url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Ind/Sub.lean | FOL.NV.Sub.Pred.One.Ind.substitution_theorem | [131, 1] | [245, 15] | simp only [Holds] | D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : β (Q : PredName) (ds : List D), Β¬(Q = P β§ ds.length = zs.length) β (I.pred_var_ Q ds β J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
aβΒΉ : IsSub P zs H h1_phi h1... | 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... | 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 ... |
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... | 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 ... | 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 ... |
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 ... | 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_va... |
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 ... | 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_va... |
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 :... | 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 :... | 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 : ... |
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 :... | 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... | 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 : ... |
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... | 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... | 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)
... |
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... | 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 h... | 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)
... |
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 h... | 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 h... | 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 d... |
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 h... | 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 h... | 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 d... |
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 h... | 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 h... | 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 d... |
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 h... | 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 h... | 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 d... |
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 h... | 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 d... |
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 :... | 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... | 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 : ... |
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 :... | 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... | 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 : ... |
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 :... | 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_... | 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 : ... |
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_... | 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_... | 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)... |
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_... | 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 ... | 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)... |
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 ... | 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 ... | 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 ... |
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 ... | 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'... | 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 ... |
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'... | 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 ... |
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 : Pred... | 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 : Pre... | 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 : Def... |
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) (d... | 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 ... |
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) (d... | 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 ... |
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 ... | 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 ... | 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 ... |
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 ... | 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
t... | 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 ... |
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 ... | 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... | 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 ... |
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... | 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 : Def... |
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... | 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... | 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 : Def... |
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... | 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... | 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 : Def... |
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... | 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 : Def... |
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 ... | 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... | 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 ... |
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... | 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 : Def... |
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... | 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... | 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 : Def... |
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... | 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 : Def... |
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'
... |
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_ := 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_ := 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_ := 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 :=
{ no... |
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_ := 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_ := 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 :=
{ no... |
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_ := 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_ := 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 :=
{ no... |
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_ := 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 :=
{ no... |
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_ := 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_v... | 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 :=
{ no... |
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_v... | 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_v... | 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 :... |
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_v... | 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_,
... | 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 :... |
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_ := 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 :=
{ no... |
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_ := 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_ := 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 :=
{ no... |
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_ := 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 :=
{ no... |
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_,
... | 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 : Interpretati... |
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_,
pr... | 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_,
pr... | 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... |
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_,
pr... | 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... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | induction h1 generalizing V V' | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
h1 : IsSubAux Ο binders F F'
h2 : β v β binders, V v = V' (Ο v)
h3 : β (v : VarName), Ο v β binders β V v = V' (Ο v)
h4 : β v β binders, v = Ο v
β’ Holds D I V E F β Holds D I V' E F' | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
Xβ : PredName
xsβ : List VarName
aβ : β v β xsβ, v β bindersβ β Οβ v β bindersβ
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : Va... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
h1 : IsSubAux Ο binders F F'
h2 : β v β binders, V v = V' (Ο v)
h3 : β (v : VarName), Ο v β binders β V v = V' (Ο v)
h4 : β v β binders,... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case true_ | false_ =>
simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | case not_ Ο' binders' phi phi' _ ih_2 =>
simp only [Holds]
congr! 1
exact ih_2 V V' h2 h3 h4 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c1 : x β binders' | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName)... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 x c1 | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignmen... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName)... | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNam... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignmen... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact ih_1 x a1 c1 | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNam... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : PredName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignm... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v... | case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c1 : x β binders' | case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 x c1 | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssign... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssign... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply ih_1 | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact c1 | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | by_cases c1 : y β binders' | case a.h.e'_3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_3
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 y c1 | case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssign... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 | case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarNa... | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssign... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply ih_1 | case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : Var... | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssi... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact c1 | case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : V... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x y : VarName
ih_1 : β (v : VarName), v = x β¨ v = y β v β binders' β Ο' v β binders'
V V' : VarAs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (Οβ v)
h4 : β v β bindersβ, v = Οβ v
β’ Holds D I V E false_ β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Οβ : VarName β VarName
bindersβ : Finset VarName
V V' : VarAssignment D
h2 : β v β bindersβ, V v = V' (Οβ v)
h3 : β (v : VarName), Οβ v β bindersβ β V v = V' (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), Ο' v β binders' ... | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact ih_2 V V' h2 h3 h4 | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (Ο' v)) β
(β (v : VarName), ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi phi' : Formula
aβ : IsSubAux Ο' binders' phi phi'
ih_2 :
β (V V' : VarAssignment D),
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β binders', V v = V' (... | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β bind... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply phi_ih_2 V V' h2 h3 h4 | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β bind... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply psi_ih_2 V V' h2 h3 h4 | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi psi'
phi_ih_2 :
β (V V' : VarAssignment D),
(β v β bind... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
phi psi phi' psi' : Formula
aβΒΉ : IsSubAux Ο' binders' phi phi'
aβ : IsSubAux Ο' binders' psi ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp at ih_2 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
ih_2 :
β (V V' : VarAssignment D),
(β v β binders' βͺ {x}, V v = V'... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
ih_2 :
β ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | have s1 : β (v : VarName), Β¬ v = x β v β binders' β Β¬ Ο' v = x | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro v a1 a2 contra | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply a1 | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [β contra] | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h4 v a2 | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 :... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | first | apply forall_congr'| apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro d | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply ih_2 | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : ... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | intro v a1 | case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
aβ : IsSubAux (Function.updateITE Ο' x x) (binders' βͺ {x}) phi phi'
V... |
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