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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | apply T_18_6 | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)) | case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | exact similar_not P_u P_v u v h1 | case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T18_9 | [1086, 1] | [1098, 13] | apply C_18_4 (exists_ u P_u) (exists_ v P_v) Q Q' Δ h2 | Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q' | case h2
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v))
case h3
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T18_9 | [1086, 1] | [1098, 13] | exact T_18_8 P_u P_v u v h3 | case h2
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T18_9 | [1086, 1] | [1098, 13] | exact h1 | case h3
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply IsDeduct.mp_ ((forall_ v P).imp_ P) | P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P) | case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P))
case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply IsDeduct.mp_ (P.imp_ (forall_ v P)) | case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P)) | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P)))
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (P.imp_ (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | simp only [def_iff_] | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P))) | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | simp only [def_and_] | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))))) | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_
(((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | SC | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_
(((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply IsDeduct.axiom_ | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (P.imp_ (forall_ v P)) | case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom (P.imp_ (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | exact IsAxiom.pred_3_ v P h1 | case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom (P.imp_ (forall_ v P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply IsDeduct.axiom_ | case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).imp_ P) | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v P).imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply IsAxiom.pred_2_ v v P P | case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v P).imp_ P) | case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastAdmits v v P
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastReplaceFree v v P = P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply fastAdmits_self | case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastAdmits v v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_1 | [1101, 1] | [1119, 33] | apply fastReplaceFree_self | case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastReplaceFree v v P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | apply IsDeduct.mp_ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))) | P : Formula
u v : VarName
⊢ IsProof ((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P))) | case a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P))))
case a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | apply IsDeduct.mp_ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))) | case a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P)))) | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P)))))
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | simp only [def_iff_] | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P))))) | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).and_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | simp only [def_and_] | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).and_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P)))))) | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | SC | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | apply T_17_10 | case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_2 | [1122, 1] | [1135, 18] | apply T_17_10 | case a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_3 | [1138, 1] | [1149, 5] | simp only [def_exists_] | P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (exists_ v P).not_) | P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (forall_ v P.not_).not_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_3 | [1138, 1] | [1149, 5] | simp only [def_iff_] | P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (forall_ v P.not_).not_.not_) | P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).and_ ((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_3 | [1138, 1] | [1149, 5] | simp only [def_and_] | P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).and_ ((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_))) | P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).imp_
((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_3 | [1138, 1] | [1149, 5] | SC | P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).imp_
((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply deduction_theorem | P : Formula
u v : VarName
⊢ IsProof ((exists_ u (forall_ v P)).imp_ (forall_ v (exists_ u P))) | case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {exists_ u (forall_ v P)}) (forall_ v (exists_ u P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {exists_ u (forall_ v P)}) (forall_ v (exists_ u P)) | case h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (forall_ v (exists_ u P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply generalization | case h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (forall_ v (exists_ u P)) | case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u P)
case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply rule_C (forall_ v P) (exists_ u P) u {exists_ u (forall_ v P)} | case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u P) | case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u (forall_ v P))
case h1.h1.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (exists_ u P)
case h1.h1.h3
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn u H
case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply IsDeduct.assume_ | case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u (forall_ v P)) | case h1.h1.h1.a
P : Formula
u v : VarName
⊢ exists_ u (forall_ v P) ∈ {exists_ u (forall_ v P)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h1.a
P : Formula
u v : VarName
⊢ exists_ u (forall_ v P) ∈ {exists_ u (forall_ v P)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply exists_intro P u u | case h1.h1.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (exists_ u P) | case h1.h1.h2.h1
P : Formula
u v : VarName
⊢ fastAdmits u u P
case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (fastReplaceFree u u P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply fastAdmits_self | case h1.h1.h2.h1
P : Formula
u v : VarName
⊢ fastAdmits u u P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [fastReplaceFree_self] | case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (fastReplaceFree u u P) | case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply specId v | case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) P | case h1.h1.h2.h2.h1
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | apply IsDeduct.assume_ | case h1.h1.h2.h2.h1
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (forall_ v P) | case h1.h1.h2.h2.h1.a
P : Formula
u v : VarName
⊢ forall_ v P ∈ {exists_ u (forall_ v P)} ∪ {forall_ v P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h2.h2.h1.a
P : Formula
u v : VarName
⊢ forall_ v P ∈ {exists_ u (forall_ v P)} ∪ {forall_ v P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h3
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn u H | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [def_exists_] | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u (forall_ v P)) | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [isFreeIn] | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P).not_).not_ | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬(¬True ∧ ¬u = v ∧ isFreeIn u P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬(¬True ∧ ¬u = v ∧ isFreeIn u P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [def_exists_] | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u P) | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u P.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [isFreeIn] | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u P.not_).not_ | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬(¬True ∧ isFreeIn u P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬(¬True ∧ isFreeIn u P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn v H | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (exists_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [def_exists_] | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (exists_ u (forall_ v P)) | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp only [isFreeIn] | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P).not_).not_ | case h1.h2
P : Formula
u v : VarName
⊢ ¬(¬v = u ∧ ¬True ∧ isFreeIn v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_4 | [1152, 1] | [1183, 9] | simp | case h1.h2
P : Formula
u v : VarName
⊢ ¬(¬v = u ∧ ¬True ∧ isFreeIn v P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | apply IsDeduct.mp_ ((forall_ v P).iff_ P) | P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))))
case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).iff_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q)))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))).imp_
(((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q)))))
case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))).imp_
(((forall_ v P).iff_ P).imp_ ((forall_ v (P.iff_ Q)).imp_ (P.iff_ (forall_ v Q))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P)))).imp_
((((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((P.imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ P))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | simp only [def_and_] | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_
(((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P)))).imp_
((((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P))).imp_
((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((P.imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ P))))) | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_).imp_
((((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_.imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((P.imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ P).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | SC | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_).imp_
((((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_.imp_
((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
((P.imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ P).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | exact T_18_1 P Q v | case a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_5 | [1186, 1] | [1200, 24] | exact T_19_1 P v h1 | case a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).iff_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v P).imp_ (exists_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply deduction_theorem | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v P).imp_ (exists_ v Q)) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)} ∪ {exists_ v P}) (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)} ∪ {exists_ v P}) (exists_ v Q) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply rule_C P (exists_ v Q) v {exists_ v P, forall_ v (P.iff_ Q)} | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v Q) | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v P)
case h1.h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (exists_ v Q)
case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v P, forall_ v (P.iff_ Q)}, ¬isFreeIn v H
case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v P, forall_ v (P.iff_ Q)} (exists_ v P) | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v P ∈ {exists_ v P, forall_ v (P.iff_ Q)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v P ∈ {exists_ v P, forall_ v (P.iff_ Q)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply exists_intro Q v v | case h1.h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (exists_ v Q) | case h1.h1.h2.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v Q
case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (fastReplaceFree v v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply fastAdmits_self | case h1.h1.h2.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [fastReplaceFree_self] | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (fastReplaceFree v v Q) | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.mp_ P | case h1.h1.h2.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) Q | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.imp_ Q)
case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.mp_ (P.iff_ Q) | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.imp_ Q) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) ((P.iff_ Q).imp_ (P.imp_ Q))
case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.iff_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_iff_] | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) ((P.iff_ Q).imp_ (P.imp_ Q)) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {P}) (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_and_] | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {P}) (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} ∪ {P})
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | SC | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} ∪ {P})
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply specId v | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (P.iff_ Q) | case h1.h1.h2.h2.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h2.h2.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) (forall_ v (P.iff_ Q)) | case h1.h1.h2.h2.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h2.h2.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | apply IsDeduct.assume_ | case h1.h1.h2.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v P, forall_ v (P.iff_ Q)} ∪ {P}) P | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ P ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h2.h2.a.a
P Q : Formula
v : VarName
⊢ P ∈ {exists_ v P, forall_ v (P.iff_ Q)} ∪ {P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_exists_] | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v P, forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v P.not_).not_, forall_ v (P.iff_ Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v P.not_).not_, forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P.not_).not_ ∧ ¬isFreeIn v (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [isFreeIn] | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P.not_).not_ ∧ ¬isFreeIn v (forall_ v (P.iff_ Q)) | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h3
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [def_exists_] | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (exists_ v Q) | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp only [isFreeIn] | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q.not_).not_ | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_left | [1203, 1] | [1237, 9] | simp | case h1.h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v Q).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v (P.iff_ Q)}) ((exists_ v Q).imp_ (exists_ v P)) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((exists_ v Q).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.mp_ (forall_ v (Q.iff_ P)) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((exists_ v Q).imp_ (exists_ v P)) | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P)))
case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (Q.iff_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply proof_imp_deduct | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((forall_ v (Q.iff_ P)).imp_ ((exists_ v Q).imp_ (exists_ v P))) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (Q.iff_ P)).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_right | [1240, 1] | [1262, 11] | apply T_19_6_left Q P v | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (Q.iff_ P)).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_right | [1240, 1] | [1262, 11] | apply generalization | case h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (Q.iff_ P)) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (Q.iff_ P)
case h1.a.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v (P.iff_ Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.mp_ (P.iff_ Q) | case h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (Q.iff_ P) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((P.iff_ Q).imp_ (Q.iff_ P))
case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (P.iff_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [def_iff_] | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} ((P.iff_ Q).imp_ (Q.iff_ P)) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ ((Q.imp_ P).and_ (P.imp_ Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [def_and_] | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ ((Q.imp_ P).and_ (P.imp_ Q))) | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ ((Q.imp_ P).imp_ (P.imp_ Q).not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | SC | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ ((Q.imp_ P).imp_ (P.imp_ Q).not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply specId v | case h1.a.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (P.iff_ Q) | case h1.a.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | apply IsDeduct.assume_ | case h1.a.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v (P.iff_ Q)} (forall_ v (P.iff_ Q)) | case h1.a.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {forall_ v (P.iff_ Q)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v (P.iff_ Q) ∈ {forall_ v (P.iff_ Q)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v (P.iff_ Q)}, ¬isFreeIn v H | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.iff_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp only [isFreeIn] | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.iff_ Q)) | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_19_6_right | [1240, 1] | [1262, 11] | simp | case h1.a.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) | no goals |
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 P).imp_ (exists_ v Q))) | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).iff_ (exists_ v Q))) | 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
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((exists_ v P).imp_ (exists_ v Q))) |
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