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/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Β¬Holds D I V E (sub Ο c phi) β Β¬Holds D I (V β Ο) E phi | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Β¬Holds D I V E (sub Ο c phi) β Β¬Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V Ο | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E 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
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName... | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName... | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName... | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignme... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V Ο | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignme... | 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
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact psi_ih V Ο | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignme... | 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
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (exists_ x phi)) β Holds D I (V β Ο) E (exists_ x phi) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E
(exists_
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (exists... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E
(exists_
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c... | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E
(exists_
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x... | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeV... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro d | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeV... | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [phi_ih] | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}, ... | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Holds_coincide_Var | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}... | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ β (v : VarName),
isFreeIn v phi β
(Function.updateITE V
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ Holds D I
(... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro v a1 | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ β (v : VarName),
isFreeIn v phi β
(Function.updateITE V
... | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
β’ (Function.updateITE V
(if β y β p... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
β’ β (v : VarName... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
β’ (Function.updateITE V
(if β y β p... | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE V
(if β y β phi.fr... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | split_ifs | case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE V
(if β y β phi.fr... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
hβ : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β (d : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSe... | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeV... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β (d : D),
Holds D I
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply exists_congr | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x... | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeV... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | obtain s0 := fresh_not_mem x c (freeVarSet (sub (Function.updateITE Ο x x) c phi)) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
β’ Function.upd... | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
s0 : fresh x c... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | generalize (fresh x c (freeVarSet (sub (Function.updateITE Ο x x) c phi))) = x' at * | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
s0 : fresh x c... | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | by_cases c2 : v = x | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : VarName
s... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [c2] | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Function.updateITE] | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | by_cases c3 : Ο v = x' | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | subst c3 | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [freeVarSet_sub_eq_freeVarSet_image] at s0 | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | have s1 : Ο v β Finset.image (Function.updateITE Ο x x) (freeVarSet phi) | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : Β¬... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Finset.mem_image_update | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : Β¬... | case s1.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | contradiction | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact c2 | case s1.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [β isFreeIn_iff_mem_freeVarSet] | case s1.h2
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 ... | case s1.h2
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h2
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact a1 | case s1.h2
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
c2 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h2
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Function.updateITE] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [if_neg c2] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [if_neg c3] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : β y β phi.freeVarSet \ {x}, Ο y = x
x' : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | by_cases c2 : v = x | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Function.up... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | subst c2 | case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {v}, Ο y = v
β’ Function.updat... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Function.updateITE] | case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {v}, Ο y = v
β’ Function.updat... | case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {v}, Ο y = v
β’ (if (if True t... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {v}, Ο y = v
β’ (if (if True t... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v p... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | have s1 : Β¬ Ο v = x | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Function.updateITE] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [if_neg c2] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [if_neg s1] | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro contra | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply c1 | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Exists.intro v | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | constructor | case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
c2 : ... | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [β isFreeIn_iff_mem_freeVarSet] | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | tauto | case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.left
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact contra | case s1.right
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
a1 : isFreeIn v phi
c1 : Β¬β y β phi.freeVarSet \ {x}, Ο y = x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.right
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
d : D
v : VarName
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | induction E | D : Type
I : Interpretation D
E : Env
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (def_ X xs)) β Holds D I (V β Ο) E (def_ X xs) | case nil
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (sub Ο c (def_ X xs)) β Holds D I (V β Ο) [] (def_ X xs)
case cons
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
head... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (def_ X xs)) β Holds D I (V β Ο) E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case nil =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (sub Ο c (def_ X xs)) β Holds D I (V β Ο) [] (def_ X xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (sub Ο c (def_ X xs)) β Holds D I (V β Ο) [] (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (sub Ο c (def_ X xs)) β Holds D I (V β Ο) [] (def_ X xs) | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (def_ X (List.map Ο xs)) β Holds D I (V β Ο) [] (def_ X xs) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (sub Ο c (def_ X xs)) β Holds D I (V β Ο) [] (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (def_ X (List.map Ο xs)) β Holds D I (V β Ο) [] (def_ X xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V [] (def_ X (List.map Ο xs)) β Holds D I (V β Ο) [] (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] at E_ih | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (sub Ο c (def_ X xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E_tl) (sub Ο c (def_ X xs)) β Holds D I (V β Ο) (E_hd :: E_tl)... | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E_tl) (sub Ο c (def_ X xs)) β Holds D I (V β Ο) (E_hd :: E_... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (sub Ο c (def_ X xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E_tl) (sub Ο c (def_ X xs)) β Holds D I (V β Ο) (E_hd :: E_... | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) β Holds D I (V β Ο) (E_hd ::... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) β Holds D I (V β Ο) (E_hd ::... | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ Holds D I V (E_hd :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D... | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | split_ifs | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.... | case pos
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.u... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
β’ (if X = E_hd.name β§... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case neg c1 =>
exact E_ih | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V E_tl (def_ X (L... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : Β¬(X = E_hd.name ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Holds_coincide_Var | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateList... | case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ β (v : VarName),
i... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro v a1 | case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ β (v : VarName),
i... | case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFreeI... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Function.updateListITE_map_mem_ext | case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFreeI... | case h1.h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFr... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp | case h1.h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | cases c1 | case h1.h2
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFr... | case h1.h2.intro
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFreeIn v E_hd.q
leftβ : X = E_hd.name
rightβ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | case _ c1_left c1_right =>
symm
exact c1_right | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFreeIn v E_hd.q
c1_left : X = E_hd.name
c1_right : xs.length = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | symm | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFreeIn v E_hd.q
c1_left : X = E_hd.name
c1_right : xs.length = ... | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFreeIn v E_hd.q
c1_left : X = E_hd.name
c1_right : xs.length = ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact c1_right | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFreeIn v E_hd.q
c1_left : X = E_hd.name
c1_right : xs.length = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
v : VarName
a1 : isFr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : isFr... | case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [β List.mem_toFinset] | case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : v β ... | case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | apply Finset.mem_of_subset E_hd.h1 a1 | case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v : VarName
a1 : v β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h3
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : X = E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact E_ih | D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V E_tl (def_ X (L... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
c : Char
X : DefName
xs : List VarName
V : VarAssignment D
Ο : VarName β VarName
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs)
c1 : Β¬(X = E_hd.name ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_is_valid | [248, 1] | [260, 11] | simp only [IsValid] at h1 | Ο : VarName β VarName
c : Char
F : Formula
h1 : F.IsValid
β’ (sub Ο c F).IsValid | Ο : VarName β VarName
c : Char
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο c F).IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
c : Char
F : Formula
h1 : F.IsValid
β’ (sub Ο c F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_is_valid | [248, 1] | [260, 11] | simp only [IsValid] | Ο : VarName β VarName
c : Char
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο c F).IsValid | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F) | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
c : Char
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο c F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_is_valid | [248, 1] | [260, 11] | intro D I V E | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F) | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F) | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_is_valid | [248, 1] | [260, 11] | simp only [substitution_theorem] | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F) | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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:
Ο : VarName β VarName
c : Char
F : Formula
h1 : 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 (sub Ο c F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_is_valid | [248, 1] | [260, 11] | apply h1 | Ο : VarName β VarName
c : Char
F : Formula
h1 : 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:
Ο : VarName β VarName
c : Char
F : Formula
h1 : 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/Parsing/DFT.lean | list_cons_to_set_union | [9, 1] | [17, 9] | ext a | Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x : Ξ±
β’ β(x :: ys).toFinset = {x} βͺ βys.toFinset | case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x : Ξ±
β’ β(x :: ys).toFinset = {x} βͺ βys.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_cons_to_set_union | [9, 1] | [17, 9] | simp | case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_append_to_set_union | [19, 1] | [26, 9] | ext a | Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
β’ β(xs ++ ys).toFinset = βxs.toFinset βͺ βys.toFinset | case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
β’ β(xs ++ ys).toFinset = βxs.toFinset βͺ βys.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_append_to_set_union | [19, 1] | [26, 9] | simp | case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | induction g | Node : Type
instβ : DecidableEq Node
g : Graph Node
β’ β (x y : Node), y β direct_succ_list g x β β ys, y β ys β§ (x, ys) β g | case nil
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
case cons
Node : Type
instβ : DecidableEq Node
headβ : Node Γ List Node
tailβ : List (Node Γ List Node)
tail_ihβ : β (x y : Node), y β direct_succ_list tailβ x β β ys, y β ys β§ (x, ys) β tailβ
β’ β (x... | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
g : Graph Node
β’ β (x y : Node), y β direct_succ_list g x β β ys, y β ys β§ (x, ys) β g
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case nil =>
simp only [direct_succ_list]
simp | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [direct_succ_list] | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β [] | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β [] | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β []
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [direct_succ_list] | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node), y β direct_succ_list (hd :: tl) x β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node), y β direct_succ_list (hd :: tl) x β β ys, y β ys β§ (x, ys) β hd :: tl
TACTIC... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | intro x y | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_lis... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | split_ifs | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | case pos
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
hβ : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl
case neg
Node : Type
instβ : DecidableEq Node
hd... | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | subst c1 | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
c1 : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
c1 : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl
TACTIC:... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [ih] | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | constructor | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mpr
Node : Type
instβ : Decidable... | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | intro a1 | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | cases a1 | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp.inl
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
hβ : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mp.inr
Node : Type
instβ : DecidableEq Node
hd : Node Γ List ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) =... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case _ left =>
apply Exists.intro hd.snd
tauto | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
left : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
left : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case _ right =>
apply Exists.elim right
intro zs a2
apply Exists.intro zs
tauto | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
right : β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
right : β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys)... |
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