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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro a a2 | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
β’ β (a : PredName Γ β),
a β phi.predVarSet β§ x β predVarFreeVarSet Ο a β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_biUnion, Finset.mem_union] | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ β a, (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply Exists.intro a | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ β a, (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | tauto | case a.h.e'_1.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β phi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β phi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply psi_ih V V' Ο | case a.h.e'_2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi) | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (x : VarName), isFreeIn x psi β V' x = V (Ο x)
case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro x a1 | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (x : VarName), isFreeIn x psi β V' x = V (Ο x) | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ V' x = V (Ο x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply h1 | case a.h.e'_2.a.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ V' x = V (Ο x) | case a.h.e'_2.a.h1.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ isFreeIn x phi β¨ isFreeIn x psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | right | case a.h.e'_2.a.h1.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ isFreeIn x phi β¨ isFreeIn x psi | case a.h.e'_2.a.h1.a.h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ isFreeIn x psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact a1 | case a.h.e'_2.a.h1.a.h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : isFreeIn x psi
β’ isFreeIn x psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro x a1 | case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x | case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β psi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply h2 | case a.h.e'_2.a.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β psi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' x = V x | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β psi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_biUnion, Finset.mem_union] at a1 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β psi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply Exists.elim a1 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
β’ β (a : PredName Γ β),
a β psi.predVarSet β§ x β predVarFreeVarSet Ο a β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro a a2 | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
β’ β (a : PredName Γ β),
a β psi.predVarSet β§ x β predVarFreeVarSet Ο a β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_biUnion, Finset.mem_union] | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο) | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ β a, (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply Exists.intro a | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ β a, (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | tauto | case a.h.e'_2.a.h2.a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
psi_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x psi β V' x = V (Ο x)) β
(β x β psi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E psi β Holds D I V E (subAux c Ο Ο psi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x phi β¨ isFreeIn x psi β V' x = V (Ο x)
h2 : β x β (phi.predVarSet βͺ psi.predVarSet).biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : β a β psi.predVarSet, x β predVarFreeVarSet Ο a
a : PredName Γ β
a2 : a β psi.predVarSet β§ x β predVarFreeVarSet Ο a
β’ (a β phi.predVarSet β¨ a β psi.predVarSet) β§ x β predVarFreeVarSet Ο a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), isFreeIn x_1 (exists_ x phi) β V' x_1 = V (Ο x_1)
h2 : β x_1 β (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x_1 = V x_1
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β Holds D I V E (subAux c Ο Ο (exists_ x phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x_1 β (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x_1 = V x_1
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β Holds D I V E (subAux c Ο Ο (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [predVarSet] at h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x_1 β (exists_ x phi).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x_1 = V x_1
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β Holds D I V E (subAux c Ο Ο (exists_ x phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β Holds D I V E (subAux c Ο Ο (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [subAux] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β Holds D I V E (subAux c Ο Ο (exists_ x phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [I'] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D
(Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds)
V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D
(Interpretation.usingPred D I fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds)
V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V' E (exists_ x phi) β
Holds D I V E
(exists_
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (β d,
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi) β
β d,
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (β d,
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi) β
β d,
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (a : D),
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x a) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
a)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro d | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (a : D),
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x a) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
a)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply ih | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) | case h.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ β (x_1 : VarName),
isFreeIn x_1 phi β
Function.updateITE V' x d x_1 =
Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
x_1)
case h.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ β x_1 β phi.predVarSet.biUnion (predVarFreeVarSet Ο),
V'' x_1 =
Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d x_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply forall_congr' | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (β (d : D),
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi) β
β (d : D),
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (a : D),
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x a) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
a)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply exists_congr | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (β d,
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x d) E phi) β
β d,
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β (a : D),
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V'' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
(Function.updateITE V' x a) E phi β
Holds D I
(Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
a)
E
(subAux c Ο
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x))
phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro v a1 | case h.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ β (x_1 : VarName),
isFreeIn x_1 phi β
Function.updateITE V' x d x_1 =
Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
x_1) | case h.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE V' x d v =
Function.updateITE V
(if x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | case h.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
β’ Function.updateITE V' x d v =
Function.updateITE V
(if x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d
(Function.updateITE Ο x
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
v) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
hβ : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Function.updateITE V' x d v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d
(Function.updateITE Ο x
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
v)
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
hβ : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Function.updateITE V' x d v = Function.updateITE V x d (Function.updateITE Ο x x v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1 =>
simp only [Function.updateITE]
split_ifs
case _ c2 c3 =>
rfl
case _ c2 c3 =>
contradiction
case _ c2 c3 =>
obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c3] at s1
simp only [Finset.mem_union] at s1
simp only [isFreeIn_iff_mem_freeVarSet] at a1
obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο x x) a1
simp only [Function.updateITE] at s2
simp only [if_neg c2] at s2
exfalso
apply s1
left
exact s2
case _ c2 c3 =>
apply h1
tauto | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Function.updateITE V' x d v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d
(Function.updateITE Ο x
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
v) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateITE] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Function.updateITE V' x d v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d
(Function.updateITE Ο x
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
v) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ (if v = x then d else V' v) =
if
(if v = x then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else Ο v) =
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) then
d
else
V
(if v = x then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ (if v = x then d else V' v) =
if
(if v = x then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else Ο v) =
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) then
d
else
V
(if v = x then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else Ο v) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβΒΉ : v = x
hβ :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d = d
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβΒΉ : v = x
hβ :
Β¬fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d =
V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβΒΉ : Β¬v = x
hβ :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβΒΉ : Β¬v = x
hβ :
Β¬Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = V (Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 c3 =>
rfl | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
c3 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 c3 =>
contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
c3 :
Β¬fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d =
V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 c3 =>
obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c3] at s1
simp only [Finset.mem_union] at s1
simp only [isFreeIn_iff_mem_freeVarSet] at a1
obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο x x) a1
simp only [Function.updateITE] at s2
simp only [if_neg c2] at s2
exfalso
apply s1
left
exact s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 c3 =>
apply h1
tauto | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Β¬Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = V (Ο v) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | rfl | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
c3 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
c3 :
Β¬fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ d =
V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s1 := fresh_not_mem x c ((Finset.image (Function.updateITE Ο x x) (freeVarSet phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο))) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [β c3] at s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_union] at s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο x x) a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Function.updateITE Ο x x v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateITE] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Function.updateITE Ο x x v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : (if v = x then x else Ο v) β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [if_neg c2] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : (if v = x then x else Ο v) β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exfalso | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ False | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : 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 β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | left | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : 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 β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : 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 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact s2 | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : 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 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply h1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Β¬Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = V (Ο v) | case a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Β¬Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ Β¬v = x β§ isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | tauto | case a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 :
Β¬Ο v =
fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ Β¬v = x β§ isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateITE] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Function.updateITE V' x d v = Function.updateITE V x d (Function.updateITE Ο x x v) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | by_cases c2 : v = x | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v)
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [if_pos c2] | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
β’ d = if True then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : v = x
β’ d = if True then d else V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [if_neg c2] | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
β’ (if v = x then d else V' v) = if (if v = x then x else Ο v) = x then d else V (if v = x then x else Ο v) | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
β’ V' v = if Ο v = x then d else V (Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
β’ V' v = if Ο v = x then d else V (Ο v) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
hβ : Ο v = x
β’ V' v = d
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
hβ : Β¬Ο v = x
β’ V' v = V (Ο v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c3 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [s1] at c1
simp only [β c3] at c1
simp only [Finset.mem_union] at c1
simp only [isFreeIn_iff_mem_freeVarSet] at a1
obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο (Ο v) (Ο v)) a1
simp only [Function.updateITE] at s2
simp only [ite_self] at s2
exfalso
apply c1
left
exact s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Ο v = x
β’ V' v = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c3 =>
apply h1
tauto | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Β¬Ο v = x
β’ V' v = V (Ο v) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Ο v = x
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [β c3] at c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_union] at c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s2 := Finset.mem_image_of_mem (Function.updateITE Ο (Ο v) (Ο v)) a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Function.updateITE Ο (Ο v) (Ο v) v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateITE] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Function.updateITE Ο (Ο v) (Ο v) v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : (if v = Ο v then Ο v else Ο v) β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [ite_self] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : (if v = Ο v then Ο v else Ο v) β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exfalso | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ V' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ False | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | left | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο) | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact s2 | case h
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
c2 : Β¬v = x
c3 : Ο v = x
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1 :
Β¬(Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet β¨
Ο v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
a1 : v β phi.freeVarSet
s2 : Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet
β’ Ο v β Finset.image (Function.updateITE Ο (Ο v) (Ο v)) phi.freeVarSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply h1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Β¬Ο v = x
β’ V' v = V (Ο v) | case a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Β¬Ο v = x
β’ Β¬v = x β§ isFreeIn v phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | tauto | case a
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : isFreeIn v phi
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 : Β¬v = x
c3 : Β¬Ο v = x
β’ Β¬v = x β§ isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro v a1 | case h.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
β’ β x_1 β phi.predVarSet.biUnion (predVarFreeVarSet Ο),
V'' x_1 =
Function.updateITE V
(if
x β
Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d x_1 | case h.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
Function.updateITE V
(if x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | case h.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
Function.updateITE V
(if x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο) then
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
else x)
d v | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβ : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d v
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβ : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1 =>
simp only [Function.updateITE]
split_ifs
case _ c2 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [β s1] at c2
obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c2] at s2
simp only [Finset.mem_union] at s2
push_neg at s2
cases s2
case _ s2_left s2_right =>
contradiction
case _ c2 =>
exact h2 v a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1 =>
simp only [Finset.mem_union] at c1
push_neg at c1
cases c1
case _ c1_left c1_right =>
have s1 : Β¬ v = x
intro contra
apply c1_right
subst contra
exact a1
simp only [Function.updateITE]
simp only [if_neg s1]
exact h2 v a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Function.updateITE] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
Function.updateITE V
(fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)))
d v | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
if
v =
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) then
d
else V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | split_ifs | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v =
if
v =
fresh x c
(Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)) then
d
else V v | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβ :
v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
hβ :
Β¬v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = V v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 =>
obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi
simp only [β s1] at c2
obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο)))
simp only [β c2] at s2
simp only [Finset.mem_union] at s2
push_neg at s2
cases s2
case _ s2_left s2_right =>
contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c2 =>
exact h2 v a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
Β¬v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s1 := Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image (Function.updateITE Ο x x) c phi | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [β s1] at c2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | obtain s2 := fresh_not_mem x c ((freeVarSet (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi)) βͺ (Finset.biUnion (predVarSet phi) (predVarFreeVarSet Ο))) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο)) β
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [β c2] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο)) β
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_union] at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
Β¬(v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet β¨
v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | push_neg at s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
Β¬(v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet β¨
v β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = d | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet β§
v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | cases s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2 :
v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet β§
v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d | case intro
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
leftβ : v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet
rightβ : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ s2_left s2_right =>
contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2_left : v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet
s2_right : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 :
(Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet =
Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c2 :
v =
fresh x c
((Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet βͺ
phi.predVarSet.biUnion (predVarFreeVarSet Ο))
s2_left : v β (Var.All.Rec.Fresh.sub (Function.updateITE Ο x x) c phi).freeVarSet
s2_right : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact h2 v a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c2 :
Β¬v = fresh x c (Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | simp only [Finset.mem_union] at c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet βͺ phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : Β¬(x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | push_neg at c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : Β¬(x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β¨ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο))
β’ V'' v = Function.updateITE V x d v | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β§ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | cases c1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1 : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet β§ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v | case intro
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
leftβ : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
rightβ : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | case _ c1_left c1_right =>
have s1 : Β¬ v = x
intro contra
apply c1_right
subst contra
exact a1
simp only [Function.updateITE]
simp only [if_neg s1]
exact h2 v a1 | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | have s1 : Β¬ v = x | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ V'' v = Function.updateITE V x d v | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Β¬v = x
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | intro contra | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ Β¬v = x
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
contra : v = x
β’ False
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | apply c1_right | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
contra : v = x
β’ False
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
contra : v = x
β’ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | subst contra | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
contra : v = x
β’ x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
h1 : β (x : VarName), Β¬x = v β§ isFreeIn x phi β V' x = V (Ο x)
c1_left : v β Finset.image (Function.updateITE Ο v v) phi.freeVarSet
c1_right : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | exact a1 | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
h1 : β (x : VarName), Β¬x = v β§ isFreeIn x phi β V' x = V (Ο x)
c1_left : v β Finset.image (Function.updateITE Ο v v) phi.freeVarSet
c1_right : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
β’ v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x phi β V' x = V (Ο x)) β
(β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E Ο) V' E phi β Holds D I V E (subAux c Ο Ο phi))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x_1 : VarName), Β¬x_1 = x β§ isFreeIn x_1 phi β V' x_1 = V (Ο x_1)
h2 : β x β phi.predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
d : D
v : VarName
a1 : v β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
c1_left : x β Finset.image (Function.updateITE Ο x x) phi.freeVarSet
c1_right : x β phi.predVarSet.biUnion (predVarFreeVarSet Ο)
s1 : Β¬v = x
β’ V'' v = Function.updateITE V x d v |
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