url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (h1_phi'.iff_ h1_psi') β Holds D J V E (h1_phi.iff_ h1_psi)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (Holds D I V E h1_phi' β Holds D I V E h1_psi') β (Holds D J V E h1_phi β Holds D J V E h1_psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (h1_phi'.iff_ h1_psi') β Holds D J V E (h1_phi.iff_ h1_psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
congr! 1
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (Holds D I V E h1_phi' β Holds D I V E h1_psi') β (Holds D J V E h1_phi β Holds D J V E h1_psi)
|
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
case a.h.e'_2.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_psi' β Holds D J V E h1_psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (Holds D I V E h1_phi' β Holds D I V E h1_psi') β (Holds D J V E h1_phi β Holds D J V E h1_psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h1_ih_1 V h2
|
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_phi' β Holds D J V E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h1_ih_2 V h2
|
case a.h.e'_2.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_psi' β Holds D J V E h1_psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1_phi'
aβ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
h1_ih_2 :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_psi' β Holds D J V E h1_psi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E h1_psi' β Holds D J V E h1_psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (exists_ h1_x h1_phi') β Holds D J V E (exists_ h1_x h1_phi)
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β d, Holds D I (Function.updateITE V h1_x d) E h1_phi') β β d, Holds D J (Function.updateITE V h1_x d) E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (exists_ h1_x h1_phi') β Holds D J V E (exists_ h1_x h1_phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
first | apply forall_congr' | apply exists_congr
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β d, Holds D I (Function.updateITE V h1_x d) E h1_phi') β β d, Holds D J (Function.updateITE V h1_x d) E h1_phi
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ β (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' β Holds D J (Function.updateITE V h1_x a) E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β d, Holds D I (Function.updateITE V h1_x d) E h1_phi') β β d, Holds D J (Function.updateITE V h1_x d) E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro d
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ β (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' β Holds D J (Function.updateITE V h1_x a) E h1_phi
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ Holds D I (Function.updateITE V h1_x d) E h1_phi' β Holds D J (Function.updateITE V h1_x d) E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ β (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' β Holds D J (Function.updateITE V h1_x a) E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply h1_ih
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ Holds D I (Function.updateITE V h1_x d) E h1_phi' β Holds D J (Function.updateITE V h1_x d) E h1_phi
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ Holds D I (Function.updateITE V h1_x d) E h1_phi' β Holds D J (Function.updateITE V h1_x d) E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro Q ds a1
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds)
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
specialize h2 Q ds a1
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
have s1 :
Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H :=
by
apply Holds_coincide_Var
intro v a1
apply Function.updateListITE_updateIte
intro contra
subst contra
contradiction
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 :
Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [h2] at s1
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 :
Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 :
Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact s1
|
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β J.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply forall_congr'
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi') β
β (d : D), Holds D J (Function.updateITE V h1_x d) E h1_phi
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ β (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' β Holds D J (Function.updateITE V h1_x a) E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi') β
β (d : D), Holds D J (Function.updateITE V h1_x d) E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply exists_congr
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β d, Holds D I (Function.updateITE V h1_x d) E h1_phi') β β d, Holds D J (Function.updateITE V h1_x d) E h1_phi
|
case h
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ β (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' β Holds D J (Function.updateITE V h1_x a) E h1_phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ (β d, Holds D I (Function.updateITE V h1_x d) E h1_phi') β β d, Holds D J (Function.updateITE V h1_x d) E h1_phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply Holds_coincide_Var
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ β (v : VarName),
isFreeIn v H β Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H β
Holds D I (Function.updateListITE V zs ds) E H
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro v a1
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ β (v : VarName),
isFreeIn v H β Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
β’ β (v : VarName),
isFreeIn v H β Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply Function.updateListITE_updateIte
|
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Β¬v = h1_x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
intro contra
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Β¬v = h1_x
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
contra : v = h1_x
β’ False
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
β’ Β¬v = h1_x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
subst contra
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
contra : v = h1_x
β’ False
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
h1_1 : Β¬isFreeIn v H
β’ False
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_x : VarName
h1_phi h1_phi' : Formula
h1_1 : Β¬isFreeIn h1_x H
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
contra : v = h1_x
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
contradiction
|
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
h1_1 : Β¬isFreeIn v H
β’ False
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_phi' : Formula
aβ : IsSub P zs H h1_phi h1_phi'
h1_ih :
β (V : VarAssignment D),
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
(Holds D I V E h1_phi' β Holds D J V E h1_phi)
V : VarAssignment D
d : D
Q : PredName
ds : List D
a1β : Q = P β§ ds.length = zs.length
h2 : Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds
v : VarName
a1 : isFreeIn v H
h1_1 : Β¬isFreeIn v H
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
cases E
|
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (def_ X xs) β Holds D J V E (def_ X xs)
|
case nil
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) [] H β J.pred_var_ P ds)
β’ Holds D I V [] (def_ X xs) β Holds D J V [] (def_ X xs)
case cons
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
headβ : Definition
tailβ : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (headβ :: tailβ) H β J.pred_var_ P ds)
β’ Holds D I V (headβ :: tailβ) (def_ X xs) β Holds D J V (headβ :: tailβ) (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)
β’ Holds D I V E (def_ X xs) β Holds D J V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
case nil =>
simp only [Holds]
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) [] H β J.pred_var_ P ds)
β’ Holds D I V [] (def_ X xs) β Holds D J V [] (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) [] H β J.pred_var_ P ds)
β’ Holds D I V [] (def_ X xs) β Holds D J V [] (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) [] H β J.pred_var_ P ds)
β’ Holds D I V [] (def_ X xs) β Holds D J V [] (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) [] H β J.pred_var_ P ds)
β’ Holds D I V [] (def_ X xs) β Holds D J V [] (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [Holds]
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
β’ Holds D I V (hd :: tl) (def_ X xs) β Holds D J V (hd :: tl) (def_ X xs)
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
β’ (if X = hd.name β§ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) β
if X = hd.name β§ xs.length = hd.args.length then
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D J V tl (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
β’ Holds D I V (hd :: tl) (def_ X xs) β Holds D J V (hd :: tl) (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
split_ifs
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
β’ (if X = hd.name β§ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) β
if X = hd.name β§ xs.length = hd.args.length then
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D J V tl (def_ X xs)
|
case pos
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
hβ : X = hd.name β§ xs.length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
hβ : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D J V tl (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
β’ (if X = hd.name β§ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) β
if X = hd.name β§ xs.length = hd.args.length then
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D J V tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply Holds_coincide_PredVar
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
|
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ I.pred_const_ = J.pred_const_
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h3_const
|
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ I.pred_const_ = J.pred_const_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ I.pred_const_ = J.pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [predVarOccursIn_iff_mem_predVarSet]
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β (I.pred_var_ P ds β J.pred_var_ P ds)
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β hd.q.predVarSet β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [hd.h2]
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β hd.q.predVarSet β (I.pred_var_ P ds β J.pred_var_ P ds)
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β β
β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β hd.q.predVarSet β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β β
β (I.pred_var_ P ds β J.pred_var_ P ds)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : X = hd.name β§ xs.length = hd.args.length
β’ β (P : PredName) (ds : List D), (P, ds.length) β β
β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
apply Holds_coincide_PredVar
|
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D J V tl (def_ X xs)
|
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ I.pred_const_ = J.pred_const_
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ Holds D I V tl (def_ X xs) β Holds D J V tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
exact h3_const
|
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ I.pred_const_ = J.pred_const_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ I.pred_const_ = J.pred_const_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp only [predVarOccursIn]
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β (I.pred_var_ P ds β J.pred_var_ P ds)
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), False β (I.pred_var_ P ds β J.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
|
[131, 1]
|
[245, 15]
|
simp
|
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), False β (I.pred_var_ P ds β J.pred_var_ P ds)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I J : Interpretation D
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
X : DefName
xs : List VarName
V : VarAssignment D
hd : Definition
tl : List Definition
h2 :
β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H β J.pred_var_ P ds)
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
β’ β (P : PredName) (ds : List D), False β (I.pred_var_ P ds β J.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [IsValid] at h2
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : F.IsValid
β’ F'.IsValid
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : F.IsValid
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [IsValid]
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
intro D I V E
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
let J : Interpretation D :=
{ nonempty := I.nonempty
pred_const_ := I.pred_const_
pred_var_ := fun (Q : PredName) (ds : List D) =>
if (Q = P β§ ds.length = zs.length)
then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds }
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
β’ Holds D I V E F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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 F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
obtain s1 := substitution_theorem D I J V E F P zs H F' h1
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
β’ Holds D I V E F'
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
I.pred_const_ = J.pred_const_ β
(β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
β’ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [Interpretation.pred_var_] at s1
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
I.pred_const_ = J.pred_const_ β
(β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F'
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β (Holds D I (Function.updateListITE V zs ds) E H β J.pred_var_ P ds)) β
I.pred_const_ = J.pred_const_ β
(β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [s2]
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
s2 : Holds D I V E F' β Holds D J V E F
β’ Holds D I V E F'
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
s2 : Holds D I V E F' β Holds D J V E F
β’ Holds D J V E F
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
s2 : Holds D I V E F' β Holds D J V E F
β’ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
apply h2
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
s2 : Holds D I V E F' β Holds D J V E F
β’ Holds D J V E F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
s2 : Holds D I V E F' β Holds D J V E F
β’ Holds D J V E F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
apply s1
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F' β Holds D J V E F
|
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)
case h3_const
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ True
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ Holds D I V E F' β Holds D J V E F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
intro Q ds a1
|
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)
|
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
cases a1
|
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
case h2.intro
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
leftβ : Q = P
rightβ : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Q = P β§ ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
case h2.intro a1_left a1_right =>
simp
simp only [if_pos a1_right]
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [if_pos a1_right]
|
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1_left : Q = P
a1_right : ds.length = zs.length
β’ Holds D I (Function.updateListITE V zs ds) E H β
if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp
|
case h3_const
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ True
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h3_const
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ True
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
intro Q ds a1
|
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)
|
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Β¬(Q = P β§ ds.length = zs.length)
β’ I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
β’ β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Pred/One/Ind/Sub.lean
|
FOL.NV.Sub.Pred.One.Ind.substitution_is_valid
|
[248, 1]
|
[282, 11]
|
simp only [if_neg a1]
|
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Β¬(Q = P β§ ds.length = zs.length)
β’ I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h3_var
F F' : Formula
P : PredName
zs : List VarName
H : Formula
h1 : IsSub P zs H F F'
h2 : 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
J : Interpretation D :=
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun Q ds =>
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }
s1 :
(β (Q : PredName) (ds : List D),
Q = P β§ ds.length = zs.length β
(Holds D I (Function.updateListITE V zs ds) E H β
if True β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) β
True β
(β (Q : PredName) (ds : List D),
Β¬(Q = P β§ ds.length = zs.length) β
(I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H
else I.pred_var_ Q ds)) β
(Holds D I V E F' β Holds D J V E F)
Q : PredName
ds : List D
a1 : Β¬(Q = P β§ ds.length = zs.length)
β’ I.pred_var_ Q ds β
if Q = P β§ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
induction h1 generalizing V V'
|
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
h1 : IsSubAux Ο binders F F'
h2 : β v β binders, V v = V' (Ο v)
h3 : β (v : VarName), Ο v β binders β V v = V' (Ο v)
h4 : β v β binders, v = Ο v
β’ Holds D I V E F β Holds D I V' E F'
|
case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
Xβ : PredName
xsβ : List VarName
aβ : β v β xsβ, v β bindersβ β Οβ v β bindersβ
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (pred_const_ Xβ xsβ) β Holds D I V' E (pred_const_ Xβ (List.map Οβ xsβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
Xβ : PredName
xsβ : List VarName
aβ : β v β xsβ, v β bindersβ β Οβ v β bindersβ
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (pred_var_ Xβ xsβ) β Holds D I V' E (pred_var_ Xβ (List.map Οβ xsβ))
case eq_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
xβ yβ : VarName
aβ : β (v : VarName), v = xβ β¨ v = yβ β v β bindersβ β Οβ v β bindersβ
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (eq_ xβ yβ) β Holds D I V' E (eq_ (Οβ xβ) (Οβ yβ))
case true_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E true_ β Holds D I V' E true_
case false_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β Holds D I V' E false_
case not_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
phiβ phi'β : Formula
aβ : IsSubAux Οβ bindersβ phiβ phi'β
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E phiβ.not_ β Holds D I V' E phi'β.not_
case imp_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSubAux Οβ bindersβ phiβ phi'β
aβ : IsSubAux Οβ bindersβ psiβ psi'β
a_ihβΒΉ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E psiβ β Holds D I V' E psi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (phiβ.imp_ psiβ) β Holds D I V' E (phi'β.imp_ psi'β)
case and_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSubAux Οβ bindersβ phiβ phi'β
aβ : IsSubAux Οβ bindersβ psiβ psi'β
a_ihβΒΉ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E psiβ β Holds D I V' E psi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (phiβ.and_ psiβ) β Holds D I V' E (phi'β.and_ psi'β)
case or_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSubAux Οβ bindersβ phiβ phi'β
aβ : IsSubAux Οβ bindersβ psiβ psi'β
a_ihβΒΉ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E psiβ β Holds D I V' E psi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (phiβ.or_ psiβ) β Holds D I V' E (phi'β.or_ psi'β)
case iff_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
phiβ psiβ phi'β psi'β : Formula
aβΒΉ : IsSubAux Οβ bindersβ phiβ phi'β
aβ : IsSubAux Οβ bindersβ psiβ psi'β
a_ihβΒΉ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ, V v = V' (Οβ v)) β
(β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)) β
(β v β bindersβ, v = Οβ v) β (Holds D I V E psiβ β Holds D I V' E psi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (phiβ.iff_ psiβ) β Holds D I V' E (phi'β.iff_ psi'β)
case forall_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
xβ : VarName
phiβ phi'β : Formula
aβ : IsSubAux (Function.updateITE Οβ xβ xβ) (bindersβ βͺ {xβ}) phiβ phi'β
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ βͺ {xβ}, V v = V' (Function.updateITE Οβ xβ xβ v)) β
(β (v : VarName), Function.updateITE Οβ xβ xβ v β bindersβ βͺ {xβ} β V v = V' (Function.updateITE Οβ xβ xβ v)) β
(β v β bindersβ βͺ {xβ}, v = Function.updateITE Οβ xβ xβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (forall_ xβ phiβ) β Holds D I V' E (forall_ xβ phi'β)
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
xβ : VarName
phiβ phi'β : Formula
aβ : IsSubAux (Function.updateITE Οβ xβ xβ) (bindersβ βͺ {xβ}) phiβ phi'β
a_ihβ :
β (V V' : VarAssignment D),
(β v β bindersβ βͺ {xβ}, V v = V' (Function.updateITE Οβ xβ xβ v)) β
(β (v : VarName), Function.updateITE Οβ xβ xβ v β bindersβ βͺ {xβ} β V v = V' (Function.updateITE Οβ xβ xβ v)) β
(β v β bindersβ βͺ {xβ}, v = Function.updateITE Οβ xβ xβ v) β (Holds D I V E phiβ β Holds D I V' E phi'β)
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (exists_ xβ phiβ) β Holds D I V' E (exists_ xβ phi'β)
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
Xβ : DefName
xsβ : List VarName
aβ : β v β xsβ, v β bindersβ β Οβ v β bindersβ
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E (def_ Xβ xsβ) β Holds D I V' E (def_ Xβ (List.map Οβ xsβ))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
h1 : IsSubAux Ο binders F F'
h2 : β v β binders, V v = V' (Ο v)
h3 : β (v : VarName), Ο v β binders β V v = V' (Ο v)
h4 : β v β binders, v = Ο v
β’ Holds D I V E F β Holds D I V' E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
case true_ | false_ =>
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β Holds D I V' E false_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β Holds D I V' E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
case not_ Ο' binders' phi phi' _ ih_2 =>
simp only [Holds]
congr! 1
exact ih_2 V V' h2 h3 h4
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi.not_ β Holds D I V' E phi'.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi.not_ β Holds D I V' E phi'.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (pred_var_ X' xs') β Holds D I V' E (pred_var_ X' (List.map Ο' xs'))
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map V' (List.map Ο' xs'))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (pred_var_ X' xs') β Holds D I V' E (pred_var_ X' (List.map Ο' xs'))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map V' (List.map Ο' xs'))
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map (V' β Ο') xs')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map V' (List.map Ο' xs'))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
congr! 1
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map (V' β Ο') xs')
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ List.map V xs' = List.map (V' β Ο') xs'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ I.pred_var_ X' (List.map V xs') β I.pred_var_ X' (List.map (V' β Ο') xs')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [List.map_eq_map_iff]
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ List.map V xs' = List.map (V' β Ο') xs'
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ β x β xs', V x = (V' β Ο') x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ List.map V xs' = List.map (V' β Ο') xs'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
intro x a1
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ β x β xs', V x = (V' β Ο') x
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = (V' β Ο') x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ β x β xs', V x = (V' β Ο') x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = (V' β Ο') x
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = V' (Ο' x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = (V' β Ο') x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
by_cases c1 : x β binders'
|
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = V' (Ο' x)
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact h2 x c1
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply h3
|
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ Ο' x β binders'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact ih_1 x a1 c1
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ Ο' x β binders'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
x : VarName
a1 : x β xs'
c1 : x β binders'
β’ Ο' x β binders'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (eq_ x y) β Holds D I V' E (eq_ (Ο' x) (Ο' y))
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V y β V' (Ο' x) = V' (Ο' y)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (eq_ x y) β Holds D I V' E (eq_ (Ο' x) (Ο' y))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
congr! 1
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V y β V' (Ο' x) = V' (Ο' y)
|
case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V' (Ο' x)
case a.h.e'_3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V y = V' (Ο' y)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V y β V' (Ο' x) = V' (Ο' y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
by_cases c1 : x β binders'
|
case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V' (Ο' x)
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact h2 x c1
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply h3
|
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ Ο' x β binders'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ V x = V' (Ο' x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply ih_1
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ Ο' x β binders'
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x = x β¨ x = y
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x β binders'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ Ο' x β binders'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x = x β¨ x = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x = x β¨ x = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact c1
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x β binders'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : x β binders'
β’ x β binders'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
by_cases c1 : y β binders'
|
case a.h.e'_3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V y = V' (Ο' y)
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ V y = V' (Ο' y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact h2 y c1
|
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply h3
|
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ Ο' y β binders'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ V y = V' (Ο' y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply ih_1
|
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ Ο' y β binders'
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y = x β¨ y = y
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y β binders'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ Ο' y β binders'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y = x β¨ y = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y = x β¨ y = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact c1
|
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y β binders'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
c1 : y β binders'
β’ y β binders'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β Holds D I V' E false_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β Holds D I V' E false_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi.not_ β Holds D I V' E phi'.not_
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Β¬Holds D I V E phi β Β¬Holds D I V' E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi.not_ β Holds D I V' E phi'.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
congr! 1
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Β¬Holds D I V E phi β Β¬Holds D I V' E phi'
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Β¬Holds D I V E phi β Β¬Holds D I V' E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact ih_2 V V' h2 h3 h4
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (phi.iff_ psi) β Holds D I V' E (phi'.iff_ psi')
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ (Holds D I V E phi β Holds D I V E psi) β (Holds D I V' E phi' β Holds D I V' E psi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (phi.iff_ psi) β Holds D I V' E (phi'.iff_ psi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
congr! 1
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ (Holds D I V E phi β Holds D I V E psi) β (Holds D I V' E phi' β Holds D I V' E psi')
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E psi β Holds D I V' E psi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ (Holds D I V E phi β Holds D I V E psi) β (Holds D I V' E phi' β Holds D I V' E psi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply phi_ih_2 V V' h2 h3 h4
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E phi β Holds D I V' E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply psi_ih_2 V V' h2 h3 h4
|
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E psi β Holds D I V' E psi'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E phi β Holds D I V' E phi')
psi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)) β
(β v β binders', v = Ο' v) β (Holds D I V E psi β Holds D I V' E psi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E psi β Holds D I V' E psi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp at ih_2
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders' βͺ {x}, V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), Function.updateITE Ο' x x v β binders' βͺ {x} β V v = V' (Function.updateITE Ο' x x v)) β
(β v β binders' βͺ {x}, v = Function.updateITE Ο' x x v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders' βͺ {x}, V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), Function.updateITE Ο' x x v β binders' βͺ {x} β V v = V' (Function.updateITE Ο' x x v)) β
(β v β binders' βͺ {x}, v = Function.updateITE Ο' x x v) β (Holds D I V E phi β Holds D I V' E phi')
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
have s1 : β (v : VarName), Β¬ v = x β v β binders' β Β¬ Ο' v = x
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
intro v a1 a2 contra
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ False
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
β’ β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply a1
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ False
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ False
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [β contra]
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = Ο' v
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = x
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
exact h4 v a2
|
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = Ο' v
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
v : VarName
a1 : Β¬v = x
a2 : v β binders'
contra : Ο' v = x
β’ v = Ο' v
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β d, Holds D I (Function.updateITE V x d) E phi) β β d, Holds D I (Function.updateITE V' x d) E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ Holds D I V E (exists_ x phi) β Holds D I V' E (exists_ x phi')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
first | apply forall_congr'| apply exists_congr
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β d, Holds D I (Function.updateITE V x d) E phi) β β d, Holds D I (Function.updateITE V' x d) E phi'
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ β (a : D), Holds D I (Function.updateITE V x a) E phi β Holds D I (Function.updateITE V' x a) E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β d, Holds D I (Function.updateITE V x d) E phi) β β d, Holds D I (Function.updateITE V' x d) E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
intro d
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ β (a : D), Holds D I (Function.updateITE V x a) E phi β Holds D I (Function.updateITE V' x a) E phi'
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ Holds D I (Function.updateITE V x d) E phi β Holds D I (Function.updateITE V' x d) E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ β (a : D), Holds D I (Function.updateITE V x a) E phi β Holds D I (Function.updateITE V' x a) E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply ih_2
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ Holds D I (Function.updateITE V x d) E phi β Holds D I (Function.updateITE V' x d) E phi'
|
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ β (v : VarName),
v β binders' β¨ v = x β Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
case h.h3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β
Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
case h.h4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ Holds D I (Function.updateITE V x d) E phi β Holds D I (Function.updateITE V' x d) E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply forall_congr'
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β (d : D), Holds D I (Function.updateITE V x d) E phi) β β (d : D), Holds D I (Function.updateITE V' x d) E phi'
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ β (a : D), Holds D I (Function.updateITE V x a) E phi β Holds D I (Function.updateITE V' x a) E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β (d : D), Holds D I (Function.updateITE V x d) E phi) β β (d : D), Holds D I (Function.updateITE V' x d) E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
apply exists_congr
|
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β d, Holds D I (Function.updateITE V x d) E phi) β β d, Holds D I (Function.updateITE V' x d) E phi'
|
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ β (a : D), Holds D I (Function.updateITE V x a) E phi β Holds D I (Function.updateITE V' x a) E phi'
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
β’ (β d, Holds D I (Function.updateITE V x d) E phi) β β d, Holds D I (Function.updateITE V' x d) E phi'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Ind/Sub.lean
|
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
|
[129, 1]
|
[273, 17]
|
intro v a1
|
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ β (v : VarName),
v β binders' β¨ v = x β Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
v : VarName
a1 : v β binders' β¨ v = x
β’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' β V v = V' (Ο' v)
h4 : β v β binders', v = Ο' v
ih_2 :
β (V V' : VarAssignment D),
(β (v : VarName), v β binders' β¨ v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName),
Function.updateITE Ο' x x v β binders' β
Β¬Function.updateITE Ο' x x v = x β V v = V' (Function.updateITE Ο' x x v)) β
(β (v : VarName), v β binders' β¨ v = x β v = Function.updateITE Ο' x x v) β
(Holds D I V E phi β Holds D I V' E phi')
s1 : β (v : VarName), Β¬v = x β v β binders' β Β¬Ο' v = x
d : D
β’ β (v : VarName),
v β binders' β¨ v = x β Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Ο' x x v)
TACTIC:
|
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