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] | 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 : PredName
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
v : VarName
a1 : v β xs
c1 : v β binders
β’ 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] | 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 : PredName
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
v : VarName
a1 : v β xs
c1 : v β binders
β’ v β binders β¨ Ο' v 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 : PredName
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
v : VarName
a1 : v β xs
c1 : 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] | simp only [h3 v 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 : PredName
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
v : VarName
a1 : v β xs
c1 : v β binders
β’ Ο' v 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 : PredName
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
v : VarName
a1 : v β xs
c1 : 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] | exact h1 v a1 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 : PredName
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
v : VarName
a1 : v β xs
c1 : 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] | 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))
x y : VarName
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
h1 : (x β binders β Ο x β binders) β§ (y β binders β Ο y 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’ V x = V y β V' (Ο' x) = V' (Ο' y) | 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))
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
leftβ : x β binders β Ο x β binders
rightβ : y β binders β Ο y β binders
β’ V x = V y β V' (Ο' x) = V' (Ο' y) |
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))
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’ V x = V y β V' (Ο' x) = V' (Ο' y) | case a.h.e'_2
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’ V x = V' (Ο' x)
case a.h.e'_3
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’ V y = V' (Ο' y) |
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 a.h.e'_2
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’ V x = V' (Ο' x) | 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))
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’ 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 c1 : x β binders | 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))
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’ x β binders β¨ Ο' x 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 : x β binders
β’ x β binders β¨ Ο' x 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 : 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
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 : x β binders
β’ x β binders β¨ Ο' x 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 : 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 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 : 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
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 : x β binders
β’ x β binders β¨ Ο' x 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 : 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 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 : x β binders
β’ Ο' x 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 : 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_left 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 : 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] | apply h2 | case a.h.e'_3
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’ V y = V' (Ο' y) | 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 |
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/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | induction F generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
β’ Holds D I (V β Ο) E F β Holds D I V E (replaceAll Ο F) | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_var_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_var_ aβΒΉ aβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (eq_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (eq_ aβΒΉ aβ))
case true_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E true_ β Holds D I V E (replaceAll Ο true_)
case false_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E false_ β Holds D I V E (replaceAll Ο false_)
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβ.not_ β Holds D I V E (replaceAll Ο aβ.not_)
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.imp_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.imp_ aβ))
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.and_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.and_ aβ))
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.or_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.or_ aβ))
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.iff_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.iff_ aβ))
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (forall_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (forall_ aβΒΉ aβ))
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (exists_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (exists_ aβΒΉ aβ))
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (def_ aβΒΉ aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | all_goals
simp only [replaceAll] | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_var_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_var_ aβΒΉ aβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (eq_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (eq_ aβΒΉ aβ))
case true_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E true_ β Holds D I V E (replaceAll Ο true_)
case false_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E false_ β Holds D I V E (replaceAll Ο false_)
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβ.not_ β Holds D I V E (replaceAll Ο aβ.not_)
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.imp_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.imp_ aβ))
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.and_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.and_ aβ))
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.or_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.or_ aβ))
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.iff_ aβ) β Holds D I V E (replaceAll Ο (aβΒΉ.iff_ aβ))
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (forall_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (forall_ aβΒΉ aβ))
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (exists_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (exists_ aβΒΉ aβ))
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (def_ aβΒΉ aβ)) | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (pred_const_ aβΒΉ (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_var_ aβΒΉ aβ) β Holds D I V E (pred_var_ aβΒΉ (List.map Ο aβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (eq_ aβΒΉ aβ) β Holds D I V E (eq_ (Ο aβΒΉ) (Ο aβ))
case true_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E true_ β Holds D I V E true_
case false_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E false_ β Holds D I V E false_
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβ.not_ β Holds D I V E (replaceAll Ο aβ).not_
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.imp_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).imp_ (replaceAll Ο aβ))
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.and_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).and_ (replaceAll Ο aβ))
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.or_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).or_ (replaceAll Ο aβ))
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.iff_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).iff_ (replaceAll Ο aβ))
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (forall_ aβΒΉ aβ) β Holds D I V E (forall_ (Ο aβΒΉ) (replaceAll Ο aβ))
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (exists_ aβΒΉ aβ) β Holds D I V E (exists_ (Ο aβΒΉ) (replaceAll Ο aβ))
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (def_ aβΒΉ (List.map Ο aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | any_goals
simp only [Holds] | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (pred_const_ aβΒΉ (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_var_ aβΒΉ aβ) β Holds D I V E (pred_var_ aβΒΉ (List.map Ο aβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (eq_ aβΒΉ aβ) β Holds D I V E (eq_ (Ο aβΒΉ) (Ο aβ))
case true_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E true_ β Holds D I V E true_
case false_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
V : VarAssignment D
β’ Holds D I (V β Ο) E false_ β Holds D I V E false_
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβ.not_ β Holds D I V E (replaceAll Ο aβ).not_
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.imp_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).imp_ (replaceAll Ο aβ))
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.and_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).and_ (replaceAll Ο aβ))
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.or_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).or_ (replaceAll Ο aβ))
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (aβΒΉ.iff_ aβ) β Holds D I V E ((replaceAll Ο aβΒΉ).iff_ (replaceAll Ο aβ))
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (forall_ aβΒΉ aβ) β Holds D I V E (forall_ (Ο aβΒΉ) (replaceAll Ο aβ))
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (exists_ aβΒΉ aβ) β Holds D I V E (exists_ (Ο aβΒΉ) (replaceAll Ο aβ))
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (def_ aβΒΉ (List.map Ο aβ)) | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ I.pred_const_ aβΒΉ (List.map (V β Ο) aβ) β I.pred_const_ aβΒΉ (List.map V (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ I.pred_var_ aβΒΉ (List.map (V β Ο) aβ) β I.pred_var_ aβΒΉ (List.map V (List.map Ο aβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : VarName
V : VarAssignment D
β’ (V β Ο) aβΒΉ = (V β Ο) aβ β V (Ο aβΒΉ) = V (Ο aβ)
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E aβ β Β¬Holds D I V E (replaceAll Ο aβ)
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβΒΉ β Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβΒΉ) β Holds D I V E (replaceAll Ο aβ)
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβΒΉ β§ Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβΒΉ) β§ Holds D I V E (replaceAll Ο aβ)
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E aβΒΉ β¨ Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβΒΉ) β¨ Holds D I V E (replaceAll Ο aβ)
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ aβ : Formula
a_ihβΒΉ : β (V : VarAssignment D), Holds D I (V β Ο) E aβΒΉ β Holds D I V E (replaceAll Ο aβΒΉ)
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ (Holds D I (V β Ο) E aβΒΉ β Holds D I (V β Ο) E aβ) β
(Holds D I V E (replaceAll Ο aβΒΉ) β Holds D I V E (replaceAll Ο aβ))
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ (β (d : D), Holds D I (Function.updateITE (V β Ο) aβΒΉ d) E aβ) β
β (d : D), Holds D I (Function.updateITE V (Ο aβΒΉ) d) E (replaceAll Ο aβ)
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) aβΒΉ d) E aβ) β
β d, Holds D I (Function.updateITE V (Ο aβΒΉ) d) E (replaceAll Ο aβ)
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (def_ aβΒΉ (List.map Ο aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case pred_const_ X xs | pred_var_ X xs =>
simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : PredName
xs : List VarName
V : VarAssignment D
β’ I.pred_var_ X (List.map (V β Ο) xs) β I.pred_var_ X (List.map V (List.map Ο xs)) | no goals |
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