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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] | 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/NV/Program/PEG.lean | InterpretationEmpty | [208, 1] | [218, 7] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
n : β
xs : List V_T
o : Option (List V_T)
h1 : Interpretation V_N V_T R (empty, xs) (n, o)
β’ n = 1 β§ o = some [] | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 = 1 β§ some [] = some [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationEmpty | [208, 1] | [218, 7] | simp | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 = 1 β§ some [] = some [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs : List V_T
o : Option (List V_T)
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, o)
β’ n > 0 | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 > 0
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case terminal_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
aβΒΉ bβ : V_T
xsβ : List V_T
aβ : Β¬aβΒΉ = bβ
β’ 1 > 0
case terminal_failure_2
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
o : Option (List V_T)
Aβ : V_N
nβ : β
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, o)
β’ nβ + 1 > 0
case seq_success
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (e2β, xs_2β ++ ysβ) (n2β, some xs_2β)
β’ n1β + n2β + 1 > 0
case seq_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
e1β e2β : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, xs) (nβ, none)
β’ nβ + 1 > 0
case seq_failure_2
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xsβ ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xsβ ++ ysβ) (n1β, some xsβ)
aβ : Interpretation V_N V_T R (e2β, ysβ) (n2β, none)
β’ n1β + n2β + 1 > 0
case choice_1
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xsβ ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, xsβ ++ ysβ) (nβ, some xsβ)
β’ nβ + 1 > 0
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
o : Option (List V_T)
e1β e2β : PE V_N V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs) (n1β, none)
aβ : Interpretation V_N V_T R (e2β, xs) (n2β, o)
β’ n1β + n2β + 1 > 0
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
β’ n1β + n2β + 1 > 0
case star_termination
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0
case notP_1
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xsβ ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xsβ ++ ysβ) (nβ, some xsβ)
β’ nβ + 1 > 0
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | all_goals
omega | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
β’ 1 > 0
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case terminal_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
aβΒΉ bβ : V_T
xsβ : List V_T
aβ : Β¬aβΒΉ = bβ
β’ 1 > 0
case terminal_failure_2
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
β’ 1 > 0
case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
o : Option (List V_T)
Aβ : V_N
nβ : β
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, o)
β’ nβ + 1 > 0
case seq_success
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (e2β, xs_2β ++ ysβ) (n2β, some xs_2β)
β’ n1β + n2β + 1 > 0
case seq_failure_1
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
e1β e2β : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, xs) (nβ, none)
β’ nβ + 1 > 0
case seq_failure_2
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xsβ ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xsβ ++ ysβ) (n1β, some xsβ)
aβ : Interpretation V_N V_T R (e2β, ysβ) (n2β, none)
β’ n1β + n2β + 1 > 0
case choice_1
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xsβ ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, xsβ ++ ysβ) (nβ, some xsβ)
β’ nβ + 1 > 0
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
o : Option (List V_T)
e1β e2β : PE V_N V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs) (n1β, none)
aβ : Interpretation V_N V_T R (e2β, xs) (n2β, o)
β’ n1β + n2β + 1 > 0
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
β’ n1β + n2β + 1 > 0
case star_termination
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0
case notP_1
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xsβ ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xsβ ++ ysβ) (nβ, some xsβ)
β’ nβ + 1 > 0
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationSteps | [221, 1] | [233, 10] | omega | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
β’ nβ + 1 > 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | EmptyStringPrefix | [236, 1] | [241, 27] | exact List.nil_prefix xs | Ξ± : Type
xs : List Ξ±
β’ [].IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | CharPrefix | [244, 1] | [250, 40] | exact List.prefix_iff_eq_take.mpr rfl | Ξ± : Type
x : Ξ±
xs : List Ξ±
β’ [x].IsPrefix (x :: xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | PrefixAppend | [253, 1] | [258, 33] | exact List.prefix_append xs ys | Ξ± : Type
xs ys : List Ξ±
β’ xs.IsPrefix (xs ++ ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | induction n using Nat.strongInductionOn generalizing e | V_N V_T : Type
R : V_N β PE V_N V_T
e : PE V_N V_T
xs ys : List V_T
n : β
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs | case ind
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
nβ : β
aβ : β m < nβ, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
e : PE V_N V_T
h1 : Interpretation V_N V_T R (e, xs) (nβ, some ys)
β’ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | cases h1 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
n : β
ih : β m < n, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
e : PE V_N V_T
h1 : Interpretation V_N V_T R (e, xs) (n, some ys)
β’ ys.IsPrefix xs | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, aβ :: xsβ) (m, some [aβ]) β [aβ].IsPrefix (aβ :: xsβ)
β’ [aβ].IsPrefix (aβ :: xsβ)
case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
Aβ : V_N
nβ : β
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, some ys)
β’ ys.IsPrefix xs
case seq_success
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (e2β, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1β ++ xs_2β ++ ysβ) (m, some (xs_1β ++ xs_2β)) β
(xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
β’ (xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
case choice_1
V_N V_T : Type
R : V_N β PE V_N V_T
ys : List V_T
e1β e2β : PE V_N V_T
ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, ys ++ ysβ) (nβ, some ys)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, ys ++ ysβ) (m, some ys) β ys.IsPrefix (ys ++ ysβ)
β’ ys.IsPrefix (ys ++ ysβ)
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1β e2β : PE V_N V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs) (n1β, none)
ih : β m < n1β + n2β + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (e2β, xs) (n2β, some ys)
β’ ys.IsPrefix xs
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1β ++ xs_2β ++ ysβ) (m, some (xs_1β ++ xs_2β)) β
(xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
β’ (xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
case star_termination
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | any_goals
first | apply EmptyStringPrefix | apply CharPrefix | apply PrefixAppend | case empty
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, aβ :: xsβ) (m, some [aβ]) β [aβ].IsPrefix (aβ :: xsβ)
β’ [aβ].IsPrefix (aβ :: xsβ)
case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
Aβ : V_N
nβ : β
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, some ys)
β’ ys.IsPrefix xs
case seq_success
V_N V_T : Type
R : V_N β PE V_N V_T
e1β e2β : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (e2β, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1β ++ xs_2β ++ ysβ) (m, some (xs_1β ++ xs_2β)) β
(xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
β’ (xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
case choice_1
V_N V_T : Type
R : V_N β PE V_N V_T
ys : List V_T
e1β e2β : PE V_N V_T
ysβ : List V_T
nβ : β
aβ : Interpretation V_N V_T R (e1β, ys ++ ysβ) (nβ, some ys)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, ys ++ ysβ) (m, some ys) β ys.IsPrefix (ys ++ ysβ)
β’ ys.IsPrefix (ys ++ ysβ)
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1β e2β : PE V_N V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs) (n1β, none)
ih : β m < n1β + n2β + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (e2β, xs) (n2β, some ys)
β’ ys.IsPrefix xs
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1β ++ xs_2β ++ ysβ) (m, some (xs_1β ++ xs_2β)) β
(xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
β’ (xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
case star_termination
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs
case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs | case nonTerminal
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
Aβ : V_N
nβ : β
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (R Aβ, xs) (nβ, some ys)
β’ ys.IsPrefix xs
case choice_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1β e2β : PE V_N V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (e1β, xs) (n1β, none)
ih : β m < n1β + n2β + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
aβ : Interpretation V_N V_T R (e2β, xs) (n2β, some ys)
β’ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | first | apply EmptyStringPrefix | apply CharPrefix | apply PrefixAppend | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply EmptyStringPrefix | case notP_2
V_N V_T : Type
R : V_N β PE V_N V_T
xs : List V_T
eβ : PE V_N V_T
nβ : β
aβ : Interpretation V_N V_T R (eβ, xs) (nβ, none)
ih : β m < nβ + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some []) β [].IsPrefix xs
β’ [].IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply CharPrefix | case terminal_success
V_N V_T : Type
R : V_N β PE V_N V_T
aβ : V_T
xsβ : List V_T
ih : β m < 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, aβ :: xsβ) (m, some [aβ]) β [aβ].IsPrefix (aβ :: xsβ)
β’ [aβ].IsPrefix (aβ :: xsβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply PrefixAppend | case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
Interpretation V_N V_T R (e, xs_1β ++ xs_2β ++ ysβ) (m, some (xs_1β ++ xs_2β)) β
(xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ)
β’ (xs_1β ++ xs_2β).IsPrefix (xs_1β ++ xs_2β ++ ysβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n _ (R A) | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_1 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n2 _ e2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | subst h2 | F F' : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
h2 : Rec.fastReplaceFree v u F = F'
β’ IsSub F v u F' | F : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
β’ IsSub F v u (Rec.fastReplaceFree v u F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders F
β’ IsSub F v u (Rec.fastReplaceFree v u F) | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_const_ aβΒΉ aβ)
β’ IsSub (pred_const_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_const_ aβΒΉ aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_var_ aβΒΉ aβ)
β’ IsSub (pred_var_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_var_ aβΒΉ aβ))
case eq_
v u aβΒΉ aβ : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (eq_ aβΒΉ aβ)
β’ IsSub (eq_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (eq_ aβΒΉ aβ))
case true_
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders true_
β’ IsSub true_ v u (Rec.fastReplaceFree v u true_)
case false_
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders false_
β’ IsSub false_ v u (Rec.fastReplaceFree v u false_)
case not_
v u : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβ.not_
β’ IsSub aβ.not_ v u (Rec.fastReplaceFree v u aβ.not_)
case imp_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.imp_ aβ)
β’ IsSub (aβΒΉ.imp_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.imp_ aβ))
case and_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.and_ aβ)
β’ IsSub (aβΒΉ.and_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.and_ aβ))
case or_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.or_ aβ)
β’ IsSub (aβΒΉ.or_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.or_ aβ))
case iff_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.iff_ aβ)
β’ IsSub (aβΒΉ.iff_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.iff_ aβ))
case forall_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (forall_ aβΒΉ aβ)
β’ IsSub (forall_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (forall_ aβΒΉ aβ))
case exists_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (exists_ aβΒΉ aβ)
β’ IsSub (exists_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (exists_ aβΒΉ aβ))
case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (def_ aβΒΉ aβ)
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | all_goals
simp only [Rec.fastAdmitsAux] at h1
simp only [Rec.fastReplaceFree] | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_const_ aβΒΉ aβ)
β’ IsSub (pred_const_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_const_ aβΒΉ aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (pred_var_ aβΒΉ aβ)
β’ IsSub (pred_var_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (pred_var_ aβΒΉ aβ))
case eq_
v u aβΒΉ aβ : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (eq_ aβΒΉ aβ)
β’ IsSub (eq_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (eq_ aβΒΉ aβ))
case true_
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders true_
β’ IsSub true_ v u (Rec.fastReplaceFree v u true_)
case false_
v u : VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders false_
β’ IsSub false_ v u (Rec.fastReplaceFree v u false_)
case not_
v u : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβ.not_
β’ IsSub aβ.not_ v u (Rec.fastReplaceFree v u aβ.not_)
case imp_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.imp_ aβ)
β’ IsSub (aβΒΉ.imp_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.imp_ aβ))
case and_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.and_ aβ)
β’ IsSub (aβΒΉ.and_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.and_ aβ))
case or_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.or_ aβ)
β’ IsSub (aβΒΉ.or_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.or_ aβ))
case iff_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (aβΒΉ.iff_ aβ)
β’ IsSub (aβΒΉ.iff_ aβ) v u (Rec.fastReplaceFree v u (aβΒΉ.iff_ aβ))
case forall_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (forall_ aβΒΉ aβ)
β’ IsSub (forall_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (forall_ aβΒΉ aβ))
case exists_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (exists_ aβΒΉ aβ)
β’ IsSub (exists_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (exists_ aβΒΉ aβ))
case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (def_ aβΒΉ aβ)
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) | case pred_const_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (pred_const_ aβΒΉ aβ) v u (pred_const_ aβΒΉ (List.map (fun x => if v = x then u else x) aβ))
case pred_var_
v u : VarName
aβΒΉ : PredName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (pred_var_ aβΒΉ aβ) v u (pred_var_ aβΒΉ (List.map (fun x => if v = x then u else x) aβ))
case eq_
v u aβΒΉ aβ : VarName
binders : Finset VarName
h1 : v = aβΒΉ β¨ v = aβ β u β binders
β’ IsSub (eq_ aβΒΉ aβ) v u (eq_ (if v = aβΒΉ then u else aβΒΉ) (if v = aβ then u else aβ))
case true_
v u : VarName
binders : Finset VarName
h1 : True
β’ IsSub true_ v u true_
case false_
v u : VarName
binders : Finset VarName
h1 : True
β’ IsSub false_ v u false_
case not_
v u : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβ
β’ IsSub aβ.not_ v u (Rec.fastReplaceFree v u aβ).not_
case imp_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβΒΉ β§ Rec.fastAdmitsAux v u binders aβ
β’ IsSub (aβΒΉ.imp_ aβ) v u ((Rec.fastReplaceFree v u aβΒΉ).imp_ (Rec.fastReplaceFree v u aβ))
case and_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβΒΉ β§ Rec.fastAdmitsAux v u binders aβ
β’ IsSub (aβΒΉ.and_ aβ) v u ((Rec.fastReplaceFree v u aβΒΉ).and_ (Rec.fastReplaceFree v u aβ))
case or_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβΒΉ β§ Rec.fastAdmitsAux v u binders aβ
β’ IsSub (aβΒΉ.or_ aβ) v u ((Rec.fastReplaceFree v u aβΒΉ).or_ (Rec.fastReplaceFree v u aβ))
case iff_
v u : VarName
aβΒΉ aβ : Formula
a_ihβΒΉ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβΒΉ β IsSub aβΒΉ v u (Rec.fastReplaceFree v u aβΒΉ)
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders aβΒΉ β§ Rec.fastAdmitsAux v u binders aβ
β’ IsSub (aβΒΉ.iff_ aβ) v u ((Rec.fastReplaceFree v u aβΒΉ).iff_ (Rec.fastReplaceFree v u aβ))
case forall_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : v = aβΒΉ β¨ Rec.fastAdmitsAux v u (binders βͺ {aβΒΉ}) aβ
β’ IsSub (forall_ aβΒΉ aβ) v u (if v = aβΒΉ then forall_ aβΒΉ aβ else forall_ aβΒΉ (Rec.fastReplaceFree v u aβ))
case exists_
v u aβΒΉ : VarName
aβ : Formula
a_ihβ : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders aβ β IsSub aβ v u (Rec.fastReplaceFree v u aβ)
binders : Finset VarName
h1 : v = aβΒΉ β¨ Rec.fastAdmitsAux v u (binders βͺ {aβΒΉ}) aβ
β’ IsSub (exists_ aβΒΉ aβ) v u (if v = aβΒΉ then exists_ aβΒΉ aβ else exists_ aβΒΉ (Rec.fastReplaceFree v u aβ))
case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (def_ aβΒΉ aβ) v u (def_ aβΒΉ (List.map (fun x => if v = x then u else x) aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case pred_const_ X xs | pred_var_ X xs =>
first | apply IsSub.pred_const_ | apply IsSub.pred_var_ | v u : VarName
X : PredName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case eq_ x y =>
apply IsSub.eq_ | v u x y : VarName
binders : Finset VarName
h1 : v = x β¨ v = y β u β binders
β’ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case true_ | false_ =>
first | apply IsSub.true_ | apply IsSub.false_ | v u : VarName
binders : Finset VarName
h1 : True
β’ IsSub false_ v u false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case not_ phi phi_ih =>
apply IsSub.not_
exact phi_ih binders h1 | v u : VarName
phi : Formula
phi_ih : β (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi β IsSub phi v u (Rec.fastReplaceFree v u phi)
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders phi
β’ IsSub phi.not_ v u (Rec.fastReplaceFree v u phi).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | case def_ X xs =>
apply IsSub.def_ | v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v β xs β u β binders
β’ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub | [125, 1] | [194, 21] | simp only [Rec.fastAdmitsAux] at h1 | case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : Rec.fastAdmitsAux v u binders (def_ aβΒΉ aβ)
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) | case def_
v u : VarName
aβΒΉ : DefName
aβ : List VarName
binders : Finset VarName
h1 : v β aβ β u β binders
β’ IsSub (def_ aβΒΉ aβ) v u (Rec.fastReplaceFree v u (def_ aβΒΉ aβ)) |
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