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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q)))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q)))))
case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_exists_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))).imp_
((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.iff_ (forall_ v Q.not_).not_)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P.not_).not_.iff_ (forall_ v Q.not_).not_)))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).and_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).and_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | SC | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_)).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_)).imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_).imp_
((forall_ v Q.not_).not_.imp_ (forall_ v P.not_).not_).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply T_19_6_right | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6 | [1265, 1] | [1280, 22] | apply T_19_6_left | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply C_18_4 (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q)))
case h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P)
case h3
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.imp_ | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v (P.imp_ Q)) (forall_ v (P.imp_ Q))
case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v P).imp_ (forall_ v Q)) (P.imp_ (forall_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.same_ | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v (P.imp_ Q)) (forall_ v (P.imp_ Q)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v (P.imp_ Q) = forall_ v (P.imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v (P.imp_ Q) = forall_ v (P.imp_ Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.imp_ | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P ((forall_ v P).imp_ (forall_ v Q)) (P.imp_ (forall_ v Q)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v P) P
case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v Q) (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.diff_ | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v P) P | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v P = forall_ v P
case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P = P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v P = forall_ v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsReplOfFormulaInFormula.same_ | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsReplOfFormulaInFormula (forall_ v P) P (forall_ v Q) (forall_ v Q) | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v Q = forall_ v Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | rfl | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ forall_ v Q = forall_ v Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | exact T_19_1 P v h1 | case h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsDeduct.axiom_ | case h3
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | case h3.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_left | [1283, 1] | [1301, 26] | apply IsAxiom.pred_1_ | case h3.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v (P.imp_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply deduction_theorem | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct (∅ ∪ {P.imp_ (forall_ v Q)}) (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct (∅ ∪ {P.imp_ (forall_ v Q)}) (forall_ v (P.imp_ Q)) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply generalization | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (forall_ v (P.imp_ Q)) | case h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (P.imp_ Q)
case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ∀ H ∈ {P.imp_ (forall_ v Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply deduction_theorem | case h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct {P.imp_ (forall_ v Q)} (P.imp_ Q) | case h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply specId v | case h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) Q | case h1.h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.mp_ P | case h1.h1.h1.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (forall_ v Q) | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (P.imp_ (forall_ v Q))
case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.assume_ | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) (P.imp_ (forall_ v Q)) | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P.imp_ (forall_ v Q) ∈ {P.imp_ (forall_ v Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P.imp_ (forall_ v Q) ∈ {P.imp_ (forall_ v Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | apply IsDeduct.assume_ | case h1.h1.h1.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ({P.imp_ (forall_ v Q)} ∪ {P}) P | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P ∈ {P.imp_ (forall_ v Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h1.h1.h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ P ∈ {P.imp_ (forall_ v Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | intro H a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ∀ H ∈ {P.imp_ (forall_ v Q)}, ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H ∈ {P.imp_ (forall_ v Q)}
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp at a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H ∈ {P.imp_ (forall_ v Q)}
⊢ ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H = P.imp_ (forall_ v Q)
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | subst a1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
H : Formula
a1 : H = P.imp_ (forall_ v Q)
⊢ ¬isFreeIn v H | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v (P.imp_ (forall_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp only [isFreeIn] | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v (P.imp_ (forall_ v Q)) | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬(isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | simp | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬(isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21_right | [1304, 1] | [1325, 13] | exact h1 | case h1.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ ¬isFreeIn v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | apply IsDeduct.mp_ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))))
case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | apply IsDeduct.mp_ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q)))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q)))))
case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_ ((forall_ v (P.imp_ Q)).iff_ (P.imp_ (forall_ v Q))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).and_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).and_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q)))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | SC | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
(((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))).imp_
((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | exact T_19_TS_21_right P Q v h1 | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v Q)).imp_ (forall_ v (P.imp_ Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_TS_21 | [1328, 1] | [1342, 35] | exact T_19_TS_21_left P Q v h1 | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.imp_ Q)).imp_ (P.imp_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply generalization | x y : VarName
⊢ IsProof (forall_ x (forall_ y ((eq_ x y).imp_ (eq_ y x)))) | case h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y ((eq_ x y).imp_ (eq_ y x)))
case h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply generalization | case h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y ((eq_ x y).imp_ (eq_ y x))) | case h1.h1
x y : VarName
⊢ IsDeduct ∅ ((eq_ x y).imp_ (eq_ y x))
case h1.h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply IsDeduct.mp_ (eq_ y y) | case h1.h1
x y : VarName
⊢ IsDeduct ∅ ((eq_ x y).imp_ (eq_ y x)) | case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))
case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ (eq_ y y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply IsDeduct.mp_ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))) | case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))) | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | simp only [def_iff_] | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).and_ (eq_ x y)).imp_ (((eq_ y x).imp_ (eq_ y y)).and_ ((eq_ y y).imp_ (eq_ y x)))).imp_
((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | simp only [def_and_] | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).and_ (eq_ x y)).imp_ (((eq_ y x).imp_ (eq_ y y)).and_ ((eq_ y y).imp_ (eq_ y x)))).imp_
((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).imp_ (eq_ x y).not_).not_.imp_
(((eq_ y x).imp_ (eq_ y y)).imp_ ((eq_ y y).imp_ (eq_ y x)).not_).not_).imp_
((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | SC | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅
((((eq_ y y).imp_ (eq_ x y).not_).not_.imp_
(((eq_ y x).imp_ (eq_ y y)).imp_ ((eq_ y y).imp_ (eq_ y x)).not_).not_).imp_
((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply specId y | case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))) | case h1.h1.a.a.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply specId y | case h1.h1.a.a.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))) | case h1.h1.a.a.h1.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply specId x | case h1.h1.a.a.h1.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))) | case h1.h1.a.a.h1.h1.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply specId y | case h1.h1.a.a.h1.h1.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))) | case h1.h1.a.a.h1.h1.h1.h1
x y : VarName
⊢ IsDeduct ∅
(forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply IsDeduct.axiom_ | case h1.h1.a.a.h1.h1.h1.h1
x y : VarName
⊢ IsDeduct ∅
(forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))) | case h1.h1.a.a.h1.h1.h1.h1.a
x y : VarName
⊢ IsAxiom (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | exact IsAxiom.eq_2_eq_ y x y y | case h1.h1.a.a.h1.h1.h1.h1.a
x y : VarName
⊢ IsAxiom (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply specId y | case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ (eq_ y y) | case h1.h1.a.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (eq_ y y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | apply IsDeduct.axiom_ | case h1.h1.a.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (eq_ y y)) | case h1.h1.a.h1.a
x y : VarName
⊢ IsAxiom (forall_ y (eq_ y y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | exact IsAxiom.eq_1_ y | case h1.h1.a.h1.a
x y : VarName
⊢ IsAxiom (forall_ y (eq_ y y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | intro H a1 | case h1.h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H | case h1.h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | simp at a1 | case h1.h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | intro H a1 | case h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H | case h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_1 | [1345, 1] | [1370, 15] | simp at a1 | case h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply generalization | x y z : VarName
⊢ IsProof (forall_ x (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))) | case h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
case h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply generalization | case h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) | case h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))
case h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply generalization | case h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))) | case h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))
case h1.h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn z H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply IsDeduct.mp_ (eq_ z z) | case h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)) | case h1.h1.h1.a
x y z : VarName
⊢ IsDeduct ∅ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))
case h1.h1.h1.a
x y z : VarName
⊢ IsDeduct ∅ (eq_ z z) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply IsDeduct.mp_ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))) | case h1.h1.h1.a
x y z : VarName
⊢ IsDeduct ∅ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))) | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))).imp_
((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | simp only [def_iff_] | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))).imp_
((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).and_ (eq_ z z)).imp_ (((eq_ x z).imp_ (eq_ y z)).and_ ((eq_ y z).imp_ (eq_ x z)))).imp_
((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | simp only [def_and_] | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).and_ (eq_ z z)).imp_ (((eq_ x z).imp_ (eq_ y z)).and_ ((eq_ y z).imp_ (eq_ x z)))).imp_
((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).imp_ (eq_ z z).not_).not_.imp_
(((eq_ x z).imp_ (eq_ y z)).imp_ ((eq_ y z).imp_ (eq_ x z)).not_).not_).imp_
((eq_ z z).imp_ (((eq_ x y).imp_ (eq_ y z).not_).not_.imp_ (eq_ x z)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | SC | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅
((((eq_ x y).imp_ (eq_ z z).not_).not_.imp_
(((eq_ x z).imp_ (eq_ y z)).imp_ ((eq_ y z).imp_ (eq_ x z)).not_).not_).imp_
((eq_ z z).imp_ (((eq_ x y).imp_ (eq_ y z).not_).not_.imp_ (eq_ x z)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply specId z | case h1.h1.h1.a.a
x y z : VarName
⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))) | case h1.h1.h1.a.a.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply specId y | case h1.h1.h1.a.a.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))) | case h1.h1.h1.a.a.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply specId z | case h1.h1.h1.a.a.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))) | case h1.h1.h1.a.a.h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply specId x | case h1.h1.h1.a.a.h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))) | case h1.h1.h1.a.a.h1.h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅
(forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply IsDeduct.axiom_ | case h1.h1.h1.a.a.h1.h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅
(forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))) | case h1.h1.h1.a.a.h1.h1.h1.h1.a
x y z : VarName
⊢ IsAxiom (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | exact IsAxiom.eq_2_eq_ x z y z | case h1.h1.h1.a.a.h1.h1.h1.h1.a
x y z : VarName
⊢ IsAxiom (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply specId z | case h1.h1.h1.a
x y z : VarName
⊢ IsDeduct ∅ (eq_ z z) | case h1.h1.h1.a.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (eq_ z z)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | apply IsDeduct.axiom_ | case h1.h1.h1.a.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (eq_ z z)) | case h1.h1.h1.a.h1.a
x y z : VarName
⊢ IsAxiom (forall_ z (eq_ z z)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | exact IsAxiom.eq_1_ z | case h1.h1.h1.a.h1.a
x y z : VarName
⊢ IsAxiom (forall_ z (eq_ z z)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | intro H a1 | case h1.h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn z H | case h1.h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn z H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | simp at a1 | case h1.h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn z H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | intro H a1 | case h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H | case h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | simp at a1 | case h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | intro H a1 | case h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H | case h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_2 | [1373, 1] | [1401, 15] | simp at a1 | case h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | induction h1 | P_r P_s : Formula
r s : VarName
h1 : IsReplOfVarInFormula r s P_r P_s
h2 : ¬isBoundIn r P_r
h3 : ¬isBoundIn s P_r
⊢ IsProof ((eq_ r s).imp_ (P_r.iff_ P_s)) | case pred_const_
P_r P_s : Formula
r s : VarName
name✝ : PredName
n✝ : ℕ
args_u✝ args_v✝ : Fin n✝ → VarName
a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s
h2 : ¬isBoundIn r (pred_const_ name✝ (List.ofFn args_u✝))
h3 : ¬isBoundIn s (pred_const_ name✝ (List.ofFn args_u✝))
⊢ IsProof ((eq_ r s).imp_ ((pred_const_ name✝ (List.ofFn args_u✝)).iff_ (pred_const_ name✝ (List.ofFn args_v✝))))
case pred_var_
P_r P_s : Formula
r s : VarName
name✝ : PredName
n✝ : ℕ
args_u✝ args_v✝ : Fin n✝ → VarName
a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s
h2 : ¬isBoundIn r (pred_var_ name✝ (List.ofFn args_u✝))
h3 : ¬isBoundIn s (pred_var_ name✝ (List.ofFn args_u✝))
⊢ IsProof ((eq_ r s).imp_ ((pred_var_ name✝ (List.ofFn args_u✝)).iff_ (pred_var_ name✝ (List.ofFn args_v✝))))
case eq_
P_r P_s : Formula
r s x_u✝ y_u✝ x_v✝ y_v✝ : VarName
a✝¹ : x_u✝ = x_v✝ ∨ x_u✝ = r ∧ x_v✝ = s
a✝ : y_u✝ = y_v✝ ∨ y_u✝ = r ∧ y_v✝ = s
h2 : ¬isBoundIn r (eq_ x_u✝ y_u✝)
h3 : ¬isBoundIn s (eq_ x_u✝ y_u✝)
⊢ IsProof ((eq_ r s).imp_ ((eq_ x_u✝ y_u✝).iff_ (eq_ x_v✝ y_v✝)))
case true_
P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_))
case false_
P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r false_
h3 : ¬isBoundIn s false_
⊢ IsProof ((eq_ r s).imp_ (false_.iff_ false_))
case not_
P_r P_s : Formula
r s : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r P_u✝.not_
h3 : ¬isBoundIn s P_u✝.not_
⊢ IsProof ((eq_ r s).imp_ (P_u✝.not_.iff_ P_v✝.not_))
case imp_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.imp_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.imp_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.imp_ Q_u✝).iff_ (P_v✝.imp_ Q_v✝)))
case and_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.and_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.and_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝)))
case or_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.or_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.or_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝)))
case iff_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.iff_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.iff_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝)))
case forall_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (forall_ x✝ P_u✝)
h3 : ¬isBoundIn s (forall_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x✝ P_u✝).iff_ (forall_ x✝ P_v✝)))
case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case true_ =>
simp only [def_iff_]
simp only [def_and_]
SC | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | all_goals
sorry | case pred_var_
P_r P_s : Formula
r s : VarName
name✝ : PredName
n✝ : ℕ
args_u✝ args_v✝ : Fin n✝ → VarName
a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s
h2 : ¬isBoundIn r (pred_var_ name✝ (List.ofFn args_u✝))
h3 : ¬isBoundIn s (pred_var_ name✝ (List.ofFn args_u✝))
⊢ IsProof ((eq_ r s).imp_ ((pred_var_ name✝ (List.ofFn args_u✝)).iff_ (pred_var_ name✝ (List.ofFn args_v✝))))
case eq_
P_r P_s : Formula
r s x_u✝ y_u✝ x_v✝ y_v✝ : VarName
a✝¹ : x_u✝ = x_v✝ ∨ x_u✝ = r ∧ x_v✝ = s
a✝ : y_u✝ = y_v✝ ∨ y_u✝ = r ∧ y_v✝ = s
h2 : ¬isBoundIn r (eq_ x_u✝ y_u✝)
h3 : ¬isBoundIn s (eq_ x_u✝ y_u✝)
⊢ IsProof ((eq_ r s).imp_ ((eq_ x_u✝ y_u✝).iff_ (eq_ x_v✝ y_v✝)))
case false_
P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r false_
h3 : ¬isBoundIn s false_
⊢ IsProof ((eq_ r s).imp_ (false_.iff_ false_))
case and_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.and_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.and_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝)))
case or_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.or_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.or_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝)))
case iff_
P_r P_s : Formula
r s : VarName
P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula
a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝
a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝
a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝))
h2 : ¬isBoundIn r (P_u✝.iff_ Q_u✝)
h3 : ¬isBoundIn s (P_u✝.iff_ Q_u✝)
⊢ IsProof ((eq_ r s).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝)))
case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_iff_] | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_)) | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).and_ (true_.imp_ true_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).and_ (true_.imp_ true_))) | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).imp_ (true_.imp_ true_).not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).imp_ (true_.imp_ true_).not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply
IsDeduct.mp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsProof ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply
IsDeduct.mp_ ((eq_ r s).imp_ (And_ (List.ofFn fun (i : Fin n) => eq_ (args_u i) (args_v i)))) | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply
IsDeduct.mp_
((And_ (List.ofFn fun (i : Fin n) => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_iff_] | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))))) | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_).imp_
(((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_
((eq_ r s).imp_
(((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_
((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply Forall_spec_id' (List.ofFn args_v) | case a.a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))) | case a.a.a.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) |
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