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/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
by_cases c1 : y β binders
|
case a.h.e'_3.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
β’ y β binders β¨ Ο' y β binders
|
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders β¨ Ο' y β binders
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders β¨ Ο' y β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
left
|
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders β¨ Ο' y β binders
|
case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact c1
|
case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
right
|
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ y β binders β¨ Ο' y β binders
|
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ Ο' y β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [h3 y c1]
|
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ Ο' y β binders
|
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ Ο y β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact h1_right c1
|
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : x β binders β Ο x β binders
h1_right : y β binders β Ο y β binders
c1 : y β binders
β’ Ο y β binders
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
congr! 1
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο binders phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ Β¬Holds D I V (headβ :: tailβ) phi β Β¬Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi)
|
case a.h.e'_1.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο binders phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact phi_ih V V' Ο Ο' binders h1 h2 h2' h3
|
case a.h.e'_1.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο binders phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
cases h1
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο binders phi β§ admitsAux Ο binders psi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ (Holds D I V (headβ :: tailβ) phi β Holds D I V (headβ :: tailβ) psi) β
(Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi) β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
|
case intro
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
leftβ : admitsAux Ο binders phi
rightβ : admitsAux Ο binders psi
β’ (Holds D I V (headβ :: tailβ) phi β Holds D I V (headβ :: tailβ) psi) β
(Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi) β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
congr! 1
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : admitsAux Ο binders phi
h1_right : admitsAux Ο binders psi
β’ (Holds D I V (headβ :: tailβ) phi β Holds D I V (headβ :: tailβ) psi) β
(Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi) β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
|
case a.h.e'_1.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : admitsAux Ο binders phi
h1_right : admitsAux Ο binders psi
β’ Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : admitsAux Ο binders phi
h1_right : admitsAux Ο binders psi
β’ Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact phi_ih V V' Ο Ο' binders h1_left h2 h2' h3
|
case a.h.e'_1.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : admitsAux Ο binders phi
h1_right : admitsAux Ο binders psi
β’ Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact psi_ih V V' Ο Ο' binders h1_right h2 h2' h3
|
case a.h.e'_2.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
psi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders psi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
h1_left : admitsAux Ο binders phi
h1_right : admitsAux Ο binders psi
β’ Holds D I V (headβ :: tailβ) psi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' psi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
first | apply forall_congr' | apply exists_congr
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ (β d, Holds D I (Function.updateITE V x d) (headβ :: tailβ) phi) β
β d, Holds D I (Function.updateITE V' x d) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ β (a : D),
Holds D I (Function.updateITE V x a) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x a) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro d
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ β (a : D),
Holds D I (Function.updateITE V x a) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x a) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ Holds D I (Function.updateITE V x d) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x d) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply phi_ih (Function.updateITE V x d) (Function.updateITE V' x d) Ο (Function.updateITE Ο' x x) (binders βͺ {x}) h1
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ Holds D I (Function.updateITE V x d) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x d) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β (v : VarName),
v β binders βͺ {x} β¨ Function.updateITE Ο' x x v β binders βͺ {x} β
Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β v β binders βͺ {x}, v = Function.updateITE Ο' x x v
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β v β binders βͺ {x}, Function.updateITE Ο' x x v = Ο v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply forall_congr'
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ (β (d : D), Holds D I (Function.updateITE V x d) (headβ :: tailβ) phi) β
β (d : D), Holds D I (Function.updateITE V' x d) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ β (a : D),
Holds D I (Function.updateITE V x a) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x a) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply exists_congr
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ (β d, Holds D I (Function.updateITE V x d) (headβ :: tailβ) phi) β
β d, Holds D I (Function.updateITE V' x d) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
case h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ β (a : D),
Holds D I (Function.updateITE V x a) (headβ :: tailβ) phi β
Holds D I (Function.updateITE V' x a) (headβ :: tailβ) (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro v a1
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β (v : VarName),
v β binders βͺ {x} β¨ Function.updateITE Ο' x x v β binders βͺ {x} β
Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x} β¨ Function.updateITE Ο' x x v β binders βͺ {x}
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [Function.updateITE] at a1
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x} β¨ Function.updateITE Ο' x x v β binders βͺ {x}
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x} β¨ (if v = x then x else Ο' v) β binders βͺ {x}
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp at a1
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x} β¨ (if v = x then x else Ο' v) β binders βͺ {x}
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [Function.updateITE]
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ (if v = x then d else V v) = if (if v = x then x else Ο' v) = x then d else V' (if v = x then x else Ο' v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
split_ifs
|
case h.h2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ (if v = x then d else V v) = if (if v = x then x else Ο' v) = x then d else V' (if v = x then x else Ο' v)
|
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
hβΒΉ : v = x
hβ : x = x
β’ d = d
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
hβΒΉ : v = x
hβ : Β¬x = x
β’ d = V' x
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
hβΒΉ : Β¬v = x
hβ : Ο' v = x
β’ V v = d
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
hβΒΉ : Β¬v = x
hβ : Β¬Ο' v = x
β’ V v = V' (Ο' v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
rfl
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : v = x
c2 : x = x
β’ d = d
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
contradiction
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : v = x
c2 : Β¬x = x
β’ d = V' x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
subst c2
tauto
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : Β¬v = x
c2 : Ο' v = x
β’ V v = d
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
simp only [if_neg c1] at a1
apply h2
tauto
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : Β¬v = x
c2 : Β¬Ο' v = x
β’ V v = V' (Ο' v)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
rfl
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : v = x
c2 : x = x
β’ d = d
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
contradiction
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : v = x
c2 : Β¬x = x
β’ d = V' x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
subst c2
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : Β¬v = x
c2 : Ο' v = x
β’ V v = d
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
h1 : admitsAux Ο (binders βͺ {Ο' v}) phi
a1 : (v β binders β¨ v = Ο' v) β¨ (if v = Ο' v then Ο' v else Ο' v) β binders β§ Β¬v = Ο' v β§ Β¬Ο' v = Ο' v
c1 : Β¬v = Ο' v
β’ V v = d
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
tauto
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
h1 : admitsAux Ο (binders βͺ {Ο' v}) phi
a1 : (v β binders β¨ v = Ο' v) β¨ (if v = Ο' v then Ο' v else Ο' v) β binders β§ Β¬v = Ο' v β§ Β¬Ο' v = Ο' v
c1 : Β¬v = Ο' v
β’ V v = d
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [if_neg c1] at a1
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : (v β binders β¨ v = x) β¨ (if v = x then x else Ο' v) β binders β§ Β¬v = x β§ Β¬Ο' v = x
c1 : Β¬v = x
c2 : Β¬Ο' v = x
β’ V v = V' (Ο' v)
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
c1 : Β¬v = x
c2 : Β¬Ο' v = x
a1 : (v β binders β¨ v = x) β¨ Ο' v β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ V v = V' (Ο' v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply h2
|
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
c1 : Β¬v = x
c2 : Β¬Ο' v = x
a1 : (v β binders β¨ v = x) β¨ Ο' v β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ V v = V' (Ο' v)
|
case a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
c1 : Β¬v = x
c2 : Β¬Ο' v = x
a1 : (v β binders β¨ v = x) β¨ Ο' v β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ v β binders β¨ Ο' v β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
tauto
|
case a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
c1 : Β¬v = x
c2 : Β¬Ο' v = x
a1 : (v β binders β¨ v = x) β¨ Ο' v β binders β§ Β¬v = x β§ Β¬Ο' v = x
β’ v β binders β¨ Ο' v β binders
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro v a1
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β v β binders βͺ {x}, v = Function.updateITE Ο' x x v
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x}
β’ v = Function.updateITE Ο' x x v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp at a1
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x}
β’ v = Function.updateITE Ο' x x v
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β¨ v = x
β’ v = Function.updateITE Ο' x x v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [Function.updateITE]
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β¨ v = x
β’ v = Function.updateITE Ο' x x v
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β¨ v = x
β’ v = if v = x then x else Ο' v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
split_ifs <;> tauto
|
case h.h2'
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β¨ v = x
β’ v = if v = x then x else Ο' v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro v a1
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
β’ β v β binders βͺ {x}, Function.updateITE Ο' x x v = Ο v
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x}
β’ Function.updateITE Ο' x x v = Ο v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp at a1
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders βͺ {x}
β’ Function.updateITE Ο' x x v = Ο v
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β§ Β¬v = x
β’ Function.updateITE Ο' x x v = Ο v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [Function.updateITE]
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β§ Β¬v = x
β’ Function.updateITE Ο' x x v = Ο v
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β§ Β¬v = x
β’ (if v = x then x else Ο' v) = Ο v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
split_ifs <;> tauto
|
case h.h3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tailβ F β Holds D I V' tailβ (fastReplaceFree Ο' F))
x : VarName
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName),
admitsAux Ο binders phi β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β
(Holds D I V (headβ :: tailβ) phi β Holds D I V' (headβ :: tailβ) (fastReplaceFree Ο' phi))
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : admitsAux Ο (binders βͺ {x}) phi
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
d : D
v : VarName
a1 : v β binders β§ Β¬v = x
β’ (if v = x then x else Ο' v) = Ο v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
split_ifs
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ (if X = hd.name β§ 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)) β
if X = hd.name β§ (List.map Ο' xs).length = hd.args.length then
Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
else Holds D I V' tl (def_ X (List.map Ο' xs))
|
case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
hβΒΉ : X = hd.name β§ xs.length = hd.args.length
hβ : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
hβΒΉ : X = hd.name β§ xs.length = hd.args.length
hβ : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β Holds D I V' tl (def_ X (List.map Ο' xs))
case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
hβΒΉ : Β¬(X = hd.name β§ xs.length = hd.args.length)
hβ : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I V tl (def_ X xs) β Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
hβΒΉ : Β¬(X = hd.name β§ xs.length = hd.args.length)
hβ : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β Holds D I V' tl (def_ X (List.map Ο' xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
simp at c2
contradiction
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I V tl (def_ X xs) β Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
case _ c1 c2 =>
specialize ih V V' Ο Ο' binders (def_ X xs)
simp only [fastReplaceFree] at ih
apply ih h1 h2 h2' h3
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
have s1 : List.map V xs = List.map (V' β Ο') xs
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ List.map V xs = List.map (V' β Ο') xs
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [s1]
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ Holds D I (Function.updateListITE V hd.args (List.map (V' β Ο') xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply Holds_coincide_Var
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ Holds D I (Function.updateListITE V hd.args (List.map (V' β Ο') xs)) tl hd.q β
Holds D I (Function.updateListITE V' hd.args (List.map (V' β Ο') xs)) tl hd.q
|
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ β (v : VarName),
isFreeIn v hd.q β
Function.updateListITE V hd.args (List.map (V' β Ο') xs) v =
Function.updateListITE V' hd.args (List.map (V' β Ο') xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro v a1
|
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
β’ β (v : VarName),
isFreeIn v hd.q β
Function.updateListITE V hd.args (List.map (V' β Ο') xs) v =
Function.updateListITE V' hd.args (List.map (V' β Ο') xs) v
|
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ Function.updateListITE V hd.args (List.map (V' β Ο') xs) v =
Function.updateListITE V' hd.args (List.map (V' β Ο') xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply Function.updateListITE_mem_eq_len
|
case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ Function.updateListITE V hd.args (List.map (V' β Ο') xs) v =
Function.updateListITE V' hd.args (List.map (V' β Ο') xs) v
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ v β hd.args
case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ hd.args.length = (List.map (V' β Ο') xs).length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [List.map_eq_map_iff]
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ List.map V xs = List.map (V' β Ο') xs
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ β x β xs, V x = (V' β Ο') x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
intro x a1
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ β x β xs, V x = (V' β Ο') x
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ V x = (V' β Ο') x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ V x = (V' β Ο') x
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ V x = V' (Ο' x)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply h2
|
case s1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ V x = V' (Ο' x)
|
case s1.a
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ x β binders β¨ Ο' x β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
by_cases c3 : x β binders
|
case s1.a
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
β’ x β binders β¨ Ο' x β binders
|
case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders β¨ Ο' x β binders
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders β¨ Ο' x β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
left
|
case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders β¨ Ο' x β binders
|
case pos.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact c3
|
case pos.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
right
|
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ x β binders β¨ Ο' x β binders
|
case neg.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ Ο' x β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [h3 x c3]
|
case neg.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ Ο' x β binders
|
case neg.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ Ο x β binders
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
exact h1 x a1 c3
|
case neg.h
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
x : VarName
a1 : x β xs
c3 : x β binders
β’ Ο x β binders
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [isFreeIn_iff_mem_freeVarSet] at a1
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ v β hd.args
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : v β hd.q.freeVarSet
β’ v β hd.args
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [β List.mem_toFinset]
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : v β hd.q.freeVarSet
β’ v β hd.args
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : v β hd.q.freeVarSet
β’ v β hd.args.toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply Finset.mem_of_subset hd.h1 a1
|
case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : v β hd.q.freeVarSet
β’ v β hd.args.toFinset
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp
|
case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ hd.args.length = (List.map (V' β Ο') xs).length
|
case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ hd.args.length = xs.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
tauto
|
case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
s1 : List.map V xs = List.map (V' β Ο') xs
v : VarName
a1 : isFreeIn v hd.q
β’ hd.args.length = xs.length
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [List.length_map] at c2
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β Holds D I V' tl (def_ X (List.map Ο' xs))
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β Holds D I V' tl (def_ X (List.map Ο' xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
contradiction
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : X = hd.name β§ xs.length = hd.args.length
c2 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β Holds D I V' tl (def_ X (List.map Ο' xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp at c2
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : X = hd.name β§ (List.map Ο' xs).length = hd.args.length
β’ Holds D I V tl (def_ X xs) β Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : X = hd.name β§ xs.length = hd.args.length
β’ Holds D I V tl (def_ X xs) β Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
contradiction
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : X = hd.name β§ xs.length = hd.args.length
β’ Holds D I V tl (def_ X xs) β Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Ο' xs))) tl hd.q
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
specialize ih V V' Ο Ο' binders (def_ X xs)
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl F β Holds D I V' tl (fastReplaceFree Ο' F))
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
ih :
admitsAux Ο binders (def_ X xs) β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl (def_ X xs) β Holds D I V' tl (fastReplaceFree Ο' (def_ X xs)))
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
simp only [fastReplaceFree] at ih
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
ih :
admitsAux Ο binders (def_ X xs) β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl (def_ X xs) β Holds D I V' tl (fastReplaceFree Ο' (def_ X xs)))
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
ih :
admitsAux Ο binders (def_ X xs) β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs)))
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux
|
[74, 1]
|
[207, 28]
|
apply ih h1 h2 h2' h3
|
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : β v β xs, v β binders β Ο v β binders
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
c2 : Β¬(X = hd.name β§ (List.map Ο' xs).length = hd.args.length)
ih :
admitsAux Ο binders (def_ X xs) β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
(β v β binders, Ο' v = Ο v) β (Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs)))
β’ Holds D I V tl (def_ X xs) β Holds D I V' tl (def_ X (List.map Ο' xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem
|
[210, 1]
|
[224, 9]
|
apply substitution_theorem_aux D I (V β Ο) V E Ο Ο β
F h1
|
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ Holds D I (V β Ο) E F β Holds D I V E (fastReplaceFree Ο F)
|
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β (v : VarName), v β β
β¨ Ο v β β
β (V β Ο) v = V (Ο v)
case h2'
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β v β β
, v = Ο v
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β v β β
, Ο v = Ο v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem
|
[210, 1]
|
[224, 9]
|
simp
|
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β (v : VarName), v β β
β¨ Ο v β β
β (V β Ο) v = V (Ο v)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem
|
[210, 1]
|
[224, 9]
|
simp
|
case h2'
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β v β β
, v = Ο v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_theorem
|
[210, 1]
|
[224, 9]
|
simp
|
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
β’ β v β β
, Ο v = Ο v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_is_valid
|
[227, 1]
|
[239, 25]
|
simp only [IsValid] at h2
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : F.IsValid
β’ (fastReplaceFree Ο F).IsValid
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (fastReplaceFree Ο F).IsValid
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_is_valid
|
[227, 1]
|
[239, 25]
|
simp only [IsValid]
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (fastReplaceFree Ο F).IsValid
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (fastReplaceFree Ο F)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_is_valid
|
[227, 1]
|
[239, 25]
|
intro D I V E
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (fastReplaceFree Ο F)
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (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
β’ Holds D I V E (fastReplaceFree Ο F)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_is_valid
|
[227, 1]
|
[239, 25]
|
simp only [β substitution_theorem D I V E Ο F h1]
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (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
β’ Holds D I V E (fastReplaceFree Ο F)
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (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
β’ Holds D I (V β Ο) E F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/Admits.lean
|
FOL.NV.Sub.Var.All.Rec.substitution_is_valid
|
[227, 1]
|
[239, 25]
|
exact h2 D I (V β Ο) E
|
Ο : VarName β VarName
F : Formula
h1 : admits Ο F
h2 : β (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
β’ Holds D I (V β Ο) E F
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
list_cons_to_set_union
|
[9, 1]
|
[17, 9]
|
ext a
|
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x : Ξ±
β’ β(x :: ys).toFinset = {x} βͺ βys.toFinset
|
case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
list_cons_to_set_union
|
[9, 1]
|
[17, 9]
|
simp
|
case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
list_append_to_set_union
|
[19, 1]
|
[26, 9]
|
ext a
|
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
β’ β(xs ++ ys).toFinset = βxs.toFinset βͺ βys.toFinset
|
case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
list_append_to_set_union
|
[19, 1]
|
[26, 9]
|
simp
|
case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
induction g
|
Node : Type
instβ : DecidableEq Node
g : Graph Node
β’ β (x y : Node), y β direct_succ_list g x β β ys, y β ys β§ (x, ys) β g
|
case nil
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
case cons
Node : Type
instβ : DecidableEq Node
headβ : Node Γ List Node
tailβ : List (Node Γ List Node)
tail_ihβ : β (x y : Node), y β direct_succ_list tailβ x β β ys, y β ys β§ (x, ys) β tailβ
β’ β (x y : Node), y β direct_succ_list (headβ :: tailβ) x β β ys, y β ys β§ (x, ys) β headβ :: tailβ
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
case nil =>
simp only [direct_succ_list]
simp
|
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
simp only [direct_succ_list]
|
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
|
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β []
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
simp
|
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β []
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
simp only [direct_succ_list]
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node), y β direct_succ_list (hd :: tl) x β β ys, y β ys β§ (x, ys) β hd :: tl
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
intro x y
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
split_ifs
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl
|
case pos
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
hβ : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl
case neg
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
hβ : Β¬hd.1 = x
β’ y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
subst c1
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
c1 : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
simp
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
simp only [ih]
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
constructor
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mpr
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)) β y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
intro a1
|
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
cases a1
|
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
case mp.inl
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
hβ : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mp.inr
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
hβ : β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
mem_direct_succ_list_iff
|
[65, 1]
|
[122, 16]
|
case _ left =>
apply Exists.intro hd.snd
tauto
|
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
left : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
|
no goals
|
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