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 | input stringlengths 73 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 (r... |
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.update... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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.update... |
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 ... | 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 (Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
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 (... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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 (r... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 Ho... |
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... | 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 : VarAssignm... | Please generate a tactic in lean4 to solve the state.
STATE:
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 Ho... |
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 : VarAssignm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : VarAssignm... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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.update... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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.update... |
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.... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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 : T... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : T... | 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 : VarNa... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : VarNa... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.Inject... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.updateI... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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.updateI... |
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 (Fun... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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.... |
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.update... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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.update... |
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 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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_h... | 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
Hol... | Please generate a tactic in lean4 to solve the state.
STATE:
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’ Ho... |
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
Hol... | 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
Hol... | Please generate a tactic in lean4 to solve the state.
STATE:
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’ (i... |
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
Hol... | 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’ Hold... | Please generate a tactic in lean4 to solve the state.
STATE:
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’ (i... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 :... |
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 (Fu... | 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.l... | Please generate a tactic in lean4 to solve the state.
STATE:
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 :... |
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’... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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_l... |
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.... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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 x... |
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.... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
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 x... |
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.... | 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.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
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 x... |
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.ar... | 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.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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.ar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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.ar... | 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.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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.ar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 :... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : F.IsValid
β’ (replaceAll Ο F).IsValid
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 ... | 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).pr... | Please generate a tactic in lean4 to solve the state.
STATE:
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 (predVarFreeVarS... |
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 (predV... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 (predV... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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 ... | 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’ (Interpr... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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’ (Interpr... | 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_c... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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_c... | 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_c... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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_c... | 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 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).pre... |
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 = ... | 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 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
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_co... |
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 = ... | 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 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
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_co... |
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 = ... | 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 = ... | Please generate a tactic in lean4 to solve the state.
STATE:
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_co... |
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 = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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_co... |
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 (predVarFr... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).predV... |
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... | 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 (pr... | Please generate a tactic in lean4 to solve the state.
STATE:
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)}.biUni... |
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 (pr... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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.len... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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 ... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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.leng... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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.leng... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = tru... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isSo... |
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 :... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isSo... |
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 ... | 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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
... | 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
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).... |
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
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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).... |
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
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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).... |
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
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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).... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 ... | 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
c... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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
c... | 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
c... | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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
c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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
c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 x... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 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 x... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 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 x... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 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 x... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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