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/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Ξ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E (false_.iff_ true_.not_) | Ξ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ False β Β¬True | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E (false_.iff_ true_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Ξ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ False β Β¬True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ False β Β¬True
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_) | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_) | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β§ Holds D I V E psi β Β¬(Holds D I V E phi β Β¬Holds D I V E psi) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β§ Holds D I V E psi β Β¬(Holds D I V E phi β Β¬Holds D I V E psi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β§ Holds D I V E psi β Β¬(Holds D I V E phi β Β¬Holds D I V E psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi)) | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi)) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi)) | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β¨ Holds D I V E psi β Β¬Holds D I V E phi β Holds D I V E psi | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β¨ Holds D I V E psi β Β¬Holds D I V E phi β Holds D I V E psi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E phi β¨ Holds D I V E psi β Β¬Holds D I V E phi β Holds D I V E psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ ... | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_ | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
((... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_ | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Β¬(((Holds D I V E phi β Holds D I V E psi) β
Β¬((Holds D I V E phi β Holds D I V E psi) β Β¬(Holds D I V E psi β Holds D I V E phi))) β
Β¬(Β¬((Holds D I V E phi β Holds D I V E psi) ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Β¬(((Holds D I V E phi β Holds D I V E psi) β
Β¬((Holds D I V E phi β Holds D I V E psi) β Β¬(Holds D I V E psi β Holds D I V E phi))) β
Β¬(Β¬((Holds D I V E phi β Holds D I V E psi) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Β¬(((Holds D I V E phi β Holds D I V E psi) β
Β¬((Holds D I V E phi β Holds D I V E psi) β Β¬(Holds D I V E psi β Holds D I V ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E _ | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_) | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_) | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ (β d, Holds D I (Function.updateITE V v d) E phi) β Β¬β (d : D), Β¬Holds D I (Function.updateITE V v d) E phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ (β d, Holds D I (Function.updateITE V v d) E phi) β Β¬β (d : D), Β¬Holds D I (Function.updateITE V v d) E phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ (β d, Holds D I (Function.updateITE V v d) E phi) β Β¬β (d : D), Β¬Holds D I (Function.updateITE V v d) E phi
TACTI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | obtain s1 := Sub.Pred.All.Rec.Option.Fresh.substitution_theorem D I V E freshChar Ο | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [β s1] at a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 ... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
s1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [β s1] | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
s1 ... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
s1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
s1 ... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.not_IsDeduct_false | [982, 1] | [988, 26] | intro contra | β’ Β¬IsDeduct [] false_ | contra : IsDeduct [] false_
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
β’ Β¬IsDeduct [] false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.not_IsDeduct_false | [982, 1] | [988, 26] | obtain s1 := soundness [] false_ contra Unit default default default | contra : IsDeduct [] false_
β’ False | contra : IsDeduct [] false_
s1 : (β H β [], Holds Unit default default default H) β Holds Unit default default default false_
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
contra : IsDeduct [] false_
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.not_IsDeduct_false | [982, 1] | [988, 26] | simp at s1 | contra : IsDeduct [] false_
s1 : (β H β [], Holds Unit default default default H) β Holds Unit default default default false_
β’ False | contra : IsDeduct [] false_
s1 : Holds Unit default default default false_
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
contra : IsDeduct [] false_
s1 : (β H β [], Holds Unit default default default H) β Holds Unit default default default false_
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.not_IsDeduct_false | [982, 1] | [988, 26] | simp only [Holds] at s1 | contra : IsDeduct [] false_
s1 : Holds Unit default default default false_
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
contra : IsDeduct [] false_
s1 : Holds Unit default default default false_
β’ False
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | induction F generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
β’ Holds D I (V β Ο) E F β Holds D I V E (replaceAll Ο F) | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β V... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
β’ Holds D I (V β Ο) E F β Holds D I V E (replaceAll Ο F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | all_goals
simp only [replaceAll] | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β V... | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (pred_const_ aβΒΉ (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β Var... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (pred_const_ aβΒΉ aβ))
case pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | any_goals
simp only [Holds] | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (pred_const_ aβΒΉ (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β Var... | case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ I.pred_const_ aβΒΉ (List.map (V β Ο) aβ) β I.pred_const_ aβΒΉ (List.map V (List.map Ο aβ))
case pred_var_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarNa... | Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (pred_const_ aβΒΉ aβ) β Holds D I V E (pred_const_ aβΒΉ (List.map Ο aβ))
case pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case pred_const_ X xs | pred_var_ X xs =>
simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : PredName
xs : List VarName
V : VarAssignment D
β’ I.pred_var_ X (List.map (V β Ο) xs) β I.pred_var_ X (List.map V (List.map Ο xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : PredName
xs : List VarName
V : VarAssignment D
β’ I.pred_var_ X (List.map (V β Ο) xs) β I.pred_var_ X (List.map V (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case eq_ x y =>
simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (replaceAll Ο phi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (r... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
first | apply forall_congr' | apply exists_congr
intro a
have s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
apply Function.updateITE_comp_right_injective
apply h1
simp only [β s1]
exact phi_ih (Function.updateITE V (Ο x) a) | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.update... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.update... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [replaceAll] | case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (def_ aβΒΉ aβ)) | case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (def_ aβΒΉ (List.map Ο aβ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case def_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ aβΒΉ aβ) β Holds D I V E (replaceAll Ο (def_ aβΒΉ aβ))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [Holds] | case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (exists_ aβΒΉ aβ) β Holds D I V E (exists_ (Ο aβΒΉ) (replaceAll ... | case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) aβΒΉ d) E aβ) β
β d, Holds D I (Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
aβΒΉ : VarName
aβ : Formula
a_ihβ : β (V : VarAssignment D), Holds D I (V β Ο) E aβ β Holds D I V E (replaceAll Ο aβ)
V : VarAssignment D
β’ Holds D I (V β Ο) E (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : PredName
xs : List VarName
V : VarAssignment D
β’ I.pred_var_ X (List.map (V β Ο) xs) β I.pred_var_ X (List.map V (List.map Ο xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : PredName
xs : List VarName
V : VarAssignment D
β’ I.pred_var_ X (List.map (V β Ο) xs) β I.pred_var_ X (List.map V (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x y : VarName
V : VarAssignment D
β’ (V β Ο) x = (V β Ο) y β V (Ο x) = V (Ο y)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (replaceAll Ο phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Β¬Holds D I (V β Ο) E phi β Β¬Holds D I V E (r... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | exact phi_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ Holds D I (V β Ο) E phi β Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | congr! 1 | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignment D
β’ (Holds D... | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignm... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | exact phi_ih V | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignm... | 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
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | exact psi_ih V | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E psi β Holds D I V E (replaceAll Ο psi)
V : VarAssignm... | 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
h1 : Function.Injective Ο
phi psi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
psi_ih : β (V : VarAssignment D), Holds D I (V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.update... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.update... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | intro a | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | have s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.updateITE... | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
case h
D : T... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Holds D I (Function... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply Function.updateITE_comp_right_injective | case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
case h
D : T... | case s1.h1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.Injective Ο
case h
D : Type
I : Interpretation D
E : Env
Ο : VarNa... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.updateITE... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply h1 | case s1.h1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.Injective Ο
case h
D : Type
I : Interpretation D
E : Env
Ο : VarNa... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I... | Please generate a tactic in lean4 to solve the state.
STATE:
case s1.h1
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
β’ Function.Inject... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [β s1] | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | exact phi_ih (Function.updateITE V (Ο x) a) | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateITE V (Ο x) a β Ο = Function.updateITE (V β Ο) x a
β’ Holds D I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
a : D
s1 : Function.updateI... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β (d : D), Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β (d : D), Holds D I (Fun... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β (d : D), Holds D I (Function.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply exists_congr | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.updateITE (V β Ο) x d) E phi) β
β d, Holds D I (Function.update... | case h
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ β (a : D),
Holds D I (Function.updateITE (V β Ο) x a) E phi β Holds D I (Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
x : VarName
phi : Formula
phi_ih : β (V : VarAssignment D), Holds D I (V β Ο) E phi β Holds D I V E (replaceAll Ο phi)
V : VarAssignment D
β’ (β d, Holds D I (Function.update... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | induction E | D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs)) | case nil
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
case cons
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) E (def_ X xs) β Holds D I V E (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case nil =>
simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I (V β Ο) [] (def_ X xs) β Holds D I V [] (def_ X (List.map Ο xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [Holds] | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Holds D I (V β Ο) (E_hd :: E_tl) (def_ X xs) β Holds D I V (E_h... | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Hol... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Hol... | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Hol... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | split_ifs | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Hol... | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Hold... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
β’ (i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | case _ c1 =>
apply E_ih | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | cases c1 | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Fu... | case intro
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
leftβ : X = E_hd.name
rightβ : xs.length = E_hd.args.l... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply Holds_coincide_Var | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_l... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | intro v a1 | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [isFreeIn_iff_mem_freeVarSet v E_hd.q] at a1 | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [<- List.mem_toFinset] | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply Finset.mem_of_subset E_hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | simp only [c1_right] | case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.ar... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_theorem | [33, 1] | [96, 19] | apply E_ih | D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
h1 : Function.Injective Ο
X : DefName
xs : List VarName
V : VarAssignment D
E_hd : Definition
E_tl : List Definition
E_ih : Holds D I (V β Ο) E_tl (def_ X xs) β Holds D I V E_tl (def_ X (List.map Ο xs))
c1 :... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | simp only [IsValid] at h2 | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : F.IsValid
β’ (replaceAll Ο F).IsValid | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replaceAll Ο F).IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : F.IsValid
β’ (replaceAll Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | simp only [IsValid] | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replaceAll Ο F).IsValid | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replaceAll Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | intro D I V E | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F)
TACTIC:... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | simp only [β substitution_theorem D I V E F Ο h1] | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F) | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 β Ο) E F | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 (replaceAll Ο F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Inj/ReplaceAll.lean | FOL.NV.Sub.Var.All.Rec.Inj.substitution_is_valid | [99, 1] | [111, 13] | apply h2 | F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 β Ο) E F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : VarName β VarName
h1 : Function.Injective Ο
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 β Ο) E F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | induction E generalizing F binders V V' Ο Ο' | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο Ο' : VarName β VarName
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ Holds D I V E F β Holds D I V' E (fast... | case nil
D : Type
I : Interpretation D
V V' : VarAssignment D
Ο Ο' : VarName β VarName
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β binders, Ο' v = Ο v
β’ Holds D I V [] F β Holds D I V' [] (f... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο Ο' : VarName β VarName
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)
h2' : β v β binders, v = Ο' v
h3 : β v β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | induction F generalizing binders V V' Ο Ο' | case cons
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v ... | case cons.pred_const_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | all_goals
simp only [admitsAux] at h1
simp only [fastReplaceFree]
simp only [Holds] | case cons.pred_const_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β... | case cons.pred_const_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.pred_const_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V V' Ο Ο' binders h1 h2 h2' h3 | D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β bind... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [admitsAux] at h1 | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [fastReplaceFree] | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [Holds] | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binder... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.def_
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | congr! 1 | D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
... | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β bind... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | intro v a1 | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | apply h2 | case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_4.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binde... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | by_cases c1 : v β binders | case a.h.e'_4.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binde... | case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binde... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | left | case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | exact c1 | case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | right | case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [h3 v c1] | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | exact h1 v a1 c1 | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | cases h1 | D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
... | case intro
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β bind... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | congr! 1 | D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v = Ο' v) β
... | case a.h.e'_2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β bind... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | apply h2 | case a.h.e'_2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_2.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binde... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | by_cases c1 : x β binders | case a.h.e'_2.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binde... | case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binde... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | left | case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | exact c1 | case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | right | case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v =... | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | simp only [h3 x c1] | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | exact h1_left c1 | case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders, v... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/Admits.lean | FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux | [74, 1] | [207, 28] | apply h2 | case a.h.e'_3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binders... | case a.h.e'_3.a
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders β¨ Ο' v β binders β V v = V' (Ο' v)) β
(β v β binde... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_3
D : Type
I : Interpretation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V V' : VarAssignment D) (Ο Ο' : VarName β VarName) (binders : Finset VarName) (F : Formula),
admitsAux Ο binders F β
(β (v : VarName), v β binders... |
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