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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] | 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 |
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_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 = if v = x 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] | simp only [if_neg 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)
s1 : Β¬v = x
β’ V'' v = if v = x then d else V 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 = 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] | 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)
s1 : Β¬v = x
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 [subAux] | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (subAux c Ο Ο (def_ X xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs)) |
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] | induction E generalizing V V' Ο | D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' E Ο) V' E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs)) | case nil
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
case cons
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' tailβ Ο) V' tailβ (def_ X xs) β Holds D I V tailβ (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (headβ :: tailβ) Ο) V' (headβ :: tailβ) (def_ X xs) β
Holds D I V (headβ :: tailβ) (def_ X (List.map Ο xs)) |
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 nil =>
simp only [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | 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 [Holds] | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' [] Ο) V' [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | 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
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) |
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
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' (E_hd :: E_tl) (def_ X xs) β
Holds D I V (E_hd :: E_tl) (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 : (List.map V' xs) = (List.map (V β Ο) xs) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 [List.map_eq_map_iff] | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ List.map V' xs = List.map (V β Ο) xs
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ β x β xs, V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 h1 x a1 | case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
x : VarName
a1 : x β xs
β’ V' x = (V β Ο) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 [s1] | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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] | clear s1 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
s1 : List.map V' xs = List.map (V β Ο) xs
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) |
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
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
else Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
else Holds D I V E_tl (def_ X (List.map Ο xs)) | case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : X = E_hd.name β§ xs.length = E_hd.args.length
hβ : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : X = E_hd.name β§ xs.length = E_hd.args.length
hβ : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs))
case pos
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q
case neg
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
hβΒΉ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 c2 =>
simp only [List.length_map] at c2
contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs)) | 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 c2 =>
simp only [List.length_map] at c2
contradiction | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | 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 s2 : Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | case s2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q |
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 Holds_coincide_Var | case s2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q |
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 s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (v : VarName),
isFreeIn v E_hd.q β
Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) v =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) v
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) x =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q |
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 Function.updateListITE_map_mem_ext | case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ Function.updateListITE V' E_hd.args (List.map (V β Ο) xs) x =
Function.updateListITE V E_hd.args (List.map (V β Ο) xs) x
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | case s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y
case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ E_hd.args.length = xs.length
case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ x β E_hd.args
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q |
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 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q |
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 s2] | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q |
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] | clear s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q |
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 Holds_coincide_PredVar | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = I.pred_const_
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length E_hd.q β ((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds) |
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 s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ β y β xs, (V β Ο) y = (V β Ο) y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Fresh/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux | [123, 1] | [434, 44] | tauto | case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ E_hd.args.length = xs.length | 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_iff_mem_freeVarSet] at a1 | case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
β’ x β E_hd.args | case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args |
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 List.mem_toFinset] | case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args | case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args.toFinset |
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 Finset.mem_of_subset E_hd.h1 a1 | case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
x : VarName
a1 : x β E_hd.q.freeVarSet
β’ x β E_hd.args.toFinset | 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 [I'] | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = I.pred_const_ | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (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_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
I.pred_const_ |
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] | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ (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_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
I.pred_const_ | 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 P ds a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length E_hd.q β ((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds) | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length E_hd.q
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds |
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 [predVarOccursIn_iff_mem_predVarSet] at a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length E_hd.q
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β E_hd.q.predVarSet
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds |
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 [E_hd.h2] at a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β E_hd.q.predVarSet
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β β
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds |
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 at a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) β β
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β I.pred_var_ P ds | 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 [List.length_map] at c2 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs)) |
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
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
c2 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) (Function.updateListITE V' E_hd.args (List.map (V β Ο) xs)) E_tl E_hd.q β
Holds D I V E_tl (def_ X (List.map Ο xs)) | 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 [List.length_map] at c2 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q |
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
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map Ο xs))) E_tl E_hd.q | 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 s2 := E_ih V V' Ο | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 s2 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β x β xs, V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) |
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] | specialize s2 h1 h2 | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 :
(β x β xs, V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) |
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 s2] | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)) | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) |
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 Holds_coincide_PredVar | D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D (I' D I V'' (E_hd :: E_tl) Ο) V' E_tl (def_ X xs) β Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = (I' D I V'' E_tl Ο).pred_const_
case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (def_ X xs) β
((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds) |
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'] | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_const_ = (I' D I V'' E_tl Ο).pred_const_ | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (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_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
(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_tl ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ |
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] | case h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (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_hd :: E_tl) ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ =
(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_tl ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds).pred_const_ | 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 P ds a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ β (P : PredName) (ds : List D),
predVarOccursIn P ds.length (def_ X xs) β
((I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds) | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds |
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 [predVarOccursIn] at a1 | case h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
β (V V' : VarAssignment D) (Ο : VarName β VarName),
(β (x : VarName), isFreeIn x (def_ X xs) β V' x = V (Ο x)) β
(β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x) β
(Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs)))
V V' : VarAssignment D
Ο : VarName β VarName
h1 : β x β xs, V' x = V (Ο x)
h2 : β x β (def_ X xs).predVarSet.biUnion (predVarFreeVarSet Ο), V'' x = V x
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : Β¬(X = E_hd.name β§ (List.map Ο xs).length = E_hd.args.length)
s2 : Holds D (I' D I V'' E_tl Ο) V' E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
β’ (I' D I V'' (E_hd :: E_tl) Ο).pred_var_ P ds β (I' D I V'' E_tl Ο).pred_var_ P ds | 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 | [437, 1] | [449, 9] | apply substitution_theorem_aux | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ Holds D (I' D I V E Ο) V E F β Holds D I V E (sub c Ο F) | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β (x : VarName), isFreeIn x F β V x = V (id x)
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β x β F.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 | [437, 1] | [449, 9] | simp | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β (x : VarName), isFreeIn x F β V x = V (id 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 | [437, 1] | [449, 9] | simp | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
β’ β x β F.predVarSet.biUnion (predVarFreeVarSet Ο), V x = 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_is_valid | [452, 1] | [464, 11] | simp only [IsValid] at h1 | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : F.IsValid
β’ (sub c Ο F).IsValid | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub c Ο F).IsValid |
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_is_valid | [452, 1] | [464, 11] | simp only [IsValid] | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub c Ο F).IsValid | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο F) |
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_is_valid | [452, 1] | [464, 11] | intro D I V E | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub c Ο F) | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E (sub c Ο F) |
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_is_valid | [452, 1] | [464, 11] | simp only [β substitution_theorem] | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E (sub c Ο F) | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D (I' D I V E Ο) V E F |
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_is_valid | [452, 1] | [464, 11] | apply h1 | c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
F : Formula
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D (I' D I V E Ο) V E F | no goals |
https://github.com/pandaman64/QuickSortInLean.git | ab0aaee0aed280959328844f9a6cd13bf00c5935 | QuickSortInLean/Permutation.lean | invertible_id | [8, 1] | [15, 19] | have : isInv (id : Ξ± β Ξ±) id := by
apply And.intro
. intro x
simp
. intro y
simp | Ξ± : Sort u_1
β’ invertible id | Ξ± : Sort u_1
this : isInv id id
β’ invertible id |
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