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 : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
β’ β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H
|
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 : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
β’ Holds D I V E H'
|
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 : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
β’ Holds D I V E H'
|
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 : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
β’ Holds D I V E H'
|
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 : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
β’ Holds D I V E H'
|
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
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
β’ H' β Ξ_1 β¨ H' = H_2 β¨ H' = H_1 β¨ H' β Ξ_2
|
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
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
β’ H' β Ξ_1 β¨ H' = H_2 β¨ H' = H_1 β¨ H' β Ξ_2
|
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
V : VarAssignment D
E : Env
H' : Formula
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
a2 : H' β Ξ_1 β¨ H' = H_1 β¨ H' = H_2 β¨ H' β Ξ_2
β’ H' β Ξ_1 β¨ H' = H_2 β¨ H' = H_1 β¨ H' β Ξ_2
|
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
V : VarAssignment D
E : Env
H' : Formula
a1 : β (H : Formula), H β Ξ_1 β¨ H = H_2 β¨ H = H_1 β¨ H β Ξ_2 β Holds D I V E H
a2 : H' β Ξ_1 β¨ H' = H_1 β¨ H' = H_2 β¨ H' β Ξ_2
β’ H' β Ξ_1 β¨ H' = H_2 β¨ H' = H_1 β¨ H' β Ξ_2
|
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 : 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) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E psi
|
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 : 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', Holds D I V E H
β’ Holds D I V E 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] 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 : 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', Holds D I V E H
β’ Holds D I V E psi
|
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', Holds D I V E H
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 β Holds D I V E psi
β’ Holds D I V E psi
|
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', Holds D I V E H
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 β Holds D I V E psi
β’ Holds D I V E psi
|
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 H β Ξ', Holds D I V E H
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 β Holds D I V E psi
β’ β H β Ξ', Holds D I V E H
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 H β Ξ', Holds D I V E H
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 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]
|
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 H β Ξ', Holds D I V E H
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 β Holds D I V E psi
β’ β H β Ξ', Holds D I V E H
|
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 H β Ξ', Holds D I V E H
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 β Holds D I V E psi
H' : Formula
a2 : H' β Ξ'
β’ Holds D I V E H'
|
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 H β Ξ', Holds D I V E H
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 β Holds D I V E psi
H' : Formula
a2 : H' β Ξ'
β’ Holds D I V E H'
|
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 H β Ξ', Holds D I V E H
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 β Holds D I V E psi
β’ Holds D I V E phi
|
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 : β H β Ξ', Holds D I V E H
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 β Holds D I V E psi
β’ β H β Ξ', Holds D I V E H
|
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 : β H β Ξ', Holds D I V E H
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 β Holds D I V E psi
β’ β H β Ξ', Holds D I V E 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 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', Holds D I V E H) β Holds D I V E (H.imp_ phi)
|
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’ Holds D I V E (H.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 : 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’ Holds D I V E (H.imp_ phi)
|
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’ 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]
|
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’ Holds D I V E H β Holds D I V E phi
|
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
a2 : 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
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
a2 : Holds D I V E H
β’ Holds D I V E phi
|
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 V E H
a2 : Holds D I V E H
β’ β H_1 β H :: Ξ', Holds D I V E H_1
|
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 V E H
a2 : Holds D I V E H
β’ β H_1 β H :: Ξ', Holds D I V E H_1
|
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 V E H
a2 : Holds D I V E H
β’ Holds D I V E H β§ β a β Ξ', Holds D I V E a
|
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 V E H
a2 : Holds D I V E H
β’ Holds D I V E H β§ β a β Ξ', Holds D I V E a
|
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 D I V E H
a2 : Holds D I V E H
β’ Holds D I V E H
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', Holds D I V E H
a2 : Holds D I V E H
β’ β a β Ξ', Holds D I V E a
|
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 D I V E H
a2 : Holds D I V E H
β’ Holds D I V E H
|
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', Holds D I V E H
a2 : Holds D I V E H
β’ β a β Ξ', Holds D I V E a
|
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 : D), Holds D I (Function.updateITE V v d) E psi
|
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 : D), Holds D I (Function.updateITE V v d) E 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
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’ Holds D I (Function.updateITE V v d) E psi
|
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’ Holds D I (Function.updateITE V v d) E 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
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’ 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]
|
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’ Holds D I (Function.updateITE V v d) 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
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 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]
|
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 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’ (β (d : D), Holds D I (Function.updateITE V v d) E phi) β
Holds D I V E (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]
|
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 (Function.updateITE V v d) E phi) β
Holds D I V E (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
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi
β’ Holds D I V E (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]
|
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 (Function.updateITE V v d) E phi
β’ Holds D I V E (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
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi
β’ Holds D I (V β Function.updateITE id v t) 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 [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 (Function.updateITE V v d) E phi
β’ Holds D I (V β Function.updateITE id v t) E 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
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi
β’ Holds D I (Function.updateITE (V β id) v (V t)) E phi
|
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 (Function.updateITE V v d) E phi
β’ Holds D I (Function.updateITE (V β id) v (V t)) E 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
a2 : β (d : D), Holds D I (Function.updateITE V v d) E phi
β’ Holds D I (Function.updateITE V v (V t)) 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 (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 (Function.updateITE V v d) E phi
β’ Holds D I (Function.updateITE V v (V t)) 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
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 : 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]
|
{
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 : 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 (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 : 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 : Β¬v' = v
β’ 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]
|
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 (forall_ v phi)
|
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 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
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 phi)
|
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 (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 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 (Function.updateITE V v d) E phi
|
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.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]
|
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.updateITE V v d) E phi
|
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 [], Holds D I (Function.updateITE V v d) E H
|
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 [], Holds D I (Function.updateITE V v d) E 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 : 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 : 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
headβ : VarName
tailβ : List VarName
tail_ihβ :
β (ys : List VarName),
tailβ.length = ys.length β
Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ tailβ ys)) β List.map V tailβ = List.map V ys
ys : List VarName
ih_1 : (headβ :: tailβ).length = ys.length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (headβ :: tailβ) ys))
β’ List.map V (headβ :: tailβ) = 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]
|
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 : PredName
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
aβ : β H β [], Holds D I V E H
headβ : VarName
tailβ : List VarName
ih_1 : [].length = (headβ :: tailβ).length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (headβ :: tailβ)))
β’ List.map V [] = List.map V (headβ :: tailβ)
|
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 V (ys_hd :: ys_tl)
|
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 V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys : List VarName
ih_1 : (xs_hd :: xs_tl).length = ys.length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) ys))
β’ List.map V (xs_hd :: xs_tl) = 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
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_tl = List.map V ys
ih_1 : (xs_hd :: xs_tl).length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) []))
β’ List.map V (xs_hd :: xs_tl) = List.map V []
case cons
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_tl = List.map V ys
headβ : VarName
tailβ : List VarName
ih_1 : (xs_hd :: xs_tl).length = (headβ :: tailβ).length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (headβ :: tailβ)))
β’ List.map V (xs_hd :: xs_tl) = List.map V (headβ :: tailβ)
|
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_tl = List.map V ys
ih_1 : (xs_hd :: xs_tl).length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) []))
β’ List.map V (xs_hd :: xs_tl) = 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
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_tl = List.map V ys
ih_1 : (xs_hd :: xs_tl).length = [].length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) []))
β’ List.map V (xs_hd :: xs_tl) = 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
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl)))
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl)))
ih_1 : xs_tl.length = ys_tl.length
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl)))
ih_1 : xs_tl.length = ys_tl.length
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
a1 : Holds D I V E ((eq_ xs_hd ys_hd).and_ (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)))
ih_1 : xs_tl.length = ys_tl.length
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
a1 : Holds D I V E ((eq_ xs_hd ys_hd).and_ (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)))
ih_1 : xs_tl.length = ys_tl.length
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1 : V xs_hd = V ys_hd β§ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1 : V xs_hd = V ys_hd β§ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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 List.map V xs_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
leftβ : V xs_hd = V ys_hd
rightβ : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ V xs_hd = V ys_hd β§ List.map V xs_tl = List.map V ys_tl
|
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_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ V xs_hd = V ys_hd β§ List.map V xs_tl = List.map V ys_tl
|
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.map V xs_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ V xs_hd = V ys_hd
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 List.map V xs_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V xs_tl = List.map V ys_tl
|
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.map V xs_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ V xs_hd = V ys_hd
|
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 List.map V xs_tl = List.map V ys
ys_hd : VarName
ys_tl : List VarName
ih_1 : xs_tl.length = ys_tl.length
a1_left : V xs_hd = V ys_hd
a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))
β’ List.map V xs_tl = List.map V ys_tl
|
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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