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] | 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/NV/Program/PEG.lean | InterpretationEmpty | [208, 1] | [218, 7] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
n : β
xs : List V_T
o : Option (List V_T)
h1 : Interpretation V_N V_T R (empty, xs) (n, o)
β’ n = 1 β§ o = some [] | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 = 1 β§ some [] = some [] | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
n : β
xs : List V_T
o : Option (List V_T)
h1 : Interpretation V_N V_T R (empty, xs) (n, o)
β’ n = 1 β§ o = some []
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationEmpty | [208, 1] | [218, 7] | simp | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 = 1 β§ some [] = some [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 = 1 β§ some [] = some []
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs : List V_T
o : Option (List V_T)
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, o)
β’ n > 0 | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 > 0
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case terminal_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
aβΒΉ bβ : V_T
xsβ : List V_T
aβ : Β¬aβΒΉ = bβ
β’ 1 > 0
case terminal_failure_2
V_N V_T : Type
R : V_N ... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs : List V_T
o : Option (List V_T)
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, o)
β’ n > 0
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | all_goals
omega | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 > 0
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case terminal_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
aβΒΉ bβ : V_T
xsβ : List V_T
aβ : Β¬aβΒΉ = bβ
β’ 1 > 0
case terminal_failure_2
V_N V_T : Type
R : V_N ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 > 0
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case terminal_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
aβΒΉ bβ : V_T
xsβ : List V_T
aβ : Β¬aβΒΉ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | omega | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | EmptyStringPrefix | [236, 1] | [241, 27] | exact List.nil_prefix xs | Ξ± : Type
xs : List Ξ±
β’ [].IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
xs : List Ξ±
β’ [].IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | CharPrefix | [244, 1] | [250, 40] | exact List.prefix_iff_eq_take.mpr rfl | Ξ± : Type
x : Ξ±
xs : List Ξ±
β’ [x].IsPrefix (x :: xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
x : Ξ±
xs : List Ξ±
β’ [x].IsPrefix (x :: xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | PrefixAppend | [253, 1] | [258, 33] | exact List.prefix_append xs ys | Ξ± : Type
xs ys : List Ξ±
β’ xs.IsPrefix (xs ++ ys) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
xs ys : List Ξ±
β’ xs.IsPrefix (xs ++ ys)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | induction n using Nat.strongInductionOn generalizing e | V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs ys : List V_T
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs | case ind
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
nβ : β
aβ : β m < nβ, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
e : PE V_N V_T
h1 : Interpretation V_N V_T R (e, xs) (nβ, some ys)
β’ ys.IsPrefix xs | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs ys : List V_T
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
n : β
ih : β m < n, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
e : PE V_N V_T
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e,... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
n : β
ih : β m < n, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
e : PE V_N V_T
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | any_goals
first | apply EmptyStringPrefix | apply CharPrefix | apply PrefixAppend | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e,... | case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
Aβ : V_N
nβ : β
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, some ys)
β’ ys.IsPrefix xs
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : L... | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | first | apply EmptyStringPrefix | apply CharPrefix | apply PrefixAppend | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
TACTI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply EmptyStringPrefix | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
TACTI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply CharPrefix | case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, aβ :: xsβ) (m, some [aβ]) β [aβ].IsPrefix (aβ :: xsβ)
β’ [aβ].IsPrefix (aβ :: xsβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, aβ :: xsβ) (m, some [aβ]) β [aβ].IsPrefix (aβ :: xsβ)
β’ [aβ].IsPrefix (aβ :: xsβ)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply PrefixAppend | case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n _ (R A) | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpreta... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_1 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n2 _ e2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1
V_N ... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretat... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretat... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | subst h2 | F F' : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
h2 : Rec.fastReplaceFree v u F = F'
β’ IsSub F v u F' | F : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
β’ IsSub F v u (Rec.fastReplaceFree v u F) | Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
h2 : Rec.fastReplaceFree v u F = F'
β’ IsSub F v u F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
β’ IsSub F v u (Rec.fastReplaceFree v u F) | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_const_ aβΒΉ aβ)
β’ IsSub (pred_const_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_const_ aβΒΉ aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Re... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
β’ IsSub F v u (Rec.fastReplaceFree v u F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | all_goals
simp only [Rec.fastAdmitsAux] at h1
simp only [Rec.fastReplaceFree] | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_const_ aβΒΉ aβ)
β’ IsSub (pred_const_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_const_ aβΒΉ aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Re... | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (pred_const_ aβΒΉ aβ) v u (pred_const_ aβΒΉ (List.map (fun x => if v = x then u else x) aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_const_ aβΒΉ aβ)
β’ IsSub (pred_const_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_const_ aβΒΉ aβ))
case pred_var_
v u : VarName
aβΒΉ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case pred_const_ X xs | pred_var_ X xs =>
first | apply IsSub.pred_const_ | apply IsSub.pred_var_ | v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case eq_ x y =>
apply IsSub.eq_ | v u x y : VarName
binders : Finset VarName
h1 : v = x β¨ v = y β u β binders
β’ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u x y : VarName
binders : Finset VarName
h1 : v = x β¨ v = y β u β binders
β’ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case true_ | false_ =>
first | apply IsSub.true_ | apply IsSub.false_ | v u : VarName
binders : Finset VarName
h1 : True
β’ IsSub false_ v u false_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
binders : Finset VarName
h1 : True
β’ IsSub false_ v u false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case not_ phi phi_ih =>
apply IsSub.not_
exact phi_ih binders h1 | v u : VarName
phi : Formula
phi_ih : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi β IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
β’ IsSub phi.not_ v u (Rec.fastReplaceFree v u phi).not_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
phi : Formula
phi_ih : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi β IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
β’ IsSub phi.not_ v u (Rec.fastReplaceFree v u ph... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case def_ X xs =>
apply IsSub.def_ | v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [Rec.fastAdmitsAux] at h1 | case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (def_ aβΒΉ aβ)
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) | case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (def_ aβΒΉ aβ)
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ))
TACTIC:
|
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