬𝓜 ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.meaningful_iff | [104, 1] | [105, 40] | exact h | case mk
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
h : ∃ f, ¬𝓜 ⊧ f
⊢ ∃ f, ¬𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.not_meaningful_iff | [107, 1] | [107, 90] | simp [meaningful_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
⊢ ¬Meaningful 𝓜 ↔ ∀ (f : F), 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realizeSet_iff | [109, 1] | [110, 58] | rintro ⟨h⟩ f hf | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
⊢ 𝓜 ⊧* T → ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f | case mk
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
h : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
f : F
hf : f ∈ T
⊢ 𝓜 ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realizeSet_iff | [109, 1] | [110, 58] | exact h hf | case mk
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
h : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
f : F
hf : f ∈ T
⊢ 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realizeSet_iff | [109, 1] | [110, 58] | intro h | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
⊢ (∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f) → 𝓜 ⊧* T | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
h : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
⊢ 𝓜 ⊧* T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.realizeSet_iff | [109, 1] | [110, 58] | exact ⟨h⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
T : Set F
h : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
⊢ 𝓜 ⊧* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.not_satisfiable_finset | [112, 1] | [114, 61] | simp [Satisfiable, realizeSet_iff, Valid, Finset.map_disj] | M : Type u_1
F : Type u_2
inst✝² : LogicalConnective F
𝓢 : Semantics F M
inst✝¹ : Tarski M
inst✝ : DecidableEq F
t : Finset F
⊢ ¬Satisfiable M ↑t ↔ Valid M (⋁Finset.image (fun x => ~x) t) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.satisfiableSet_iff_models_nonempty | [116, 1] | [118, 71] | rintro ⟨𝓜, h𝓜⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
⊢ Satisfiable M T → (models M T).Nonempty | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
𝓜 : M
h𝓜 : 𝓜 ⊧* T
⊢ (models M T).Nonempty |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.satisfiableSet_iff_models_nonempty | [116, 1] | [118, 71] | exact ⟨𝓜, h𝓜⟩ | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
𝓜 : M
h𝓜 : 𝓜 ⊧* T
⊢ (models M T).Nonempty | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.satisfiableSet_iff_models_nonempty | [116, 1] | [118, 71] | rintro ⟨𝓜, h𝓜⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
⊢ (models M T).Nonempty → Satisfiable M T | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
𝓜 : M
h𝓜 : 𝓜 ∈ models M T
⊢ Satisfiable M T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.satisfiableSet_iff_models_nonempty | [116, 1] | [118, 71] | exact ⟨𝓜, h𝓜⟩ | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
𝓜 : M
h𝓜 : 𝓜 ∈ models M T
⊢ Satisfiable M T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.empty | [133, 1] | [133, 60] | simp | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
𝓜 : M
⊢ ∀ ⦃f : F⦄, f ∈ ∅ → 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.singleton_iff | [135, 1] | [136, 49] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
f : F
𝓜 : M
⊢ 𝓜 ⊧* {f} ↔ 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.insert_iff | [138, 1] | [140, 24] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T : Set F
f : F
𝓜 : M
⊢ 𝓜 ⊧* insert f T ↔ 𝓜 ⊧ f ∧ 𝓜 ⊧* T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.union_iff | [142, 1] | [147, 61] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
⊢ 𝓜 ⊧* T ∪ U ↔ 𝓜 ⊧* T ∧ 𝓜 ⊧* U | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
⊢ (∀ ⦃f : F⦄, f ∈ T ∨ f ∈ U → 𝓜 ⊧ f) ↔ (∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f) ∧ ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.union_iff | [142, 1] | [147, 61] | exact
⟨ fun h => ⟨fun _ hf => h (Or.inl hf), fun _ hf => h (Or.inr hf)⟩,
by rintro ⟨h₁, h₂⟩ f (h | h); exact h₁ h; exact h₂ h ⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
⊢ (∀ ⦃f : F⦄, f ∈ T ∨ f ∈ U → 𝓜 ⊧ f) ↔ (∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f) ∧ ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.union_iff | [142, 1] | [147, 61] | rintro ⟨h₁, h₂⟩ f (h | h) | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
⊢ ((∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f) ∧ ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f) → ∀ ⦃f : F⦄, f ∈ T ∨ f ∈ U → 𝓜 ⊧ f | case intro.inl
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
h₂ : ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f
f : F
h : f ∈ T
⊢ 𝓜 ⊧ f
case intro.inr
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.union_iff | [142, 1] | [147, 61] | exact h₁ h | case intro.inl
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
h₂ : ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f
f : F
h : f ∈ T
⊢ 𝓜 ⊧ f
case intro.inr
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ... | case intro.inr
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
h₂ : ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f
f : F
h : f ∈ U
⊢ 𝓜 ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.union_iff | [142, 1] | [147, 61] | exact h₂ h | case intro.inr
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
𝓜 : M
h₁ : ∀ ⦃f : F⦄, f ∈ T → 𝓜 ⊧ f
h₂ : ∀ ⦃f : F⦄, f ∈ U → 𝓜 ⊧ f
f : F
h : f ∈ U
⊢ 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.image_iff | [149, 1] | [150, 65] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
ι : Type u_3
f : ι → F
A : Set ι
𝓜 : M
⊢ 𝓜 ⊧* f '' A ↔ ∀ i ∈ A, 𝓜 ⊧ f i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.range_iff | [152, 1] | [153, 66] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
ι : Sort u_3
f : ι → F
𝓜 : M
⊢ 𝓜 ⊧* Set.range f ↔ ∀ (i : ι), 𝓜 ⊧ f i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.RealizeSet.setOf_iff | [155, 1] | [156, 64] | simp [realizeSet_iff] | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
P : F → Prop
𝓜 : M
⊢ 𝓜 ⊧* setOf P ↔ ∀ (f : F), P f → 𝓜 ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.valid_neg_iff | [160, 1] | [160, 107] | simp [Valid, Satisfiable] | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : Tarski M
f : F
⊢ Valid M (~f) ↔ ¬Satisfiable M {f} | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.Satisfiable.of_subset | [162, 1] | [163, 61] | rcases h with ⟨𝓜, h⟩ | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
h : Satisfiable M U
ss : T ⊆ U
⊢ Satisfiable M T | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
ss : T ⊆ U
𝓜 : M
h : 𝓜 ⊧* U
⊢ Satisfiable M T |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.Satisfiable.of_subset | [162, 1] | [163, 61] | exact ⟨𝓜, RealizeSet.of_subset h ss⟩ | case intro
M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
T U : Set F
ss : T ⊆ U
𝓜 : M
h : 𝓜 ⊧* U
⊢ Satisfiable M T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.empty_models | [169, 1] | [169, 74] | rintro h | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
f : F
⊢ ∅ ⊧ f | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
f : F
h : M
⊢ h ∈ ∅ → h ⊧ f |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.empty_models | [169, 1] | [169, 74] | simp | M : Type u_1
F : Type u_2
inst✝ : LogicalConnective F
𝓢 : Semantics F M
f : F
h : M
⊢ h ∈ ∅ → h ⊧ f | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | rintro ⟨f, hf⟩ | M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
⊢ Meaningful s → s.Nonempty | case mk.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
f : F
hf : ¬s ⊧ f
⊢ s.Nonempty |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | by_contra A | case mk.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
f : F
hf : ¬s ⊧ f
⊢ s.Nonempty | case mk.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
f : F
hf : ¬s ⊧ f
A : ¬s.Nonempty
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Logic/Semantics.lean | LO.Semantics.set_meaningful_iff_nonempty | [182, 1] | [186, 62] | rcases Set.not_nonempty_iff_eq_empty.mp A | case mk.intro
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
s : Set M
f : F
hf : ¬s ⊧ f
A : ¬s.Nonempty
⊢ False | case mk.intro.refl
M : Type u_1
F : Type u_2
inst✝¹ : LogicalConnective F
𝓢 : Semantics F M
inst✝ : ∀ (𝓜 : M), Meaningful 𝓜
f : F
hf : ¬∅ ⊧ f
A : ¬∅.Nonempty
⊢ False |
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