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/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (phi.iff_ psi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔ Holds D I V E_ref (replace P zs H (phi.iff_ psi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔ Holds D I V E_ref (replace P zs H (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔ Holds D I V E_ref (replace P zs H (phi.iff_ psi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔
Holds D I V E_ref ((replace P zs H phi).iff_ (replace P zs H psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔
Holds D I V E_ref ((replace P zs H phi).iff_ (replace P zs H psi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
(Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
(Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi)) | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
left✝ : admitsAux P zs H binders phi
right✝ : admitsAux P zs H binders psi
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
(Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
(Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi)) | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact phi_ih V binders h1_left h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact psi_ih V binders h1_right h2 | case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders psi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (exists_ x phi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (exists_ x (replace P zs H phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (exists_ x (replace P zs H phi)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d, Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d, Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi ↔
Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi ↔
Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro v a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ Function.updateITE V x d v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Function.updateITE] | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ Function.updateITE V x d v = V' v | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ (if v = x then d else V v) = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ (if v = x then d else V v) = V' v | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ ¬v = x
⊢ (if v = x then d else V v) = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | push_neg at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ ¬v = x
⊢ (if v = x then d else V v) = V' v | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ v ≠ x
⊢ (if v = x then d else V v) = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∧ v ≠ x
⊢ (if v = x then d else V v) = V' v | case h.intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
left✝ : v ∉ binders
right✝ : v ≠ x
⊢ (if v = x then d else V v) = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ (if v = x then d else V v) = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∀ (d : D), Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
∀ (d : D), Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d, Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
∃ d, Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi) | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [if_neg a1_right] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ (if v = x then d else V v) = V' v | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ V v = V' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
admitsAux P zs H binders phi →
(∀ x ∉ binders, V x = V' x) →
(Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1_left : v ∉ binders
a1_right : v ≠ x
⊢ V v = V' v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (replace P zs H (def_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [E_ref] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (def_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' [] P zs H) V [] (def_ X xs) ↔ Holds D I V [] (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' [] P zs H) V [] (def_ X xs) ↔ Holds D I V [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (replace P zs H (def_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [E_ref] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Holds D I V E_ref (def_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' (hd :: tl) P zs H) V (hd :: tl) (def_ X xs) ↔ Holds D I V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' (hd :: tl) P zs H) V (hd :: tl) (def_ X xs) ↔ Holds D I V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D (I' D I V' (hd :: tl) P zs H) V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = hd.name ∧ xs.length = hd.args.length then
Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D (I' D I V' (hd :: tl) P zs H) V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
h✝ : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) V tl (def_ X xs) ↔ Holds D I V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) (Function.updateListITE V hd.args (List.map V xs)) E_ref hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) E_ref (replace P zs H hd.q)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace_no_predVar P zs H hd.q hd.h2] at ih | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) (Function.updateListITE V hd.args (List.map V xs)) E_ref hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) E_ref (replace P zs H hd.q)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ (I' D I V' (hd :: tl) P zs H).pred_const_ = I.pred_const_
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
predVarOccursIn P_1 ds.length hd.q → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ (I' D I V' (hd :: tl) P zs H).pred_const_ = I.pred_const_ | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) (hd :: tl) H
else I.pred_var_ Q ds).pred_const_ =
I.pred_const_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) (hd :: tl) H
else I.pred_var_ Q ds).pred_const_ =
I.pred_const_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [predVarOccursIn_iff_mem_predVarSet] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
predVarOccursIn P_1 ds.length hd.q → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
(P_1, ds.length) ∈ hd.q.predVarSet → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [hd.h2] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
(P_1, ds.length) ∈ hd.q.predVarSet → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
(P_1, ds.length) ∈ ∅ → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
∀ (binders : Finset VarName),
admitsAux P zs H binders hd.q →
(∀ x ∉ binders, Function.updateListITE V hd.args (List.map V xs) x = V' x) →
(Holds D (I' D I V' tl P zs H) (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q)
⊢ ∀ (P_1 : PredName) (ds : List D),
(P_1, ds.length) ∈ ∅ → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D (I' D I V' (hd :: tl) P zs H) V tl (def_ X xs) ↔ Holds D I V tl (def_ X xs) | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ (I' D I V' (hd :: tl) P zs H).pred_const_ = I.pred_const_
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P_1 : PredName) (ds : List D),
predVarOccursIn P_1 ds.length (def_ X xs) → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ (I' D I V' (hd :: tl) P zs H).pred_const_ = I.pred_const_ | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) (hd :: tl) H
else I.pred_var_ Q ds).pred_const_ =
I.pred_const_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ (Interpretation.usingPred D I fun Q ds =>
if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) (hd :: tl) H
else I.pred_var_ Q ds).pred_const_ =
I.pred_const_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [predVarOccursIn] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P_1 : PredName) (ds : List D),
predVarOccursIn P_1 ds.length (def_ X xs) → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P_1 : PredName) (ds : List D), False → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P_1 : PredName) (ds : List D), False → ((I' D I V' (hd :: tl) P zs H).pred_var_ P_1 ds ↔ I.pred_var_ P_1 ds) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem | [337, 1] | [352, 9] | apply substitution_theorem_aux D I V V E F P zs H ∅ | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
⊢ Holds D (I' D I V E P zs H) V E F ↔ Holds D I V E (replace P zs H F) | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
⊢ admitsAux P zs H ∅ F
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
⊢ ∀ x ∉ ∅, V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem | [337, 1] | [352, 9] | exact h1 | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
⊢ admitsAux P zs H ∅ F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem | [337, 1] | [352, 9] | simp | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
⊢ ∀ x ∉ ∅, V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_is_valid | [355, 1] | [369, 11] | simp only [IsValid] at h2 | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : F.IsValid
⊢ (replace P zs H F).IsValid | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (replace P zs H F).IsValid |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_is_valid | [355, 1] | [369, 11] | simp only [IsValid] | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (replace P zs H F).IsValid | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace P zs H F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_is_valid | [355, 1] | [369, 11] | intro D I V E | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace P zs H F) | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (replace P zs H F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_is_valid | [355, 1] | [369, 11] | simp only [← substitution_theorem D I V E F P zs H h1] | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (replace P zs H F) | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D (I' D I V E P zs H) V E F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_is_valid | [355, 1] | [369, 11] | apply h2 | F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D (I' D I V E P zs H) V E F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | induction F | v : Var
F : Formula
⊢ occursIn v F ↔ v ∈ F.varSet | case pred_
v : Var
a✝¹ : String
a✝ : List Var
⊢ occursIn v (pred_ a✝¹ a✝) ↔ v ∈ (pred_ a✝¹ a✝).varSet
case not_
v : Var
a✝ : Formula
a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet
⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet
case imp_
v : Var
a✝¹ a✝ : Formula
a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet
a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet
⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet
case forall_
v : Var
a✝¹ : String
a✝ : Formula
a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet
⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | case pred_ X vs =>
simp only [occursIn]
simp only [varSet]
simp | v : Var
X : String
vs : List Var
⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | case not_ phi phi_ih =>
simp only [occursIn]
simp only [varSet]
exact phi_ih | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | case imp_ phi psi phi_ih psi_ih =>
simp only [occursIn]
simp only [varSet]
simp
congr! | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | case forall_ _ phi phi_ih =>
simp only [occursIn]
simp only [varSet]
exact phi_ih | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [occursIn] | v : Var
X : String
vs : List Var
⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet | v : Var
X : String
vs : List Var
⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [varSet] | v : Var
X : String
vs : List Var
⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet | v : Var
X : String
vs : List Var
⊢ v ∈ vs ↔ v ∈ vs.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp | v : Var
X : String
vs : List Var
⊢ v ∈ vs ↔ v ∈ vs.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [occursIn] | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.not_.varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [varSet] | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.not_.varSet | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | exact phi_ih | v : Var
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [occursIn] | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [varSet] | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | congr! | v : Var
phi psi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
psi_ih : occursIn v psi ↔ v ∈ psi.varSet
⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [occursIn] | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | simp only [varSet] | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.varSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.occursIn_iff_mem_varSet | [186, 1] | [208, 17] | exact phi_ih | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ↔ v ∈ phi.varSet
⊢ occursIn v phi ↔ v ∈ phi.varSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | induction F | v : Var
F : Formula
⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet | case pred_
v : Var
a✝¹ : String
a✝ : List Var
⊢ occursIn v (pred_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (pred_ a✝¹ a✝).freeVarSet
case not_
v : Var
a✝ : Formula
a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet
⊢ occursIn v a✝.not_ ∧ v.isFree ↔ v ∈ a✝.not_.freeVarSet
case imp_
v : Var
a✝¹ a✝ : Formula
a_ih✝¹ : occursIn v a✝¹ ∧ v.isFree ↔ v ∈ a✝¹.freeVarSet
a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet
⊢ occursIn v (a✝¹.imp_ a✝) ∧ v.isFree ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet
case forall_
v : Var
a✝¹ : String
a✝ : Formula
a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet
⊢ occursIn v (forall_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case not_ phi phi_ih =>
simp only [Formula.freeVarSet]
simp only [occursIn]
exact phi_ih | v : Var
phi : Formula
phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet
⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case imp_ phi psi phi_ih psi_ih =>
simp only [Formula.freeVarSet]
simp only [occursIn]
simp
tauto | v : Var
phi psi : Formula
phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet
psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet
⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case forall_ _ phi phi_ih =>
simp only [Formula.freeVarSet]
simp only [occursIn]
exact phi_ih | v : Var
a✝ : String
phi : Formula
phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet
⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp only [Formula.freeVarSet] | v : Var
X : String
vs : List Var
⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ (pred_ X vs).freeVarSet | v : Var
X : String
vs : List Var
⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp only [occursIn] | v : Var
X : String
vs : List Var
⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet | v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp | v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet | v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | constructor | v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet | case mp
v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet
case mpr
v : Var
X : String
vs : List Var
⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | intro a1 | case mp
v : Var
X : String
vs : List Var
⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet | case mp
v : Var
X : String
vs : List Var
a1 : v ∈ vs ∧ v.isFree
⊢ ∃ a ∈ vs, v ∈ a.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | apply Exists.intro v | case mp
v : Var
X : String
vs : List Var
a1 : v ∈ vs ∧ v.isFree
⊢ ∃ a ∈ vs, v ∈ a.freeVarSet | case mp
v : Var
X : String
vs : List Var
a1 : v ∈ vs ∧ v.isFree
⊢ v ∈ vs ∧ v ∈ v.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | cases v | case mp
v : Var
X : String
vs : List Var
a1 : v ∈ vs ∧ v.isFree
⊢ v ∈ vs ∧ v ∈ v.freeVarSet | case mp.free_
X : String
vs : List Var
a✝ : String
a1 : free_ a✝ ∈ vs ∧ (free_ a✝).isFree
⊢ free_ a✝ ∈ vs ∧ free_ a✝ ∈ (free_ a✝).freeVarSet
case mp.bound_
X : String
vs : List Var
a✝ : ℕ
a1 : bound_ a✝ ∈ vs ∧ (bound_ a✝).isFree
⊢ bound_ a✝ ∈ vs ∧ bound_ a✝ ∈ (bound_ a✝).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ x =>
simp only [Var.freeVarSet]
simp
cases a1
case _ a1_left a1_right =>
exact a1_left | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ i =>
simp only [isFree] at a1
cases a1
case _ a1_left a1_right =>
contradiction | X : String
vs : List Var
i : ℕ
a1 : bound_ i ∈ vs ∧ (bound_ i).isFree
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp only [Var.freeVarSet] | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x} | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | cases a1 | X : String
vs : List Var
x : String
a1 : free_ x ∈ vs ∧ (free_ x).isFree
⊢ free_ x ∈ vs | case intro
X : String
vs : List Var
x : String
left✝ : free_ x ∈ vs
right✝ : (free_ x).isFree
⊢ free_ x ∈ vs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ a1_left a1_right =>
exact a1_left | X : String
vs : List Var
x : String
a1_left : free_ x ∈ vs
a1_right : (free_ x).isFree
⊢ free_ x ∈ vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | exact a1_left | X : String
vs : List Var
x : String
a1_left : free_ x ∈ vs
a1_right : (free_ x).isFree
⊢ free_ x ∈ vs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp only [isFree] at a1 | X : String
vs : List Var
i : ℕ
a1 : bound_ i ∈ vs ∧ (bound_ i).isFree
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet | X : String
vs : List Var
i : ℕ
a1 : bound_ i ∈ vs ∧ False
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | cases a1 | X : String
vs : List Var
i : ℕ
a1 : bound_ i ∈ vs ∧ False
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet | case intro
X : String
vs : List Var
i : ℕ
left✝ : bound_ i ∈ vs
right✝ : False
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ a1_left a1_right =>
contradiction | X : String
vs : List Var
i : ℕ
a1_left : bound_ i ∈ vs
a1_right : False
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | contradiction | X : String
vs : List Var
i : ℕ
a1_left : bound_ i ∈ vs
a1_right : False
⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | intro a1 | case mpr
v : Var
X : String
vs : List Var
⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
⊢ v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | apply Exists.elim a1 | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
⊢ v ∈ vs ∧ v.isFree | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | intro u a2 | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
u : Var
a2 : u ∈ vs ∧ v ∈ u.freeVarSet
⊢ v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | cases u | case mpr
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
u : Var
a2 : u ∈ vs ∧ v ∈ u.freeVarSet
⊢ v ∈ vs ∧ v.isFree | case mpr.free_
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
a✝ : String
a2 : free_ a✝ ∈ vs ∧ v ∈ (free_ a✝).freeVarSet
⊢ v ∈ vs ∧ v.isFree
case mpr.bound_
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
a✝ : ℕ
a2 : bound_ a✝ ∈ vs ∧ v ∈ (bound_ a✝).freeVarSet
⊢ v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ x =>
simp only [Var.freeVarSet] at a2
simp at a2
cases a2
case _ a2_left a2_right =>
subst a2_right
simp only [isFree]
simp
exact a2_left | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet
⊢ v ∈ vs ∧ v.isFree | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | case _ i =>
simp only [Var.freeVarSet] at a2
simp at a2 | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
i : ℕ
a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet
⊢ v ∈ vs ∧ v.isFree | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp only [Var.freeVarSet] at a2 | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet
⊢ v ∈ vs ∧ v.isFree | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v ∈ {free_ x}
⊢ v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | simp at a2 | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v ∈ {free_ x}
⊢ v ∈ vs ∧ v.isFree | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v = free_ x
⊢ v ∈ vs ∧ v.isFree |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Binders.lean | LN.isFreeIn_iff_mem_freeVarSet | [211, 1] | [264, 17] | cases a2 | v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
a2 : free_ x ∈ vs ∧ v = free_ x
⊢ v ∈ vs ∧ v.isFree | case intro
v : Var
X : String
vs : List Var
a1 : ∃ a ∈ vs, v ∈ a.freeVarSet
x : String
left✝ : free_ x ∈ vs
right✝ : v = free_ x
⊢ v ∈ vs ∧ v.isFree |
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