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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | exact h1 | case refine'_1.a
P Q : Formula
v t : VarName
Δ : Set Formula
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
h1 : IsDeduct Δ (forall_ v P.not_).not_
⊢ IsDeduct Δ (forall_ v P.not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | intro contra | case refine'_2
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬isFreeIn t Q | case refine'_2
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply h4 | case refine'_2
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ False | case refine'_2
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ occursIn t Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply isFreeIn_imp_occursIn | case refine'_2
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ occursIn t Q | case refine'_2.h1
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ isFreeIn t Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | exact contra | case refine'_2.h1
P Q : Formula
v t : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {fastReplaceFree v t P}) Q
h3 : ¬occursIn t P
h4 : ¬occursIn t Q
h5 : ∀ H ∈ Δ, ¬isFreeIn t H
contra : isFreeIn t Q
⊢ isFreeIn t Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((exists_ v (P.and_ Q)).imp_ ((exists_ v P).and_ (exists_ v Q))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {exists_ v (P.and_ Q)}) ((exists_ v P).and_ (exists_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {exists_ v (P.and_ Q)}) ((exists_ v P).and_ (exists_ v Q)) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v (P.and_ Q)} ((exists_ v P).and_ (exists_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply rule_C (P.and_ Q) ((exists_ v P).and_ (exists_ v Q)) v | case h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v (P.and_ Q)} ((exists_ v P).and_ (exists_ v Q)) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v (P.and_ Q)} (exists_ v (P.and_ Q))
case h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((exists_ v P).and_ (exists_ v Q))
case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v (P.and_ Q)}, ¬isFreeIn v H
case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((exists_ v P).and_ (exists_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.assume_ | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {exists_ v (P.and_ Q)} (exists_ v (P.and_ Q)) | case h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v (P.and_ Q) ∈ {exists_ v (P.and_ Q)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1.h1.a
P Q : Formula
v : VarName
⊢ exists_ v (P.and_ Q) ∈ {exists_ v (P.and_ Q)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.mp_ (exists_ v Q) | case h1.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((exists_ v P).and_ (exists_ v Q)) | case h1.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((exists_ v Q).imp_ ((exists_ v P).and_ (exists_ v Q)))
case h1.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (exists_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.mp_ (exists_ v P) | case h1.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((exists_ v Q).imp_ ((exists_ v P).and_ (exists_ v Q))) | case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q})
((exists_ v P).imp_ ((exists_ v Q).imp_ ((exists_ v P).and_ (exists_ v Q))))
case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (exists_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_and_] | case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q})
((exists_ v P).imp_ ((exists_ v Q).imp_ ((exists_ v P).and_ (exists_ v Q)))) | case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_})
((exists_ v P).imp_ ((exists_ v Q).imp_ ((exists_ v P).imp_ (exists_ v Q).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | SC | case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_})
((exists_ v P).imp_ ((exists_ v Q).imp_ ((exists_ v P).imp_ (exists_ v Q).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply exists_intro P v v | case h1.h2.a.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (exists_ v P) | case h1.h2.a.a.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v P
case h1.h2.a.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (fastReplaceFree v v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply fastAdmits_self | case h1.h2.a.a.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [fastReplaceFree_self] | case h1.h2.a.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (fastReplaceFree v v P) | case h1.h2.a.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.mp_ (P.and_ Q) | case h1.h2.a.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) P | case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((P.and_ Q).imp_ P)
case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (P.and_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_and_] | case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((P.and_ Q).imp_ P) | case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_}) ((P.imp_ Q.not_).not_.imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | SC | case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_}) ((P.imp_ Q.not_).not_.imp_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.assume_ | case h1.h2.a.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (P.and_ Q) | case h1.h2.a.a.h2.a.a
P Q : Formula
v : VarName
⊢ P.and_ Q ∈ {exists_ v (P.and_ Q)} ∪ {P.and_ Q} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1.h2.a.a.h2.a.a
P Q : Formula
v : VarName
⊢ P.and_ Q ∈ {exists_ v (P.and_ Q)} ∪ {P.and_ Q} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply exists_intro Q v v | case h1.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (exists_ v Q) | case h1.h2.a.h1
P Q : Formula
v : VarName
⊢ fastAdmits v v Q
case h1.h2.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (fastReplaceFree v v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply fastAdmits_self | case h1.h2.a.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_17_14 | [606, 1] | [651, 9] | simp only [fastReplaceFree_self] | case h1.h2.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (fastReplaceFree v v Q) | case h1.h2.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.mp_ (P.and_ Q) | case h1.h2.a.h2
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) Q | case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((P.and_ Q).imp_ Q)
case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (P.and_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_and_] | case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) ((P.and_ Q).imp_ Q) | case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_}) ((P.imp_ Q.not_).not_.imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | SC | case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.imp_ Q.not_).not_} ∪ {(P.imp_ Q.not_).not_}) ((P.imp_ Q.not_).not_.imp_ Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | apply IsDeduct.assume_ | case h1.h2.a.h2.a
P Q : Formula
v : VarName
⊢ IsDeduct ({exists_ v (P.and_ Q)} ∪ {P.and_ Q}) (P.and_ Q) | case h1.h2.a.h2.a.a
P Q : Formula
v : VarName
⊢ P.and_ Q ∈ {exists_ v (P.and_ Q)} ∪ {P.and_ Q} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1.h2.a.h2.a.a
P Q : Formula
v : VarName
⊢ P.and_ Q ∈ {exists_ v (P.and_ Q)} ∪ {P.and_ Q} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_and_] | case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v (P.and_ Q)}, ¬isFreeIn v H | case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v (P.imp_ Q.not_).not_}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_exists_] | case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {exists_ v (P.imp_ Q.not_).not_}, ¬isFreeIn v H | case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v (P.imp_ Q.not_).not_.not_).not_}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1.h3
P Q : Formula
v : VarName
⊢ ∀ H ∈ {(forall_ v (P.imp_ Q.not_).not_.not_).not_}, ¬isFreeIn v H | case h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.imp_ Q.not_).not_.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [isFreeIn] | case h1.h3
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v (P.imp_ Q.not_).not_.not_).not_ | case h1.h3
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_17_14 | [606, 1] | [651, 9] | simp | case h1.h3
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_17_14 | [606, 1] | [651, 9] | simp only [def_and_] | case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((exists_ v P).and_ (exists_ v Q)) | case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((exists_ v P).imp_ (exists_ v Q).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [def_exists_] | case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((exists_ v P).imp_ (exists_ v Q).not_).not_ | case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp only [isFreeIn] | case h1.h4
P Q : Formula
v : VarName
⊢ ¬isFreeIn v ((forall_ v P.not_).not_.imp_ (forall_ v Q.not_).not_.not_).not_ | case h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_14 | [606, 1] | [651, 9] | simp | case h1.h4
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P ∨ ¬True ∧ isFreeIn v Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp only [def_iff_] | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P).imp_ (forall_ v Q))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v P).imp_ (forall_ v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply deduction_theorem | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v P).imp_ (forall_ v Q)) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {forall_ v P}) (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {forall_ v P}) (forall_ v Q) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply generalization | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q) | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q
case h1.h1.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply IsDeduct.mp_ P | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (P.imp_ Q)
case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply IsDeduct.mp_ ((P.imp_ Q).and_ (Q.imp_ P)) | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (P.imp_ Q) | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q))
case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp only [def_and_] | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((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_18_1_left | [654, 1] | [678, 9] | SC | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((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_18_1_left | [654, 1] | [678, 9] | apply specId v | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P)) | case h1.h1.h1.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply IsDeduct.assume_ | case h1.h1.h1.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) | case h1.h1.h1.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp | case h1.h1.h1.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply specId v | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P | case h1.h1.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | apply IsDeduct.assume_ | case h1.h1.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P) | case h1.h1.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v P ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp | case h1.h1.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v P ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, ¬isFreeIn v H | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp only [isFreeIn] | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v P) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_left | [654, 1] | [678, 9] | simp | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp only [def_iff_] | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))) | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply deduction_theorem | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))) | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v Q).imp_ (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply deduction_theorem | case h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v Q).imp_ (forall_ v P)) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {forall_ v Q}) (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ∪ {forall_ v Q}) (forall_ v P) | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply generalization | case h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P) | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P
case h1.h1.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply IsDeduct.mp_ Q | case h1.h1.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (Q.imp_ P)
case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply IsDeduct.mp_ ((P.imp_ Q).and_ (Q.imp_ P)) | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (Q.imp_ P) | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (Q.imp_ P))
case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp only [def_and_] | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (Q.imp_ P)) | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (Q.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | SC | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_}
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (Q.imp_ P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply specId v | case h1.h1.h1.a.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P)) | case h1.h1.h1.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply IsDeduct.assume_ | case h1.h1.h1.a.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) | case h1.h1.h1.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp | case h1.h1.h1.a.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply specId v | case h1.h1.h1.a
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q | case h1.h1.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | apply IsDeduct.assume_ | case h1.h1.h1.a.h1
P Q : Formula
v : VarName
⊢ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q) | case h1.h1.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v Q ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp | case h1.h1.h1.a.h1.a
P Q : Formula
v : VarName
⊢ forall_ v Q ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ∀ H ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, ¬isFreeIn v H | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp only [isFreeIn] | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬isFreeIn v (forall_ v Q) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P))) | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1_right | [681, 1] | [705, 9] | simp | case h1.h1.h2
P Q : Formula
v : VarName
⊢ ¬(¬True ∧ isFreeIn v Q) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))) | P Q : Formula
v : VarName
⊢ IsProof ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))))
case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q)))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q)))))
case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | simp only [def_iff_] | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_
(((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((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)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | 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 P).imp_ (forall_ v Q))).imp_
(((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((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)))))) | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((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_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | SC | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_
(((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_
((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_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_1 | [708, 1] | [721, 23] | apply T_18_1_left | case a.a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ 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_18_1 | [708, 1] | [721, 23] | apply T_18_1_right | case a
P Q : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | induction xs | xs : List VarName
P : Formula
⊢ IsProof ((Forall_ xs P).imp_ P) | case nil
P : Formula
⊢ IsProof ((Forall_ [] P).imp_ P)
case cons
P : Formula
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : IsProof ((Forall_ tail✝ P).imp_ P)
⊢ IsProof ((Forall_ (head✝ :: tail✝) P).imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | case nil =>
simp only [Forall_]
apply prop_id | P : Formula
⊢ IsProof ((Forall_ [] P).imp_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | case cons xs_hd xs_tl xs_ih =>
simp only [Forall_]
apply deduction_theorem
simp
apply IsDeduct.mp_ (Forall_ xs_tl P)
apply proof_imp_deduct
exact xs_ih
apply specId xs_hd
apply IsDeduct.assume_
simp
rfl | P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((Forall_ (xs_hd :: xs_tl) P).imp_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | simp only [Forall_] | P : Formula
⊢ IsProof ((Forall_ [] P).imp_ P) | P : Formula
⊢ IsProof ((List.foldr forall_ P []).imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply prop_id | P : Formula
⊢ IsProof ((List.foldr forall_ P []).imp_ P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | simp only [Forall_] | P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((Forall_ (xs_hd :: xs_tl) P).imp_ P) | P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((List.foldr forall_ P (xs_hd :: xs_tl)).imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply deduction_theorem | P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((List.foldr forall_ P (xs_hd :: xs_tl)).imp_ P) | case h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct (∅ ∪ {List.foldr forall_ P (xs_hd :: xs_tl)}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | simp | case h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct (∅ ∪ {List.foldr forall_ P (xs_hd :: xs_tl)}) P | case h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply IsDeduct.mp_ (Forall_ xs_tl P) | case h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} P | case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} ((Forall_ xs_tl P).imp_ P)
case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply proof_imp_deduct | case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} ((Forall_ xs_tl P).imp_ P)
case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) | case h1.a.h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((Forall_ xs_tl P).imp_ P)
case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | exact xs_ih | case h1.a.h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsProof ((Forall_ xs_tl P).imp_ P)
case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) | case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply specId xs_hd | case h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P) | case h1.a.h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (forall_ xs_hd (Forall_ xs_tl P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | apply IsDeduct.assume_ | case h1.a.h1
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (forall_ xs_hd (Forall_ xs_tl P)) | case h1.a.h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ forall_ xs_hd (Forall_ xs_tl P) ∈ {forall_ xs_hd (List.foldr forall_ P xs_tl)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | simp | case h1.a.h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ forall_ xs_hd (Forall_ xs_tl P) ∈ {forall_ xs_hd (List.foldr forall_ P xs_tl)} | case h1.a.h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ Forall_ xs_tl P = List.foldr forall_ P xs_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id | [724, 1] | [743, 8] | rfl | case h1.a.h1.a
P : Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsProof ((Forall_ xs_tl P).imp_ P)
⊢ Forall_ xs_tl P = List.foldr forall_ P xs_tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id' | [746, 1] | [764, 13] | induction xs | xs : List VarName
P : Formula
Δ : Set Formula
h1 : IsDeduct Δ (Forall_ xs P)
⊢ IsDeduct Δ P | case nil
P : Formula
Δ : Set Formula
h1 : IsDeduct Δ (Forall_ [] P)
⊢ IsDeduct Δ P
case cons
P : Formula
Δ : Set Formula
head✝ : VarName
tail✝ : List VarName
tail_ih✝ : IsDeduct Δ (Forall_ tail✝ P) → IsDeduct Δ P
h1 : IsDeduct Δ (Forall_ (head✝ :: tail✝) P)
⊢ IsDeduct Δ P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id' | [746, 1] | [764, 13] | case nil =>
simp only [Forall_] at h1
simp at h1
exact h1 | P : Formula
Δ : Set Formula
h1 : IsDeduct Δ (Forall_ [] P)
⊢ IsDeduct Δ P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.Forall_spec_id' | [746, 1] | [764, 13] | case cons xs_hd xs_tl xs_ih =>
simp only [Forall_] at h1
simp at h1
apply xs_ih
simp only [Forall_]
apply specId xs_hd
exact h1 | P : Formula
Δ : Set Formula
xs_hd : VarName
xs_tl : List VarName
xs_ih : IsDeduct Δ (Forall_ xs_tl P) → IsDeduct Δ P
h1 : IsDeduct Δ (Forall_ (xs_hd :: xs_tl) P)
⊢ IsDeduct Δ P | no goals |
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