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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V... | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V... | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply a1 | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V... | case a.a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a2 | case a.a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D... | case a.a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | tauto | case a.a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D... | no goals |
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 phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [phi], Holds D I V E H) β Holds D I V E phi | Ξ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β [phi], 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 phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β [phi], Holds D I V E H
β’ Holds D I V E phi | Ξ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : Holds D I V E phi
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 | Ξ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : Holds D I V E phi
β’ Holds D I V E phi | no goals |
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
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 true_ | 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 true_ |
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 true_ | no goals |
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.imp_ (psi.imp_ phi)) | 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.imp_ (psi.imp_ phi)) |
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.imp_ (psi.imp_ phi)) | 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 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] | 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 Holds D I V E phi | no goals |
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 chi : 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.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | Ξ : List Formula
F phi psi chi : 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.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) |
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 chi : 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.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | Ξ : List Formula
F phi psi chi : 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 Holds D I V E chi) β
(Holds D I V E phi β Holds D I V E psi) β Holds D I V E phi β Holds D I V E chi |
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 chi : 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 Holds D I V E chi) β
(Holds D I V E phi β Holds D I V E psi) β Holds D I V E phi β Holds D I V E chi | no goals |
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.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | 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.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) |
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.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | 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 Β¬Holds D I V E psi) β Holds D I V E psi 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] | 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 Β¬Holds D I V E psi) β Holds D I V E psi β Holds D I V E phi | no goals |
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 psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_major :
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.imp_ psi)
ih_minor :
β (D : Type) (I : Interpretation D) (V : Va... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_major :
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.imp_ psi)
ih_minor :
β (D : Type) (I : Interpretation D) (V : Va... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] at ih_major | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_major :
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.imp_ psi)
ih_minor :
β (D : Type) (I : Interpretation D) (V : Va... | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β H β Ξ'... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_major | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β H β Ξ'... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1 H' a2 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_minor | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 : β... | case a.a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 :... |
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.a
Ξ : List Formula
F : Formula
Ξ' : List Formula
phi psi : Formula
aβΒΉ : IsDeduct Ξ' (phi.imp_ psi)
aβ : IsDeduct Ξ' phi
ih_minor :
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 :... | no goals |
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
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ', Hold... | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
β’... |
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
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
β’... | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
β’... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
β’... | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
a... |
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
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I V E H
a... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I ... | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | constructor | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds D I ... | case a.left
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a2 | case a.left
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Holds... | no goals |
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.right
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phi : Formula
aβ : IsDeduct (H :: Ξ') phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ', Hold... | no goals |
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 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 ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | Ξ : List Formula
F : Formula
v : VarName
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 ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) |
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 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 ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | Ξ : List Formula
F : Formula
v : VarName
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 (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi) β
(β (d : D), Holds D I (Function.updateITE V v d) E phi) β β (d ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 a3 d | Ξ : List Formula
F : Formula
v : VarName
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 (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi) β
(β (d : D), Holds D I (Function.updateITE V v d) E phi) β β (d ... | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi
a3 : β (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
β’ H... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply a2 d | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi
a3 : β (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
β’ H... | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi
a3 : β (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
β’ H... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a3 d | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi β Holds D I (Function.updateITE V v d) E psi
a3 : β (d : D), Holds D I (Function.updateITE V v d) E phi
d : D
β’ H... | no goals |
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 t : 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 ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | Ξ : List Formula
F : Formula
v t : 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 ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi | Ξ : List Formula
F : Formula
v t : 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 ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
β’ Holds D I V E ((forall_ v p... |
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 t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
β’ Holds D I V E ((forall_ v p... | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
β’ (β (d : D), Holds D I (Func... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
β’ (β (d : D), Holds D I (Func... | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [s1] | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Function.updateITE_comp_left] | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... |
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 t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a2 (V t) | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
s1 :
Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) β
Holds D I (V β Function.updateITE id v t) E phi
a2 : β (d : D), Holds D I (Fu... | no goals |
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
ih : Β¬isFreeIn v phi
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.imp_ (forall_ v phi)) | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
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.imp_ (forall_ v phi)) |
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
ih : Β¬isFreeIn v phi
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.imp_ (forall_ v phi)) | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
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 β (d : D), Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 d | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
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 β (d : D), Holds D I (Function.updateITE V v d) E phi | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ Holds D I (Function.updateITE V v d) E phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | have s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ Holds D I (Function.updateITE V v d) E phi | case s1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
Ξ : List Formula
F : Formula
v : VarName
phi : Formu... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | {
apply Holds_coincide_Var
intro v' a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
subst c1
contradiction
case _ c1 =>
rfl
} | case s1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
Ξ : List Formula
F : Formula
v : VarName
phi : Formu... | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
β’ Holds D I (Function.updateITE V v d) 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 [s1] | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
β’ Holds D I (Function.updateITE V v d) E phi | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
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 a2 | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
β’ Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply Holds_coincide_Var | case s1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ β (v_1 : VarName), isFreeIn v_1 phi β Function.updateITE V v d v_1 = V v_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro v' a1 | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
β’ β (v_1 : VarName), isFreeIn v_1 phi β Function.updateITE V v d v_1 = V v_1 | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
β’ Function.updateITE V v d v' = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Function.updateITE] | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
β’ Function.updateITE V v d v' = V v' | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
β’ (if v' = v then d else V v') = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | split_ifs | case s1.h1
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
β’ (if v' = v then d else V v') = V v' | case pos
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
hβ : v' = v
β’ d = V v'
case neg
Ξ : List Formula
F : Formula
v : VarName
phi : For... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ c1 =>
subst c1
contradiction | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : v' = v
β’ d = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ c1 =>
rfl | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : Β¬v' = v
β’ V v' = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | subst c1 | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : v' = v
β’ d = V v' | Ξ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
ih : Β¬isFreeIn v' phi
β’ d = V v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | contradiction | Ξ : List Formula
F phi : Formula
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
ih : Β¬isFreeIn v' phi
β’ d = V v' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | rfl | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : Holds D I V E phi
d : D
v' : VarName
a1 : isFreeIn v' phi
c1 : Β¬v' = v
β’ V v' = V v' | no goals |
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
aβ : IsDeduct [] phi
ih_2 :
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
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... | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
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 (forall_ v p... |
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
aβΒΉ : IsDeduct [] phi
ih_2 :
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
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 (forall_ v p... | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ β (d : D), Holds D I (Func... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro d | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ β (d : D), Holds D I (Func... | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
d : D
β’ Holds D I (Function.... |
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
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
d : D
β’ Holds D I (Function.... | case a
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
d : D
β’ β H β [], Hol... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp | case a
Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβΒΉ : IsDeduct [] phi
ih_2 :
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
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
d : D
β’ β H β [], Hol... | no goals |
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
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 (eq_ v v) | Ξ : List Formula
F : Formula
v : VarName
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 (eq_ v v) |
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
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 (eq_ v v) | no goals |
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
x_0 x_1 y_0 y_1 : VarName
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 (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
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 (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) |
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
x_0 x_1 y_0 y_1 : VarName
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 (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ V x_0 = V y_0 β§ V x_1 = V y_1 β (V x_0 = V x_1 β V y_0 = V y_1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a2 | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
β’ V x_0 = V y_0 β§ V x_1 = V y_1 β (V x_0 = V x_1 β V y_0 = V y_1) | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : V x_0 = V y_0 β§ V x_1 = V y_1
β’ V x_0 = V x_1 β V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a2 | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2 : V x_0 = V y_0 β§ V x_1 = V y_1
β’ V x_0 = V x_1 β V y_0 = V y_1 | case intro
Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
leftβ : V x_0 = V y_0
rightβ : V x_1 = V y_1
β’ V x_0 = V x_1 β V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2_left a2_right =>
simp only [a2_left]
simp only [a2_right] | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
β’ V x_0 = V x_1 β V y_0 = V y_1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2_left] | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
β’ V x_0 = V x_1 β V y_0 = V y_1 | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
β’ V y_0 = V x_1 β V y_0 = V y_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2_right] | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a2_left : V x_0 = V y_0
a2_right : V x_1 = V y_1
β’ V y_0 = V x_1 β V y_0 = V y_1 | no goals |
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
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
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 ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) | Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
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 ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) |
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
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
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 ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys))) | Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
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 (List.foldr and_ true_ (List.zipWith eq_ xs ys)) β
(I.pred_var_ X (List.map V xs) β I.pred_var_ X (List.map V ys)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro a1 | Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
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 (List.foldr and_ true_ (List.zipWith eq_ xs ys)) β
(I.pred_var_ X (List.map V xs) β I.pred_var_ X (List.map V ys)) | Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys))
β’ I.pred_var_ X (List.map V xs) β I.pred_var_ X (List.map V ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | congr! 1 | Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys))
β’ I.pred_var_ X (List.map V xs) β I.pred_var_ X (List.map V ys) | case a.h.e'_4
Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys))
β’ List.map V xs = List.map V ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | induction xs generalizing ys | case a.h.e'_4
Ξ : List Formula
F : Formula
X : PredName
xs ys : List VarName
ih_1 : xs.length = ys.length
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys))
β’ List.map V xs = List.map V ys | case a.h.e'_4.nil
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ys : List VarName
ih_1 : [].length = ys.length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys))
β’ List.map V [] = List.map V ys
case a.h.e'_4.cons
Ξ : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ =>
cases ys
case _ =>
simp only [Holds]
case cons ys_hd ys_tl =>
simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ys : List VarName
ih_1 : [].length = ys.length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys))
β’ List.map V [] = List.map V ys | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases ys | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ys : List VarName
ih_1 : [].length = ys.length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys))
β’ List.map V [] = List.map V ys | case nil
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ih_1 : [].length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] []))
β’ List.map V [] = List.map V []
case cons
Ξ : List Formula
F : Formula
X : PredNam... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ =>
simp only [Holds] | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ih_1 : [].length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] []))
β’ List.map V [] = List.map V [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case cons ys_hd ys_tl =>
simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ys_hd : VarName
ys_tl : List VarName
ih_1 : [].length = (ys_hd :: ys_tl).length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (ys_hd :: ys_tl)))
β’ List.map V [] = List.map ... | no goals |
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
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ih_1 : [].length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] []))
β’ List.map V [] = List.map V [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
ys_hd : VarName
ys_tl : List VarName
ih_1 : [].length = (ys_hd :: ys_tl).length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (ys_hd :: ys_tl)))
β’ List.map V [] = List.map ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases ys | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | case nil
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case nil =>
simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at ih_1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... |
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
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [Holds] at a1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a1 | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | case intro
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β Lis... |
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
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | constructor | Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List.map V xs_... | case left
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_left | case left
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β List... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply xs_ih ys_tl ih_1 a1_right | case right
Ξ : List Formula
F : Formula
X : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
xs_hd : VarName
xs_tl : List VarName
xs_ih :
β (ys : List VarName),
xs_tl.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) β Lis... | no goals |
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
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 (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’ Holds D I V E (false_.iff_ true_.not_) |
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