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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Β¬Holds D I V E (sub Ο c phi) β Β¬Holds D I (V β Ο) E phi | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V Ο | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (phi.iff_ psi)) β Holds D I (V β Ο) E (phi.iff_ psi) | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E ((sub Ο c phi).iff_ (sub Ο c psi)) β Holds D I (V β Ο) E (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E ((sub Ο c phi).iff_ (sub Ο c psi)) β Holds D I (V β Ο) E (phi.iff_ psi) | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ (Holds D I V E (sub Ο c phi) β Holds D I V E (sub Ο c psi)) β (Holds D I (V β Ο) E phi β Holds D I (V β Ο) E psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ (Holds D I V E (sub Ο c phi) β Holds D I V E (sub Ο c psi)) β (Holds D I (V β Ο) E phi β Holds D I (V β Ο) E psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact phi_ih V Ο | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | exact psi_ih V Ο | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
c : Char
phi psi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
psi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c psi) β Holds D I (V β Ο) E psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [sub] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E (sub Ο c (exists_ x phi)) β Holds D I (V β Ο) E (exists_ x phi) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E
(exists_
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
Holds D I (V β Ο) E (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ Holds D I V E
(exists_
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
Holds D I (V β Ο) E (exists_ x phi) | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
β d, Holds D I (Function.updateITE (V β Ο) x d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ (β d,
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
β d, Holds D I (Function.updateITE (V β Ο) x d) E phi | 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.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | intro d | case h
D : Type
I : Interpretation D
E : Env
c : Char
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D) (Ο : VarName β VarName), Holds D I V E (sub Ο c phi) β Holds D I (V β Ο) E phi
V : VarAssignment D
Ο : VarName β VarName
β’ β (a : D),
Holds D I
(Function.updateITE V
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x a) E phi | 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}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | simp only [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}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x d) E phi | 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}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
E phi β
Holds D I (Function.updateITE (V β Ο) x d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | 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}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
E phi β
Holds D I (Function.updateITE (V β Ο) x d) E phi | 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
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
v =
Function.updateITE (V β Ο) x d v |
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
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
v =
Function.updateITE (V β Ο) x d 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 phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
v =
Function.updateITE (V β Ο) x d v |
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 phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d β
Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
v =
Function.updateITE (V β Ο) x d 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 phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x) d
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
v) =
Function.updateITE (V β Ο) x d v |
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.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x) d
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet else x)
v) =
Function.updateITE (V β Ο) x d v | 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’ Function.updateITE V (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) d
(Function.updateITE Ο x (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) v) =
Function.updateITE (V β Ο) x d v
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
hβ : Β¬β y β phi.freeVarSet \ {x}, Ο y = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v |
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.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
β (d : D), Holds D I (Function.updateITE (V β Ο) x d) E phi | 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.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | 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}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
d)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi)) β
β d, Holds D I (Function.updateITE (V β Ο) x d) E phi | 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.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x)
a)
E
(sub
(Function.updateITE Ο x
(if β y β phi.freeVarSet \ {x}, Ο y = x then fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet
else x))
c phi) β
Holds D I (Function.updateITE (V β Ο) x a) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | 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.updateITE V (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) d
(Function.updateITE Ο x (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) v) =
Function.updateITE (V β Ο) x d v | 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 (sub (Function.updateITE Ο x x) c phi).freeVarSet β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) d
(Function.updateITE Ο x (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) v) =
Function.updateITE (V β Ο) x d v |
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 (sub (Function.updateITE Ο x x) c phi).freeVarSet β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) d
(Function.updateITE Ο x (fresh x c (sub (Function.updateITE Ο x x) c phi).freeVarSet) v) =
Function.updateITE (V β Ο) x d v | 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
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v |
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
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' x) = Function.updateITE (V β Ο) x d x |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' x) = Function.updateITE (V β Ο) x d 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ (if (if True then x' else Ο x) = x' then d else V (if True then x' else Ο x)) = if True then d else (V β Ο) x |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : v = x
β’ (if (if True then x' else Ο x) = x' then d else V (if True then x' else Ο x)) = if True then d else (V β Ο) x | no goals |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Ο v = x'
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Ο v = x'
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v | 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 : Β¬v = x
s0 : Ο v β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v |
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 : Β¬v = x
s0 : Ο v β (sub (Function.updateITE Ο x x) c phi).freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v | 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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v |
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d 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 y β phi.freeVarSet \ {x}, Ο y = x
c2 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v |
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¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v | 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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Β¬v = x
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ v β phi.freeVarSet
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v |
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Function.updateITE V (Ο v) d (Function.updateITE Ο x (Ο v) v) = Function.updateITE (V β Ο) x d v | no goals |
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ Β¬v = x | no goals |
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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ v β phi.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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean | FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem | [128, 1] | [245, 19] | 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 : Β¬v = x
s0 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ isFreeIn v phi | no goals |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ Function.updateITE V x' d (Function.updateITE Ο x x' v) = Function.updateITE (V β Ο) x d v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ (if (if v = x then x' else Ο v) = x' then d else V (if v = x then x' else Ο v)) = if v = x then d else (V β Ο) v |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ (if (if v = x then x' else Ο v) = x' then d else V (if v = x then x' else Ο v)) = if v = x then d else (V β Ο) v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ (if Ο v = x' then d else V (Ο v)) = (V β Ο) v |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ (if Ο v = x' then d else V (Ο v)) = (V β Ο) v | 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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ V (Ο v) = (V β Ο) v |
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' : VarName
s0 : x' β (sub (Function.updateITE Ο x x) c phi).freeVarSet
c2 : Β¬v = x
c3 : Β¬Ο v = x'
β’ V (Ο v) = (V β Ο) v | no goals |
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.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v | 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 : v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v
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 : Β¬v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v |
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 : v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v | 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.updateITE V v d (Function.updateITE Ο v v v) = Function.updateITE (V β Ο) v d v |
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.updateITE V v d (Function.updateITE Ο v v v) = Function.updateITE (V β Ο) v d v | 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 then v else Ο v) = v then d else V (if True then v else Ο v)) = if True then d else (V β Ο) v |
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 then v else Ο v) = v then d else V (if True then v else Ο v)) = if True then d else (V β Ο) v | no goals |
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 : Β¬v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d 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 : Β¬v = x
β’ 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 : Β¬v = x
s1 : Β¬Ο v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v |
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 : Β¬v = x
s1 : Β¬Ο v = x
β’ Function.updateITE V x d (Function.updateITE Ο x x v) = Function.updateITE (V β Ο) x d v | 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 : Β¬v = x
s1 : Β¬Ο v = x
β’ (if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v)) = if v = x then d else (V β Ο) v |
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 : Β¬v = x
s1 : Β¬Ο v = x
β’ (if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v)) = if v = x then d else (V β Ο) v | 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 : Β¬v = x
s1 : Β¬Ο v = x
β’ (if Ο v = x then d else V (Ο v)) = (V β Ο) v |
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 : Β¬v = x
s1 : Β¬Ο v = x
β’ (if Ο v = x then d else V (Ο v)) = (V β Ο) v | 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 : Β¬v = x
s1 : Β¬Ο v = x
β’ V (Ο v) = (V β Ο) v |
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 : Β¬v = x
s1 : Β¬Ο v = x
β’ V (Ο v) = (V β Ο) v | no goals |
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 : Β¬v = x
β’ Β¬Ο v = x | 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 : Β¬v = x
contra : Ο v = x
β’ False |
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 : Β¬v = x
contra : Ο v = x
β’ False | 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 : Β¬v = x
contra : Ο v = x
β’ β y β phi.freeVarSet \ {x}, Ο y = x |
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 : Β¬v = x
contra : Ο v = x
β’ β y β phi.freeVarSet \ {x}, Ο y = x | 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 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet \ {x} β§ Ο v = x |
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 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet \ {x} β§ Ο v = 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
c2 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet \ {x}
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
c2 : Β¬v = x
contra : Ο v = x
β’ Ο v = x |
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
c2 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet \ {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
c2 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet β§ Β¬v = x |
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
c2 : Β¬v = x
contra : Ο v = x
β’ v β phi.freeVarSet β§ Β¬v = 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
c2 : Β¬v = x
contra : Ο v = x
β’ isFreeIn v phi β§ Β¬v = x |
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
c2 : Β¬v = x
contra : Ο v = x
β’ isFreeIn v phi β§ Β¬v = x | no goals |
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
c2 : Β¬v = x
contra : Ο v = x
β’ Ο v = x | no goals |
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
headβ : Definition
tailβ : List Definition
tail_ihβ : Holds D I V tailβ (sub Ο c (def_ X xs)) β Holds D I (V β Ο) tailβ (def_ X xs)
β’ Holds D I V (headβ :: tailβ) (sub Ο c (def_ X xs)) β Holds D I (V β Ο) (headβ :: tailβ) (def_ X xs) |
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 |
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) |
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 |
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) (def_ X xs) | 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_tl) (def_ X xs) |
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_tl) (def_ X xs) | 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 :: E_tl) (def_ X xs) |
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 :: E_tl) (def_ X xs) | 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 I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs) |
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 I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs) | 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.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs) |
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.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs))) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D I (V β Ο) E_tl (def_ X xs) | 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.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
case neg
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 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V E_tl (def_ X (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs) |
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 (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs) | no goals |
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.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | 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),
isFreeIn v E_hd.q β
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v |
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),
isFreeIn v E_hd.q β
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v | 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 : isFreeIn v E_hd.q
β’ Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v |
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 : isFreeIn v E_hd.q
β’ Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE (V β Ο) E_hd.args (List.map (V β Ο) xs) v | 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 : isFreeIn v E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y
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 : isFreeIn v E_hd.q
β’ E_hd.args.length = xs.length
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 : isFreeIn v E_hd.q
β’ v β E_hd.args |
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 : isFreeIn v E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y | no goals |
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 : isFreeIn v E_hd.q
β’ E_hd.args.length = xs.length | 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 : xs.length = E_hd.args.length
β’ E_hd.args.length = xs.length |
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 = E_hd.args.length
β’ E_hd.args.length = xs.length | no goals |
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 = E_hd.args.length
β’ E_hd.args.length = 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 = E_hd.args.length
β’ xs.length = E_hd.args.length |
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 = E_hd.args.length
β’ xs.length = E_hd.args.length | no goals |
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 : isFreeIn v E_hd.q
β’ v β E_hd.args | 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 E_hd.q.freeVarSet
β’ v β E_hd.args |
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 E_hd.q.freeVarSet
β’ v β E_hd.args | 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 E_hd.q.freeVarSet
β’ v β E_hd.args.toFinset |
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 E_hd.q.freeVarSet
β’ v β E_hd.args.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/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 (List.map Ο xs)) β Holds D I (V β Ο) E_tl (def_ X xs) | no goals |
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 |
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) |
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) |
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 |
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 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_cons_to_set_union | [9, 1] | [17, 9] | ext a | Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x : Ξ±
β’ β(x :: ys).toFinset = {x} βͺ βys.toFinset | case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_cons_to_set_union | [9, 1] | [17, 9] | simp | case h
Ξ± : Type
inst : DecidableEq Ξ±
ys : List Ξ±
x a : Ξ±
β’ a β β(x :: ys).toFinset β a β {x} βͺ βys.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_append_to_set_union | [19, 1] | [26, 9] | ext a | Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
β’ β(xs ++ ys).toFinset = βxs.toFinset βͺ βys.toFinset | case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | list_append_to_set_union | [19, 1] | [26, 9] | simp | case h
Ξ± : Type
inst : DecidableEq Ξ±
xs ys : List Ξ±
a : Ξ±
β’ a β β(xs ++ ys).toFinset β a β βxs.toFinset βͺ βys.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | induction g | Node : Type
instβ : DecidableEq Node
g : Graph Node
β’ β (x y : Node), y β direct_succ_list g x β β ys, y β ys β§ (x, ys) β g | case nil
Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β []
case cons
Node : Type
instβ : DecidableEq Node
headβ : Node Γ List Node
tailβ : List (Node Γ List Node)
tail_ihβ : β (x y : Node), y β direct_succ_list tailβ x β β ys, y β ys β§ (x, ys) β tailβ
β’ β (x y : Node), y β direct_succ_list (headβ :: tailβ) x β β ys, y β ys β§ (x, ys) β headβ :: tailβ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case nil =>
simp only [direct_succ_list]
simp | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [direct_succ_list] | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β direct_succ_list [] x β β ys, y β ys β§ (x, ys) β [] | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp | Node : Type
instβ : DecidableEq Node
β’ β (x y : Node), y β [] β β ys, y β ys β§ (x, ys) β [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [direct_succ_list] | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node), y β direct_succ_list (hd :: tl) x β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | intro x y | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
β’ β (x y : Node),
(y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | split_ifs | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
β’ (y β if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) β β ys, y β ys β§ (x, ys) β hd :: tl | case pos
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
hβ : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl
case neg
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
hβ : Β¬hd.1 = x
β’ y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | subst c1 | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
x y : Node
c1 : hd.1 = x
β’ y β hd.2 ++ direct_succ_list tl x β β ys, y β ys β§ (x, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 ++ direct_succ_list tl hd.1 β β ys, y β ys β§ (hd.1, ys) β hd :: tl | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | simp only [ih] | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ y β hd.2 β¨ y β direct_succ_list tl hd.1 β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | constructor | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mpr
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)) β y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | intro a1 | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
β’ (y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl) β β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | cases a1 | case mp
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
a1 : y β hd.2 β¨ β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | case mp.inl
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
hβ : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl)
case mp.inr
Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
hβ : β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case _ left =>
apply Exists.intro hd.snd
tauto | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
left : y β hd.2
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/DFT.lean | mem_direct_succ_list_iff | [65, 1] | [122, 16] | case _ right =>
apply Exists.elim right
intro zs a2
apply Exists.intro zs
tauto | Node : Type
instβ : DecidableEq Node
hd : Node Γ List Node
tl : List (Node Γ List Node)
ih : β (x y : Node), y β direct_succ_list tl x β β ys, y β ys β§ (x, ys) β tl
y : Node
right : β ys, y β ys β§ (hd.1, ys) β tl
β’ β ys, y β ys β§ ((hd.1, ys) = hd β¨ (hd.1, ys) β tl) | no goals |
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