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/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [isBoundIn] at h2
|
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v (h1_P.imp_ h1_Q) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
|
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_
((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))))
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_
((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q'))))
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')).imp_
(((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_
((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))))
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [def_iff_]
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q')).imp_
(((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_
((Forall_ l (U.iff_ V)).imp_ ((h1_P.imp_ h1_Q).iff_ (h1_P'.imp_ h1_Q')))))
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_Q.imp_ h1_Q').and_ (h1_Q'.imp_ h1_Q))).imp_
(((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_
((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_
(((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).and_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q))))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [def_and_]
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_Q.imp_ h1_Q').and_ (h1_Q'.imp_ h1_Q))).imp_
(((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_
((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_
(((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).and_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q))))))
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_Q.imp_ h1_Q').imp_ (h1_Q'.imp_ h1_Q).not_).not_).imp_
(((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_
((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_
((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_
(((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).imp_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)).not_).not_)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
SC
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
(((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_Q.imp_ h1_Q').imp_ (h1_Q'.imp_ h1_Q).not_).not_).imp_
(((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_
((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_
((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_
(((h1_P.imp_ h1_Q).imp_ (h1_P'.imp_ h1_Q')).imp_ ((h1_P'.imp_ h1_Q').imp_ (h1_P.imp_ h1_Q)).not_).not_)))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h1_ih_2
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
intro v a2
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a2 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q
β’ v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h2 v
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a2 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q
β’ v β l
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a2 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
tauto
|
case a.a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a2 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h1_ih_1
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ IsDeduct β
((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
intro v a1
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h2 v
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ v β l
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
cases a1
|
case a
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
case a.intro
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
leftβ : isFreeIn v U β¨ isFreeIn v V
rightβ : isBoundIn v h1_P
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
constructor
|
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q)
|
case left
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isFreeIn v U β¨ isFreeIn v V
case right
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P β¨ isBoundIn v h1_Q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
exact a1_left
|
case left
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isFreeIn v U β¨ isFreeIn v V
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
left
|
case right
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P β¨ isBoundIn v h1_Q
|
case right.h
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
exact a1_right
|
case right.h
U V P_U P_V : Formula
l : List VarName
h1_P h1_Q h1_P' h1_Q' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_2 : IsReplOfFormulaInFormula U V h1_Q h1_Q'
h1_ih_1 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h1_ih_2 :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_Q β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_Q.iff_ h1_Q'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (isBoundIn v h1_P β¨ isBoundIn v h1_Q) β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [isBoundIn] at h2
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v (forall_ h1_x h1_P) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (v = h1_x β¨ isBoundIn v h1_P) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp at h2
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ (v = h1_x β¨ isBoundIn v h1_P) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply deduction_theorem
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
case h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct (β
βͺ {Forall_ l (U.iff_ V)}) ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp
|
case h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct (β
βͺ {Forall_ l (U.iff_ V)}) ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
|
case h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply IsDeduct.mp_ (forall_ h1_x (h1_P.iff_ h1_P'))
|
case h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P'))
|
case h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)}
((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
case h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (forall_ h1_x (h1_P.iff_ h1_P'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply proof_imp_deduct
|
case h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)}
((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
case h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply T_18_1
|
case h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((forall_ h1_x (h1_P.iff_ h1_P')).imp_ ((forall_ h1_x h1_P).iff_ (forall_ h1_x h1_P')))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply generalization
|
case h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (forall_ h1_x (h1_P.iff_ h1_P'))
|
case h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (h1_P.iff_ h1_P')
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ β H β {Forall_ l (U.iff_ V)}, Β¬isFreeIn h1_x H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply IsDeduct.mp_ (Forall_ l (U.iff_ V))
|
case h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (h1_P.iff_ h1_P')
|
case h1.a.h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
case h1.a.h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (Forall_ l (U.iff_ V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply proof_imp_deduct
|
case h1.a.h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h1_ih
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
intro v a1
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
cases a1
|
case h1.a.h1.a.h1
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P
β’ v β l
|
case h1.a.h1.a.h1.intro
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
leftβ : isFreeIn v U β¨ isFreeIn v V
rightβ : isBoundIn v h1_P
β’ v β l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
case _ a1_left a1_right =>
apply h2 v a1_left
right
apply a1_right
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ v β l
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply h2 v a1_left
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ v β l
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ v = h1_x β¨ isBoundIn v h1_P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
right
|
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ v = h1_x β¨ isBoundIn v h1_P
|
case h
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply a1_right
|
case h
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
v : VarName
a1_left : isFreeIn v U β¨ isFreeIn v V
a1_right : isBoundIn v h1_P
β’ isBoundIn v h1_P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
apply IsDeduct.assume_
|
case h1.a.h1.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ IsDeduct {Forall_ l (U.iff_ V)} (Forall_ l (U.iff_ V))
|
case h1.a.h1.a.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Forall_ l (U.iff_ V) β {Forall_ l (U.iff_ V)}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp
|
case h1.a.h1.a.a
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Forall_ l (U.iff_ V) β {Forall_ l (U.iff_ V)}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
intro H a1
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ β H β {Forall_ l (U.iff_ V)}, Β¬isFreeIn h1_x H
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
H : Formula
a1 : H β {Forall_ l (U.iff_ V)}
β’ Β¬isFreeIn h1_x H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp at a1
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
H : Formula
a1 : H β {Forall_ l (U.iff_ V)}
β’ Β¬isFreeIn h1_x H
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
H : Formula
a1 : H = Forall_ l (U.iff_ V)
β’ Β¬isFreeIn h1_x H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
subst a1
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
H : Formula
a1 : H = Forall_ l (U.iff_ V)
β’ Β¬isFreeIn h1_x H
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬isFreeIn h1_x (Forall_ l (U.iff_ V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [Forall_isFreeIn]
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬isFreeIn h1_x (Forall_ l (U.iff_ V))
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x (U.iff_ V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [def_iff_]
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x (U.iff_ V))
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x ((U.imp_ V).and_ (V.imp_ U)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [def_and_]
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x ((U.imp_ V).and_ (V.imp_ U)))
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x ((U.imp_ V).imp_ (V.imp_ U).not_).not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
simp only [isFreeIn]
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ isFreeIn h1_x ((U.imp_ V).imp_ (V.imp_ U).not_).not_)
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ ((isFreeIn h1_x U β¨ isFreeIn h1_x V) β¨ isFreeIn h1_x V β¨ isFreeIn h1_x U))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
sorry
|
case h1.a.h2
U V P_U P_V : Formula
l : List VarName
h1_x : VarName
h1_P h1_P' : Formula
h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P'
h1_ih :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v h1_P β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
h2 : β (v : VarName), isFreeIn v U β¨ isFreeIn v V β v = h1_x β¨ isBoundIn v h1_P β v β l
β’ Β¬(h1_x β l β§ ((isFreeIn h1_x U β¨ isFreeIn h1_x V) β¨ isFreeIn h1_x V β¨ isFreeIn h1_x U))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_2
|
[802, 1]
|
[886, 10]
|
sorry
|
case exists_
U V P_U P_V : Formula
l : List VarName
xβ : VarName
P_uβ P_vβ : Formula
aβ : IsReplOfFormulaInFormula U V P_uβ P_vβ
a_ihβ :
(β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_uβ β v β l) β
IsProof ((Forall_ l (U.iff_ V)).imp_ (P_uβ.iff_ P_vβ))
h2 : β (v : VarName), (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v (exists_ xβ P_uβ) β v β l
β’ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ xβ P_uβ).iff_ (exists_ xβ P_vβ)))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
apply
IsDeduct.mp_
(Forall_ ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList (U.iff_ V))
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsProof (P_U.iff_ P_V)
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
((Forall_ ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList (U.iff_ V)).imp_ (P_U.iff_ P_V))
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(Forall_ ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList (U.iff_ V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
apply T_18_2 U V P_U P_V ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList h1
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
((Forall_ ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList (U.iff_ V)).imp_ (P_U.iff_ P_V))
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ β (v : VarName),
(isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U β v β ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
intro v a1
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ β (v : VarName),
(isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U β v β ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U
β’ v β ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U
β’ v β ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp only [isFreeIn_iff_mem_freeVarSet] at a1
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (isFreeIn v U β¨ isFreeIn v V) β§ isBoundIn v P_U
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (v β U.freeVarSet β¨ v β V.freeVarSet) β§ isBoundIn v P_U
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp only [isBoundIn_iff_mem_boundVarSet] at a1
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (v β U.freeVarSet β¨ v β V.freeVarSet) β§ isBoundIn v P_U
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
exact a1
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
v : VarName
a1 : (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
β’ (v β U.freeVarSet β¨ v β V.freeVarSet) β§ v β P_U.boundVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp only [Formula.Forall_]
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(Forall_ ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList (U.iff_ V))
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
induction ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList
|
case a
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) ((U.freeVarSet βͺ V.freeVarSet) β© P_U.boundVarSet).toList)
|
case a.nil
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) [])
case a.cons
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
headβ : VarName
tailβ : List VarName
tail_ihβ : IsDeduct β
(List.foldr forall_ (U.iff_ V) tailβ)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) (headβ :: tailβ))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
case _ =>
simp
exact h2
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) [])
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) [])
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(U.iff_ V)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
exact h2
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
β’ IsDeduct β
(U.iff_ V)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) (hd :: tl))
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ IsDeduct β
(forall_ hd (List.foldr forall_ (U.iff_ V) tl))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
apply generalization
|
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ IsDeduct β
(forall_ hd (List.foldr forall_ (U.iff_ V) tl))
|
case h1
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
case h2
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ β H β β
, Β¬isFreeIn hd H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
exact ih
|
case h1
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_3
|
[889, 1]
|
[913, 13]
|
simp
|
case h2
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct β
(List.foldr forall_ (U.iff_ V) tl)
β’ β H β β
, Β¬isFreeIn hd H
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
apply IsDeduct.mp_ P_U
|
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ P_V
|
case a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (P_U.imp_ P_V)
case a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ P_U
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
apply IsDeduct.mp_ (P_U.iff_ P_V)
|
case a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (P_U.imp_ P_V)
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ ((P_U.iff_ P_V).imp_ (P_U.imp_ P_V))
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (P_U.iff_ P_V)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
simp only [def_iff_]
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ ((P_U.iff_ P_V).imp_ (P_U.imp_ P_V))
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (((P_U.imp_ P_V).and_ (P_V.imp_ P_U)).imp_ (P_U.imp_ P_V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
simp only [def_and_]
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (((P_U.imp_ P_V).and_ (P_V.imp_ P_U)).imp_ (P_U.imp_ P_V))
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (((P_U.imp_ P_V).imp_ (P_V.imp_ P_U).not_).not_.imp_ (P_U.imp_ P_V))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
SC
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (((P_U.imp_ P_V).imp_ (P_V.imp_ P_U).not_).not_.imp_ (P_U.imp_ P_V))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
apply proof_imp_deduct
|
case a.a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ (P_U.iff_ P_V)
|
case a.a.h1
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsProof (P_U.iff_ P_V)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
exact C_18_3 U V P_U P_V h1 h2
|
case a.a.h1
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsProof (P_U.iff_ P_V)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.C_18_4
|
[917, 1]
|
[935, 13]
|
exact h3
|
case a
U V P_U P_V : Formula
Ξ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Ξ P_U
β’ IsDeduct Ξ P_U
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_exists_]
|
P : Formula
v : VarName
β’ IsProof ((forall_ v P).iff_ (exists_ v P.not_).not_)
|
P : Formula
v : VarName
β’ IsProof ((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply C_18_4 P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_)
|
P : Formula
v : VarName
β’ IsProof ((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_)
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_)
((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_)
case h2
P : Formula
v : VarName
β’ IsProof (P.iff_ P.not_.not_)
case h3
P : Formula
v : VarName
β’ IsDeduct β
((forall_ v P).iff_ (forall_ v P).not_.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_iff_]
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_)
((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_)
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).and_ ((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_and_]
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).and_ ((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)))
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_).not_
|
case h1.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_)
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.imp_
|
case h1.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_)
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_)
|
case h1.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).imp_ (forall_ v P).not_.not_)
((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_)
case h1.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.imp_
|
case h1.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).imp_ (forall_ v P).not_.not_)
((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_)
|
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P)
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.same_
|
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P)
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ forall_ v P = forall_ v P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ forall_ v P = forall_ v P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.forall_
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_)
|
case h1.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.diff_
|
case h1.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_
|
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P = P
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P.not_.not_ = P.not_.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P = P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P.not_.not_ = P.not_.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_
|
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P))
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.imp_
|
case h1.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P))
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P))
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.not_
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_
|
case h1.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.forall_
|
case h1.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_)
|
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.diff_
|
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_
|
case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P = P
case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P.not_.not_ = P.not_.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P = P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
β’ P.not_.not_ = P.not_.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
apply IsReplOfFormulaInFormula.same_
|
case h1.a.a.a.a
P : Formula
v : VarName
β’ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P)
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ forall_ v P = forall_ v P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
rfl
|
case h1.a.a.a.a.a
P : Formula
v : VarName
β’ forall_ v P = forall_ v P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_iff_]
|
case h2
P : Formula
v : VarName
β’ IsProof (P.iff_ P.not_.not_)
|
case h2
P : Formula
v : VarName
β’ IsProof ((P.imp_ P.not_.not_).and_ (P.not_.not_.imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_and_]
|
case h2
P : Formula
v : VarName
β’ IsProof ((P.imp_ P.not_.not_).and_ (P.not_.not_.imp_ P))
|
case h2
P : Formula
v : VarName
β’ IsProof ((P.imp_ P.not_.not_).imp_ (P.not_.not_.imp_ P).not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
SC
|
case h2
P : Formula
v : VarName
β’ IsProof ((P.imp_ P.not_.not_).imp_ (P.not_.not_.imp_ P).not_).not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_iff_]
|
case h3
P : Formula
v : VarName
β’ IsDeduct β
((forall_ v P).iff_ (forall_ v P).not_.not_)
|
case h3
P : Formula
v : VarName
β’ IsDeduct β
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_5
|
[938, 1]
|
[980, 7]
|
simp only [def_and_]
|
case h3
P : Formula
v : VarName
β’ IsDeduct β
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
|
case h3
P : Formula
v : VarName
β’ IsDeduct β
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
|
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