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 [s1] at 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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β Holds D J V E (pred_var_ P h1_ts)
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 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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β J.pred_var_ P (List.map V h1_ts)
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
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ (List.map V h1_ts).length = zs.length
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_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_X : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
|
no goals
|
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 : PredName
h1_ts : List VarName
h1_1 : h1_X = P β§ h1_ts.length = zs.length
h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H
V : VarAssignment D
s2 : V β Function.updateListITE id zs h1_ts = Function.updateListITE (V β id) zs (List.map V h1_ts)
s1 :
Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H β
Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H)
h2 :
h1_X = P β§ (List.map V h1_ts).length = zs.length β
(Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) β
J.pred_var_ P (List.map V h1_ts))
β’ h1_X = P β§ h1_ts.length = zs.length
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 h1_y : 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 (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
|
no goals
|
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 h1_y : 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 (eq_ h1_x h1_y) β Holds D J V E (eq_ h1_x h1_y)
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)
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 false_ β Holds D J V E false_
|
no goals
|
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)
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 false_ β Holds D J V E 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]
|
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_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
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'.not_ β Holds D J V E h1_phi.not_
|
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
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¬Holds D I V E h1_phi' β Β¬Holds D J V 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_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
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'.not_ β Holds D J V E h1_phi.not_
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_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
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¬Holds D I V E h1_phi' β Β¬Holds D J V E h1_phi
|
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_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
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
|
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_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
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¬Holds D I V E h1_phi' β 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 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_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
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_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
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]
|
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/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
induction s
|
Ξ± : Type
s : Str Ξ±
β’ β n, List.reverse s β exp Ξ± n
|
case nil
Ξ± : Type
β’ β n, [].reverse β exp Ξ± n
case cons
Ξ± : Type
headβ : Ξ±
tailβ : List Ξ±
tail_ihβ : β n, tailβ.reverse β exp Ξ± n
β’ β n, (headβ :: tailβ).reverse β exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ β n, List.reverse s β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
case nil =>
apply Exists.intro 0
exact exp.zero
|
Ξ± : Type
β’ β n, [].reverse β exp Ξ± n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ β n, [].reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
case cons hd tl ih =>
apply Exists.elim ih
intro n a1
apply Exists.intro (n + 1)
simp
exact exp.succ n hd tl.reverse a1
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
apply Exists.intro 0
|
Ξ± : Type
β’ β n, [].reverse β exp Ξ± n
|
Ξ± : Type
β’ [].reverse β exp Ξ± 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ β n, [].reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
exact exp.zero
|
Ξ± : Type
β’ [].reverse β exp Ξ± 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ [].reverse β exp Ξ± 0
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
apply Exists.elim ih
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β (a : β), tl.reverse β exp Ξ± a β β n, (hd :: tl).reverse β exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
intro n a1
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β (a : β), tl.reverse β exp Ξ± a β β n, (hd :: tl).reverse β exp Ξ± n
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
β’ β (a : β), tl.reverse β exp Ξ± a β β n, (hd :: tl).reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
apply Exists.intro (n + 1)
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ (hd :: tl).reverse β exp Ξ± (n + 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ β n, (hd :: tl).reverse β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
simp
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ (hd :: tl).reverse β exp Ξ± (n + 1)
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ tl.reverse ++ [hd] β exp Ξ± (n + 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ (hd :: tl).reverse β exp Ξ± (n + 1)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp
|
[53, 1]
|
[67, 40]
|
exact exp.succ n hd tl.reverse a1
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ tl.reverse ++ [hd] β exp Ξ± (n + 1)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : β n, tl.reverse β exp Ξ± n
n : β
a1 : tl.reverse β exp Ξ± n
β’ tl.reverse ++ [hd] β exp Ξ± (n + 1)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp
|
[70, 1]
|
[77, 13]
|
obtain s1 := rev_str_mem_exp s.reverse
|
Ξ± : Type
s : Str Ξ±
β’ β n, s β exp Ξ± n
|
Ξ± : Type
s : Str Ξ±
s1 : β n, (List.reverse s).reverse β exp Ξ± n
β’ β n, s β exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ β n, s β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp
|
[70, 1]
|
[77, 13]
|
simp only [List.reverse_reverse] at s1
|
Ξ± : Type
s : Str Ξ±
s1 : β n, (List.reverse s).reverse β exp Ξ± n
β’ β n, s β exp Ξ± n
|
Ξ± : Type
s : Str Ξ±
s1 : β n, s β exp Ξ± n
β’ β n, s β exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
s1 : β n, (List.reverse s).reverse β exp Ξ± n
β’ β n, s β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp
|
[70, 1]
|
[77, 13]
|
exact s1
|
Ξ± : Type
s : Str Ξ±
s1 : β n, s β exp Ξ± n
β’ β n, s β exp Ξ± n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
s1 : β n, s β exp Ξ± n
β’ β n, s β exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
induction s
|
Ξ± : Type
s : Str Ξ±
β’ List.reverse s β exp Ξ± (List.length s)
|
case nil
Ξ± : Type
β’ [].reverse β exp Ξ± [].length
case cons
Ξ± : Type
headβ : Ξ±
tailβ : List Ξ±
tail_ihβ : tailβ.reverse β exp Ξ± tailβ.length
β’ (headβ :: tailβ).reverse β exp Ξ± (headβ :: tailβ).length
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ List.reverse s β exp Ξ± (List.length s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
case nil =>
simp
exact exp.zero
|
Ξ± : Type
β’ [].reverse β exp Ξ± [].length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ [].reverse β exp Ξ± [].length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
case cons hd tl ih =>
simp
exact exp.succ tl.length hd tl.reverse ih
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ (hd :: tl).reverse β exp Ξ± (hd :: tl).length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ (hd :: tl).reverse β exp Ξ± (hd :: tl).length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
simp
|
Ξ± : Type
β’ [].reverse β exp Ξ± [].length
|
Ξ± : Type
β’ [] β exp Ξ± 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ [].reverse β exp Ξ± [].length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
exact exp.zero
|
Ξ± : Type
β’ [] β exp Ξ± 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
β’ [] β exp Ξ± 0
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
simp
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ (hd :: tl).reverse β exp Ξ± (hd :: tl).length
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ tl.reverse ++ [hd] β exp Ξ± (tl.length + 1)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ (hd :: tl).reverse β exp Ξ± (hd :: tl).length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.rev_str_mem_exp_str_len
|
[80, 1]
|
[91, 48]
|
exact exp.succ tl.length hd tl.reverse ih
|
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ tl.reverse ++ [hd] β exp Ξ± (tl.length + 1)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
hd : Ξ±
tl : List Ξ±
ih : tl.reverse β exp Ξ± tl.length
β’ tl.reverse ++ [hd] β exp Ξ± (tl.length + 1)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp_str_len
|
[94, 1]
|
[101, 13]
|
obtain s1 := rev_str_mem_exp_str_len s.reverse
|
Ξ± : Type
s : Str Ξ±
β’ s β exp Ξ± (List.length s)
|
Ξ± : Type
s : Str Ξ±
s1 : (List.reverse s).reverse β exp Ξ± (List.reverse s).length
β’ s β exp Ξ± (List.length s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ s β exp Ξ± (List.length s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp_str_len
|
[94, 1]
|
[101, 13]
|
simp at s1
|
Ξ± : Type
s : Str Ξ±
s1 : (List.reverse s).reverse β exp Ξ± (List.reverse s).length
β’ s β exp Ξ± (List.length s)
|
Ξ± : Type
s : Str Ξ±
s1 : s β exp Ξ± (List.length s)
β’ s β exp Ξ± (List.length s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
s1 : (List.reverse s).reverse β exp Ξ± (List.reverse s).length
β’ s β exp Ξ± (List.length s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.str_mem_exp_str_len
|
[94, 1]
|
[101, 13]
|
exact s1
|
Ξ± : Type
s : Str Ξ±
s1 : s β exp Ξ± (List.length s)
β’ s β exp Ξ± (List.length s)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
s1 : s β exp Ξ± (List.length s)
β’ s β exp Ξ± (List.length s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
induction h1
|
Ξ± : Type
s : Str Ξ±
n : β
h1 : s β exp Ξ± n
β’ List.length s = n
|
case zero
Ξ± : Type
s : Str Ξ±
n : β
β’ [].length = 0
case succ
Ξ± : Type
s : Str Ξ±
n nβ : β
aβΒΉ : Ξ±
sβ : Str Ξ±
aβ : sβ β exp Ξ± nβ
a_ihβ : List.length sβ = nβ
β’ List.length (sβ ++ [aβΒΉ]) = nβ + 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
n : β
h1 : s β exp Ξ± n
β’ List.length s = n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
case zero =>
simp
|
Ξ± : Type
s : Str Ξ±
n : β
β’ [].length = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
n : β
β’ [].length = 0
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
case succ m a s ih_1 ih_2 =>
simp
exact ih_2
|
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length (s ++ [a]) = m + 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length (s ++ [a]) = m + 1
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
simp
|
Ξ± : Type
s : Str Ξ±
n : β
β’ [].length = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
n : β
β’ [].length = 0
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
simp
|
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length (s ++ [a]) = m + 1
|
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length s = m
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length (s ++ [a]) = m + 1
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.mem_exp_imp_str_len_eq
|
[104, 1]
|
[116, 17]
|
exact ih_2
|
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length s = m
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
sβ : Str Ξ±
n m : β
a : Ξ±
s : Str Ξ±
ih_1 : s β exp Ξ± m
ih_2 : List.length s = m
β’ List.length s = m
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.all_str_mem_kleene_closure
|
[153, 1]
|
[160, 24]
|
simp only [kleene_closure]
|
Ξ± : Type
s : Str Ξ±
β’ s β kleene_closure Ξ±
|
Ξ± : Type
s : Str Ξ±
β’ s β β n, exp Ξ± n
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ s β kleene_closure Ξ±
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.all_str_mem_kleene_closure
|
[153, 1]
|
[160, 24]
|
simp
|
Ξ± : Type
s : Str Ξ±
β’ s β β n, exp Ξ± n
|
Ξ± : Type
s : Str Ξ±
β’ β i, s β exp Ξ± i
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ s β β n, exp Ξ± n
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.all_str_mem_kleene_closure
|
[153, 1]
|
[160, 24]
|
exact str_mem_exp s
|
Ξ± : Type
s : Str Ξ±
β’ β i, s β exp Ξ± i
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Str Ξ±
β’ β i, s β exp Ξ± i
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.thm_2
|
[192, 1]
|
[198, 36]
|
symm
|
Ξ± : Type
s t u : Str Ξ±
β’ s ++ (t ++ u) = s ++ t ++ u
|
Ξ± : Type
s t u : Str Ξ±
β’ s ++ t ++ u = s ++ (t ++ u)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s t u : Str Ξ±
β’ s ++ (t ++ u) = s ++ t ++ u
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Strings.thm_2
|
[192, 1]
|
[198, 36]
|
exact (List.append_assoc s t u)
|
Ξ± : Type
s t u : Str Ξ±
β’ s ++ t ++ u = s ++ (t ++ u)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s t u : Str Ξ±
β’ s ++ t ++ u = s ++ (t ++ u)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Languages.thm_3_a
|
[237, 1]
|
[243, 9]
|
simp only [concat]
|
Ξ± : Type
L : Language Ξ±
β’ concat L β
= β
|
Ξ± : Type
L : Language Ξ±
β’ {x | β s β L, β t β β
, s ++ t = x} = β
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
L : Language Ξ±
β’ concat L β
= β
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Languages.thm_3_a
|
[237, 1]
|
[243, 9]
|
simp
|
Ξ± : Type
L : Language Ξ±
β’ {x | β s β L, β t β β
, s ++ t = x} = β
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
L : Language Ξ±
β’ {x | β s β L, β t β β
, s ++ t = x} = β
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Languages.thm_3_b
|
[246, 1]
|
[252, 9]
|
simp only [concat]
|
Ξ± : Type
L : Language Ξ±
β’ concat β
L = β
|
Ξ± : Type
L : Language Ξ±
β’ {x | β s β β
, β t β L, s ++ t = x} = β
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
L : Language Ξ±
β’ concat β
L = β
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Languages.thm_3_b
|
[246, 1]
|
[252, 9]
|
simp
|
Ξ± : Type
L : Language Ξ±
β’ {x | β s β β
, β t β L, s ++ t = x} = β
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
L : Language Ξ±
β’ {x | β s β β
, β t β L, s ++ t = x} = β
TACTIC:
|
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