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/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case eq_ x y =>
simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y) | 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] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (replaceAll Ο phi) | 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] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
first | apply forall_congr' | apply exists_congr
intro a
have s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
apply Function.updateITE_comp_right_injective
apply h1
simp only [β s1]
exact phi_ih (Function.updateITE V (Ο x) a) | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.updateITE V (Ο x) d) E (replaceAll Ο phi) | 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] | simp only [replaceAll] | 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 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] | simp only [Holds] | 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 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) |
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] | 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 |
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] | simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y) | 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] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (replaceAll Ο phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi) |
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] | exact phi_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi) | 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] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ (Holds D I (V β Ο) E phi β Holds D I (V β Ο) E psi) β
(Holds D I V E (replaceAll Ο phi) β Holds D I V E (replaceAll Ο psi)) | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi) |
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] | exact phi_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi) | 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] | exact psi_ih V | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi) | 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] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.updateITE V (Ο x) d) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | intro a | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | have s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | apply Function.updateITE_comp_right_injective | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | case s1.h1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.Injective Ο
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | apply h1 | case s1.h1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.Injective Ο
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | simp only [β s1] | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE V (Ο x) a β Ο) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | exact phi_ih (Function.updateITE V (Ο x) a) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I (Function.updateITE V (Ο x) a β Ο) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) | 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] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β (d : D), Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β (d : D), Holds D I (Function.updateITE V (Ο x) d) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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] | apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.updateITE V (Ο x) d) E (replaceAll Ο phi) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE V (Ο x) a) E (replaceAll Ο phi) |
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 E | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs)) | case nil
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
case cons
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
headβ : Definition
tailβ : List Definition
tail_ihβ : Holds D I (V β Ο) tailβ (def_ X xs) β Holds D I V tailβ (def_ X (List.map Ο xs))
β’ Holds D I (V β Ο) (headβ :: tailβ) (def_ X xs) β Holds D I V (headβ :: tailβ) (def_ X (List.map Ο xs)) |
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 nil =>
simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | 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] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | 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] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D I (V β Ο) (E_hd :: E_tl) (def_ X xs) β Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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] | simp | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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] | split_ifs | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
hβ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 _ c1 =>
apply E_ih | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | 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] | cases c1 | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | case intro
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
leftβ : X = E_hd.name
rightβ : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q |
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] | apply Holds_coincide_Var | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v |
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] | intro v a1 | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : isFreeIn v E_hd.q
β’ Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v |
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] | simp only [isFreeIn_iff_mem_freeVarSet v E_hd.q] at a1 | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : isFreeIn v E_hd.q
β’ Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v |
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] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ v β E_hd.args
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ E_hd.args.length = (List.map (V β Ο) xs).length |
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] | simp only [<- List.mem_toFinset] | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ v β E_hd.args | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ v β E_hd.args.toFinset |
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] | apply Finset.mem_of_subset E_hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ v β E_hd.args.toFinset | 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] | simp | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ E_hd.args.length = (List.map (V β Ο) xs).length | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ E_hd.args.length = xs.length |
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] | simp only [c1_right] | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.q.freeVarSet
β’ E_hd.args.length = xs.length | 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] | apply E_ih | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | 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_is_valid | [99, 1] | [111, 13] | simp only [IsValid] at h2 | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : F.IsValid
β’ (replaceAll Ο F).IsValid | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replaceAll Ο F).IsValid |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | simp only [IsValid] | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replaceAll Ο F).IsValid | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | intro D I V E | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | simp only [β substitution_theorem D I V E F Ο h1] | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | apply h2 | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | induction F generalizing V V' Ο | D : Type
I : Interpretation D
V V' V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
Ο : VarName β VarName
F : Formula
h1 : β (x : VarName), isFreeIn x F β V' x = V (Ο x)
h2 : β x β F.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E F β Holds D I V E (subAux c Ο Ο F) | case pred_const_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : PredName
aβ : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (pred_const_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (pred_const_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : PredName
aβ : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (pred_var_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (pred_var_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (pred_var_ aβΒΉ aβ))
case eq_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (eq_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (eq_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (eq_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (eq_ aβΒΉ aβ))
case true_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x true_ β V' x = V (Ο x)
h2 : β x β true_.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E true_ β Holds D I V E (subAux c Ο Ο true_)
case false_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x false_ β V' x = V (Ο x)
h2 : β x β false_.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E false_ β Holds D I V E (subAux c Ο Ο false_)
case not_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : Formula
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x aβ.not_ β V' x = V (Ο x)
h2 : β x β aβ.not_.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E aβ.not_ β Holds D I V E (subAux c Ο Ο aβ.not_)
case imp_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβΒΉ β V' x = V (Ο x)) β
(β x β aβΒΉ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβΒΉ β Holds D I V E (subAux c Ο Ο aβΒΉ))
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (aβΒΉ.imp_ aβ) β V' x = V (Ο x)
h2 : β x β (aβΒΉ.imp_ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (aβΒΉ.imp_ aβ) β Holds D I V E (subAux c Ο Ο (aβΒΉ.imp_ aβ))
case and_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβΒΉ β V' x = V (Ο x)) β
(β x β aβΒΉ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβΒΉ β Holds D I V E (subAux c Ο Ο aβΒΉ))
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (aβΒΉ.and_ aβ) β V' x = V (Ο x)
h2 : β x β (aβΒΉ.and_ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (aβΒΉ.and_ aβ) β Holds D I V E (subAux c Ο Ο (aβΒΉ.and_ aβ))
case or_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβΒΉ β V' x = V (Ο x)) β
(β x β aβΒΉ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβΒΉ β Holds D I V E (subAux c Ο Ο aβΒΉ))
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (aβΒΉ.or_ aβ) β V' x = V (Ο x)
h2 : β x β (aβΒΉ.or_ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (aβΒΉ.or_ aβ) β Holds D I V E (subAux c Ο Ο (aβΒΉ.or_ aβ))
case iff_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβΒΉ β V' x = V (Ο x)) β
(β x β aβΒΉ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβΒΉ β Holds D I V E (subAux c Ο Ο aβΒΉ))
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (aβΒΉ.iff_ aβ) β V' x = V (Ο x)
h2 : β x β (aβΒΉ.iff_ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (aβΒΉ.iff_ aβ) β Holds D I V E (subAux c Ο Ο (aβΒΉ.iff_ aβ))
case forall_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (forall_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (forall_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (forall_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (forall_ aβΒΉ aβ))
case exists_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x aβ β V' x = V (Ο x)) β
(β x β aβ.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E aβ β Holds D I V E (subAux c Ο Ο aβ))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (exists_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (exists_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (exists_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (exists_ aβΒΉ aβ))
case def_
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : DefName
aβ : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ aβΒΉ aβ) β V' x = V (Ο x)
h2 : β x β (def_ aβΒΉ aβ).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ aβΒΉ aβ) β Holds D I V E (subAux c Ο Ο (def_ aβΒΉ aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case pred_const_ X xs =>
simp only [isFreeIn] at h1
simp only [subAux]
simp only [Holds]
simp only [I']
simp only [Interpretation.usingPred]
simp
congr! 1
simp only [List.map_eq_map_iff]
intro x a1
simp
exact h1 x a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (pred_const_ X xs) β V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (subAux c Ο Ο (pred_const_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case eq_ x y =>
simp only [isFreeIn] at h1
simp only [subAux]
simp only [Holds]
have s1 : V' x = V (Ο x)
apply h1
simp
simp only [s1]
have s2 : V' y = V (Ο y)
apply h1
simp
simp only [s2] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x y : VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), isFreeIn x_1 (eq_ x y) β V' x_1 = V (Ο x_1)
h2 : β x_1 β (eq_ x y).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x_1 = V x_1
β’ Holds D (I' D I V'' E Ο) V' E (eq_ x y) β Holds D I V E (subAux c Ο Ο (eq_ x y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case true_ | false_ =>
simp only [subAux]
simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x false_ β V' x = V (Ο x)
h2 : β x β false_.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E false_ β Holds D I V E (subAux c Ο Ο false_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case not_ phi phi_ih =>
simp only [isFreeIn] at h1
simp only [predVarSet] at h2
simp only [subAux]
simp only [Holds]
congr! 1
exact phi_ih V V' Ο h1 h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi.not_ β V' x = V (Ο x)
h2 : β x β phi.not_.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E phi.not_ β Holds D I V E (subAux c Ο Ο phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (pred_const_ X xs) β V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (subAux c Ο Ο (pred_const_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (subAux c Ο Ο (pred_const_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [subAux] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (subAux c Ο Ο (pred_const_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (pred_const_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_const_ X xs) β Holds D I V E (pred_const_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (I' D I V'' E Ο).pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map V (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [I'] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (I' D I V'' E Ο).pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map V (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_
X (List.map V' xs) β
I.pred_const_ X (List.map V (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_
X (List.map V' xs) β
I.pred_const_ X (List.map V (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ I.pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map V (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ I.pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map V (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ I.pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map (V β Ο) xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | congr! 1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ I.pred_const_ X (List.map V' xs) β I.pred_const_ X (List.map (V β Ο) xs) | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = V (Ο x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact h1 x a1 | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_const_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = V (Ο x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (pred_var_ X xs) β V' x = V (Ο x)
h2 : β x β (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [predVarSet] at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (pred_var_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β {(X, xs.length)}.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β {(X, xs.length)}.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β predVarFreeVarSet Ο (X, xs.length), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [predVarFreeVarSet] at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β predVarFreeVarSet Ο (X, xs.length), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [subAux] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β Holds D I V E (subAux c Ο Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (pred_var_ X xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (I' D I V'' E Ο).pred_var_ X (List.map V' xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [I'] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (I' D I V'' E Ο).pred_var_ X (List.map V' xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_var_
X (List.map V' xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_var_
X (List.map V' xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (if h : (Ο X (List.map V' xs).length).isSome = true then
if (List.map V' xs).length = ((Ο X (List.map V' xs).length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X (List.map V' xs).length).get β―).1 (List.map V' xs)) E
((Ο X (List.map V' xs).length).get β―).2
else I.pred_var_ X (List.map V' xs)
else I.pred_var_ X (List.map V' xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (if h : (Ο X (List.map V' xs).length).isSome = true then
if (List.map V' xs).length = ((Ο X (List.map V' xs).length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X (List.map V' xs).length).get β―).1 (List.map V' xs)) E
((Ο X (List.map V' xs).length).get β―).2
else I.pred_var_ X (List.map V' xs)
else I.pred_var_ X (List.map V' xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2
else I.pred_var_ X (List.map V' xs)
else I.pred_var_ X (List.map V' xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
β’ (if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2
else I.pred_var_ X (List.map V' xs)
else I.pred_var_ X (List.map V' xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2
else pred_var_ X (List.map Ο xs)
else pred_var_ X (List.map Ο xs)) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
hβΒΉ : (Ο X xs.length).isSome = true
hβ : xs.length = ((Ο X xs.length).get β―).1.length
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2)
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
hβΒΉ : (Ο X xs.length).isSome = true
hβ : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β Holds D I V E (pred_var_ X (List.map Ο xs))
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
hβ : Β¬(Ο X xs.length).isSome = true
β’ I.pred_var_ X (List.map V' xs) β Holds D I V E (pred_var_ X (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1 c2 =>
simp only [Holds]
simp
congr! 1
simp only [List.map_eq_map_iff]
intro x a1
simp
exact h1 x a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β Holds D I V E (pred_var_ X (List.map Ο xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1 =>
simp only [Holds]
simp
congr! 1
simp only [List.map_eq_map_iff]
intro x a1
simp
exact h1 x a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : Β¬(Ο X xs.length).isSome = true
β’ I.pred_var_ X (List.map V' xs) β Holds D I V E (pred_var_ X (List.map Ο xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [c1] at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
h2 :
β x β if h : True then ((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset else β
, V'' x = V x
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
h2 :
β x β if h : True then ((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset else β
, V'' x = V x
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
h2 : β x β ((Ο X xs.length).get β―).2.freeVarSet, x β ((Ο X xs.length).get β―).1 β V'' x = V x
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | set zs := (Option.get (Ο X (List.length xs)) (_ : Option.isSome (Ο X (List.length xs)) = true)).1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
h2 : β x β ((Ο X xs.length).get β―).2.freeVarSet, x β ((Ο X xs.length).get β―).1 β V'' x = V x
β’ Holds D I (Function.updateListITE V'' ((Ο X xs.length).get β―).1 (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 (List.map Ο xs)) c
((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
h2 : β x β ((Ο X xs.length).get β―).2.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c ((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | set H := (Option.get (Ο X (List.length xs)) (_ : Option.isSome (Ο X (List.length xs)) = true)).2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
h2 : β x β ((Ο X xs.length).get β―).2.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c ((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s1 := Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateListITE id zs (xs.map Ο)) c H | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (V β Function.updateListITE id zs (List.map Ο xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateListITE_comp] at s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (V β Function.updateListITE id zs (List.map Ο xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE (V β id) zs (List.map V (List.map Ο xs))) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp at s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE (V β id) zs (List.map V (List.map Ο xs))) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [s1] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | clear s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
s1 :
Holds D I V E (Var.All.Rec.Fresh.sub (Function.updateListITE id zs (List.map Ο xs)) c H) β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ Holds D I (Function.updateListITE V'' zs (List.map V' xs)) E H β
Holds D I (Function.updateListITE V zs (List.map (V β Ο) xs)) E H | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ β (v : VarName),
isFreeIn v H β
Function.updateListITE V'' zs (List.map V' xs) v = Function.updateListITE V zs (List.map (V β Ο) xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro x a1 | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
β’ β (v : VarName),
isFreeIn v H β
Function.updateListITE V'' zs (List.map V' xs) v = Function.updateListITE V zs (List.map (V β Ο) xs) v | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | by_cases c3 : x β zs | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply Function.updateListITE_map_mem_ext | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x | case pos.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ β y β xs, V' y = (V β Ο) y
case pos.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ zs.length = xs.length
case pos.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β zs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | case pos.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ β y β xs, V' y = (V β Ο) y | case pos.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ β y β xs, V' y = V (Ο y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact h1 | case pos.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ β y β xs, V' y = V (Ο y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [β c2] | case pos.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ zs.length = xs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact c3 | case pos.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β zs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateListITE_not_mem V'' x zs (List.map V' xs) c3] | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ Function.updateListITE V'' zs (List.map V' xs) x = Function.updateListITE V zs (List.map (V β Ο) xs) x | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ V'' x = Function.updateListITE V zs (List.map (V β Ο) xs) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateListITE_not_mem V x zs (List.map (V β Ο ) xs) c3] | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ V'' x = Function.updateListITE V zs (List.map (V β Ο) xs) x | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ V'' x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply h2 | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ V'' x = V x | case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β H.freeVarSet
case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β zs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β H.freeVarSet | case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
c3 : x β zs
a1 : x β H.freeVarSet
β’ x β H.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact a1 | case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
c3 : x β zs
a1 : x β H.freeVarSet
β’ x β H.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact c3 | case neg.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
c1 : (Ο X xs.length).isSome = true
zs : List VarName := ((Ο X xs.length).get β―).1
c2 : xs.length = zs.length
H : Formula := ((Ο X xs.length).get β―).2
h2 : β x β H.freeVarSet, x β zs β V'' x = V x
x : VarName
a1 : isFreeIn x H
c3 : x β zs
β’ x β zs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β Holds D I V E (pred_var_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β I.pred_var_ X (List.map V (List.map Ο xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β I.pred_var_ X (List.map V (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β I.pred_var_ X (List.map (V β Ο) xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | congr! 1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V' xs) β I.pred_var_ X (List.map (V β Ο) xs) | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ List.map V' xs = List.map (V β Ο) xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ List.map V' xs = List.map (V β Ο) xs | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ β x β xs, V' x = (V β Ο) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ β x β xs, V' x = (V β Ο) x | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x | case a.h.e'_4
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 :
β
x β
if h : (Ο X xs.length).isSome = true then
((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset
else β
,
V'' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
x : VarName
a1 : x β xs
β’ V' x = V (Ο x) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.